TPTP Problem File: ITP275^3.p

%------------------------------------------------------------------------------
% File     : ITP275^3 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Uniqueness 00039_002338
% 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    : 0075_VEBT_Uniqueness_00039_002338 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11199 (5763 unt; 942 typ;   0 def)
%            Number of atoms       : 28640 (13092 equ;   0 cnn)
%            Maximal formula atoms :   71 (   2 avg)
%            Number of connectives : 115952 (3150   ~; 567   |;2000   &;99447   @)
%                                         (   0 <=>;10788  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   39 (   6 avg)
%            Number of types       :   85 (  84 usr)
%            Number of type conns  : 3432 (3432   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  861 ( 858 usr;  63 con; 0-8 aty)
%            Number of variables   : 25577 (1817   ^;22995   !; 765   ?;25577   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 15:20:13.443
%------------------------------------------------------------------------------
% Could-be-implicit typings (84)
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(ty_n_t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
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thf(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__String__Ochar_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__Code____Numeral__Ointeger,type,
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thf(ty_n_t__Extended____Nat__Oenat,type,
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thf(ty_n_t__List__Olist_I_Eo_J,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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% Explicit typings (858)
thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,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_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_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_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_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_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_Opush__bit_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,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_Ounset__bit_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,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_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Otake__bit__num,type,
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thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num,type,
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thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oxor__num,type,
    bit_un6178654185764691216or_num: num > num > option_num ).

thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
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thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
    bit_un2480387367778600638or_num: num > num > option_num ).

thf(sy_c_Code__Numeral_Obit__cut__integer,type,
    code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).

thf(sy_c_Code__Numeral_Odivmod__abs,type,
    code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).

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

thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
    code_int_of_integer: code_integer > int ).

thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: int > code_integer ).

thf(sy_c_Code__Numeral_Ointeger__of__num,type,
    code_integer_of_num: num > code_integer ).

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

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

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
    comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex ).

thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
    complete_Inf_Inf_nat: set_nat > nat ).

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

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

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

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

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

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

thf(sy_c_Complex_Ocnj,type,
    cnj: complex > complex ).

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

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

thf(sy_c_Complex_Ocsqrt,type,
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thf(sy_c_Complex_Oimaginary__unit,type,
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thf(sy_c_Deriv_Odifferentiable_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Deriv_Ohas__derivative_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
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thf(sy_c_Divides_Oadjust__div,type,
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thf(sy_c_Divides_Oadjust__mod,type,
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thf(sy_c_Divides_Odivmod__nat,type,
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thf(sy_c_Divides_Oeucl__rel__int,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Int__Oint,type,
    unique6319869463603278526ux_int: product_prod_int_int > $o ).

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

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Int__Oint,type,
    unique5052692396658037445od_int: num > num > product_prod_int_int ).

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

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
    unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).

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

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

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

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
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thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
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thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
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thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
    inverse_inverse_real: real > real ).

thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
    at_bot_real: filter_real ).

thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
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thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
    at_top_real: filter_real ).

thf(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
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thf(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
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thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
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thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Finite__Set_Ocard_001_Eo,type,
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thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
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thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ocard_001t__String__Ochar,type,
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thf(sy_c_Finite__Set_Ofinite_001_Eo,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Complex__Ocomplex_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Num__Onum,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Rat__Orat,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
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thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Complex__Ocomplex,type,
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thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Fun_Ocomp_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Nat__Onat,type,
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thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__String__Ochar,type,
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thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Fun_Ostrict__mono__on_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Fun_Othe__inv__into_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_GCD_OGcd__class_OGcd_001t__Int__Oint,type,
    gcd_Gcd_int: set_int > int ).

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

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

thf(sy_c_GCD_Obezw__rel,type,
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thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Int__Oint,type,
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thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Nat__Onat,type,
    gcd_gcd_nat: nat > nat > nat ).

thf(sy_c_GCD_Ogcd__nat__rel,type,
    gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(sy_c_Groups_Oabs__class_Oabs_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups_Oabs__class_Oabs_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
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thf(sy_c_Groups_Oabs__class_Oabs_001t__Rat__Orat,type,
    abs_abs_rat: rat > rat ).

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

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Complex__Ocomplex_M_Eo_J,type,
    minus_8727706125548526216plex_o: ( complex > $o ) > ( complex > $o ) > complex > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Int__Oint_M_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    minus_1139252259498527702_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Real__Oreal_M_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Rat__Orat,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Rat__Orat,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
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thf(sy_c_Groups_Osgn__class_Osgn_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Osgn__class_Osgn_001t__Int__Oint,type,
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thf(sy_c_Groups_Osgn__class_Osgn_001t__Rat__Orat,type,
    sgn_sgn_rat: rat > rat ).

thf(sy_c_Groups_Osgn__class_Osgn_001t__Real__Oreal,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Rat__Orat,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Real__Oreal_M_Eo_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Rat__Orat,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Rat__Orat,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_If_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_If_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Int_OAbs__Integ,type,
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thf(sy_c_Int_ORep__Integ,type,
    rep_Integ: int > product_prod_nat_nat ).

thf(sy_c_Int_Onat,type,
    nat2: int > nat ).

thf(sy_c_Int_Opower__int_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,
    ring_1_of_int_int: int > int ).

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_001_062_It__Complex__Ocomplex_M_Eo_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    inf_inf_list_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Real__Oreal_M_Eo_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
    inf_inf_nat: nat > nat > nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Rat__Orat,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Real__Oreal,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Int__Oint_M_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    sup_sup_list_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Real__Oreal_M_Eo_J,type,
    sup_sup_real_o: ( real > $o ) > ( real > $o ) > real > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    sup_sup_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Extended____Nat__Oenat,type,
    sup_su3973961784419623482d_enat: extended_enat > extended_enat > extended_enat ).

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

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

thf(sy_c_Lattices_Osup__class_Osup_001t__Rat__Orat,type,
    sup_sup_rat: rat > rat > rat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Real__Oreal,type,
    sup_sup_real: real > real > real ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_Eo_J,type,
    sup_sup_set_o: set_o > set_o > set_o ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    sup_sup_set_complex: set_complex > set_complex > set_complex ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Int__Oint_J,type,
    sup_sup_set_int: set_int > set_int > set_int ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    sup_sup_set_list_nat: set_list_nat > set_list_nat > set_list_nat ).

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

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Real__Oreal_J,type,
    sup_sup_set_real: set_real > set_real > set_real ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    sup_su6272177626956685416T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT ).

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
    nil_int: list_int ).

thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

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

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

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

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

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

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

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

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

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

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

thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
    tl_nat: list_nat > list_nat ).

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

thf(sy_c_List_Olist__update_001t__Complex__Ocomplex,type,
    list_update_complex: list_complex > nat > complex > list_complex ).

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

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

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

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

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

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

thf(sy_c_List_Onth_001t__Code____Numeral__Ointeger,type,
    nth_Code_integer: list_Code_integer > nat > code_integer ).

thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
    nth_complex: list_complex > nat > complex ).

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

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

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
    nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
    nth_Pr5826913651314560976_o_nat: list_P6285523579766656935_o_nat > nat > product_prod_o_nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
    nth_Pr8522763379788166057eger_o: list_P8526636022914148096eger_o > nat > produc6271795597528267376eger_o ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    nth_Pr6456567536196504476um_num: list_P3744719386663036955um_num > nat > product_prod_num_num ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
    nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).

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

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

thf(sy_c_List_Onth_001t__Real__Oreal,type,
    nth_real: list_real > nat > real ).

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

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

thf(sy_c_List_Oproduct_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,
    product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).

thf(sy_c_List_Oproduct_001t__Code____Numeral__Ointeger_001_Eo,type,
    produc3607205314601156340eger_o: list_Code_integer > list_o > list_P8526636022914148096eger_o ).

thf(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
    product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).

thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).

thf(sy_c_List_Oproduct_001t__Num__Onum_001t__Num__Onum,type,
    product_num_num: list_num > list_num > list_P3744719386663036955um_num ).

thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).

thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).

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

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

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

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

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

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

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

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

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

thf(sy_c_List_Ounion_001t__Int__Oint,type,
    union_int: list_int > list_int > list_int ).

thf(sy_c_List_Ounion_001t__Nat__Onat,type,
    union_nat: list_nat > list_nat > list_nat ).

thf(sy_c_List_Ounion_001t__VEBT____Definitions__OVEBT,type,
    union_VEBT_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_VEBT_VEBT ).

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

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

thf(sy_c_List_Oupto__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_ONats_001t__Complex__Ocomplex,type,
    semiri3842193898606819883omplex: set_complex ).

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

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

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

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

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

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__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__Nat__Onat_J,type,
    size_size_list_nat: list_nat > nat ).

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

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

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

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

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
    size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Nat__Bijection_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_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__Nat__Onat,type,
    none_nat: option_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
    none_num: option_num ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

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

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

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

thf(sy_c_Option_Ooption_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__Nat__Onat,type,
    size_option_nat: ( nat > nat ) > option_nat > nat ).

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

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

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

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

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

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

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

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    bot_bot_list_nat_o: list_nat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
    bot_bot_nat_o: nat > $o ).

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

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    bot_bot_set_nat_o: set_nat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
    bot_bo4199563552545308370d_enat: extended_enat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    bot_bot_filter_nat: filter_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
    bot_bot_set_o: set_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    bot_bot_set_complex: set_complex ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    bot_bo7653980558646680370d_enat: set_Extended_enat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    bot_bot_set_list_nat: set_list_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
    bot_bot_set_num: set_num ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
    bot_bot_set_rat: set_rat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
    bot_bot_set_real: set_real ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bot_set_set_nat: set_set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    bot_bo8194388402131092736T_VEBT: set_VEBT_VEBT ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__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__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__Nat__Onat_J_J,type,
    ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__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__Real__Oreal_M_Eo_J,type,
    ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Omin_001t__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_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__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_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
    produc4035269172776083154on_nat: ( nat > nat > $o ) > produc4953844613479565601on_nat > produc2233624965454879586on_nat ).

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__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
    produc8929957630744042906on_nat: ( nat > nat > nat ) > produc4953844613479565601on_nat > produc8306885398267862888on_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Num__Onum_M_062_It__Num__Onum_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    produc3576312749637752826on_num: ( num > num > $o ) > produc3447558737645232053on_num > produc7036089656553540234on_num ).

thf(sy_c_Product__Type_OPair_001_062_It__Num__Onum_M_062_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
    produc5778274026573060048on_num: ( num > num > num ) > produc3447558737645232053on_num > produc1193250871479095198on_num ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
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thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
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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,
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thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
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thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J,type,
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thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001_Eo,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_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__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,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
    product_fst_int_int: product_prod_int_int > int ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__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,
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thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
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thf(sy_c_Rat_Onormalize,type,
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thf(sy_c_Rat_Oquotient__of,type,
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thf(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
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thf(sy_c_Real__Vector__Spaces_Obounded__linear_001t__Real__Oreal_001t__Real__Oreal,type,
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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,
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thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
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thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
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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,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__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__Extended____Nat__Oenat,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
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thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
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thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
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thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
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thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
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thf(sy_c_Series_Osums_001t__Real__Oreal,type,
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thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
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thf(sy_c_Set_OCollect_001t__Int__Oint,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set_OPow_001t__Nat__Onat,type,
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thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
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thf(sy_c_Set_Oinsert_001_Eo,type,
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thf(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
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thf(sy_c_Set_Oinsert_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
    insert_int: int > set_int > set_int ).

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__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__Nat__Onat_J,type,
    set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
    set_or5849166863359141190n_real: real > set_real ).

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

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
    set_or5984915006950818249n_real: real > set_real ).

thf(sy_c_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_Ochar_Osize__char,type,
    size_char: char > nat ).

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

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

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

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

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

thf(sy_c_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__Real__Oreal,type,
    cos_real: real > real ).

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

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

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

thf(sy_c_Transcendental_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__Real__Oreal,type,
    sin_real: real > real ).

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

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

thf(sy_c_Transcendental_Otan_001t__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_Otrancl_001t__Nat__Onat,type,
    transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_VEBT__InsertCorrectness_OVEBT__internal_Oinsert_H,type,
    vEBT_VEBT_insert: vEBT_VEBT > nat > vEBT_VEBT ).

thf(sy_c_VEBT__InsertCorrectness_OVEBT__internal_Oinsert_H__rel,type,
    vEBT_VEBT_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_member_001t__List__Olist_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__Nat__Onat_J,type,
    member_set_nat: set_nat > set_set_nat > $o ).

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

thf(sy_v_deg____,type,
    deg: nat ).

thf(sy_v_i____,type,
    i: nat ).

thf(sy_v_m____,type,
    m: nat ).

thf(sy_v_na____,type,
    na: nat ).

thf(sy_v_sa____,type,
    sa: vEBT_VEBT ).

thf(sy_v_summary_H____,type,
    summary: vEBT_VEBT ).

thf(sy_v_summary____,type,
    summary2: vEBT_VEBT ).

thf(sy_v_treeList_H____,type,
    treeList: list_VEBT_VEBT ).

thf(sy_v_treeList____,type,
    treeList2: list_VEBT_VEBT ).

thf(sy_v_x____,type,
    x: nat ).

% Relevant facts (10213)
thf(fact_0_ac,axiom,
    ( ( size_s6755466524823107622T_VEBT @ treeList )
    = ( size_s6755466524823107622T_VEBT @ treeList2 ) ) ).

% ac
thf(fact_1__C2_Ohyps_C_I3_J,axiom,
    m = na ).

% "2.hyps"(3)
thf(fact_2__C2_Ohyps_C_I2_J,axiom,
    ( ( size_s6755466524823107622T_VEBT @ treeList2 )
    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).

% "2.hyps"(2)
thf(fact_3_less__exp,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% less_exp
thf(fact_4_semiring__norm_I85_J,axiom,
    ! [M: num] :
      ( ( bit0 @ M )
     != one ) ).

% semiring_norm(85)
thf(fact_5_semiring__norm_I83_J,axiom,
    ! [N: num] :
      ( one
     != ( bit0 @ N ) ) ).

% semiring_norm(83)
thf(fact_6_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_7_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_8_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_9_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_10_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_11_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N: nat] :
      ( ( vEBT_vebt_member @ Tree @ X )
     => ( ( vEBT_invar_vebt @ Tree @ N )
       => ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% member_bound
thf(fact_12_verit__eq__simplify_I8_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( ( bit0 @ X2 )
        = ( bit0 @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% verit_eq_simplify(8)
thf(fact_13_semiring__norm_I87_J,axiom,
    ! [M: num,N: num] :
      ( ( ( bit0 @ M )
        = ( bit0 @ N ) )
      = ( M = N ) ) ).

% semiring_norm(87)
thf(fact_14_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numera1916890842035813515d_enat @ M )
        = ( numera1916890842035813515d_enat @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_15_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numera6690914467698888265omplex @ M )
        = ( numera6690914467698888265omplex @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_16_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_real @ M )
        = ( numeral_numeral_real @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_17_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_nat @ M )
        = ( numeral_numeral_nat @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_18_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_int @ M )
        = ( numeral_numeral_int @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_19_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_rat @ M )
        = ( numeral_numeral_rat @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_20_verit__eq__simplify_I10_J,axiom,
    ! [X2: num] :
      ( one
     != ( bit0 @ X2 ) ) ).

% verit_eq_simplify(10)
thf(fact_21_local_Opower__def,axiom,
    ( vEBT_VEBT_power
    = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).

% local.power_def
thf(fact_22_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ~ ( vEBT_vebt_member @ T @ X ) ) ).

% min_Null_member
thf(fact_23__C2_Ohyps_C_I1_J,axiom,
    vEBT_invar_vebt @ summary2 @ m ).

% "2.hyps"(1)
thf(fact_24_insert_H__pres__valid,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( vEBT_invar_vebt @ ( vEBT_VEBT_insert @ T @ X ) @ N ) ) ).

% insert'_pres_valid
thf(fact_25_semiring__norm_I78_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(78)
thf(fact_26_semiring__norm_I75_J,axiom,
    ! [M: num] :
      ~ ( ord_less_num @ M @ one ) ).

% semiring_norm(75)
thf(fact_27_semiring__norm_I76_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).

% semiring_norm(76)
thf(fact_28_member__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_vebt_member @ T @ X )
        = ( member_nat @ X @ ( vEBT_set_vebt @ T ) ) ) ) ).

% member_correct
thf(fact_29_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_30_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_31_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_32_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_33_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_34_verit__comp__simplify1_I1_J,axiom,
    ! [A: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_35_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
         => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X ) @ Y )
           => ( ( vEBT_vebt_member @ T @ Y )
              | ( X = Y ) ) ) ) ) ) ).

% post_member_pre_member
thf(fact_36_valid__pres__insert,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( vEBT_invar_vebt @ ( vEBT_vebt_insert @ T @ X ) @ N ) ) ) ).

% valid_pres_insert
thf(fact_37_enat__ord__number_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).

% enat_ord_number(2)
thf(fact_38_aa,axiom,
    ! [X3: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ treeList ) )
     => ( vEBT_invar_vebt @ X3 @ na ) ) ).

% aa
thf(fact_39__C2_Oprems_C_I1_J,axiom,
    vEBT_invar_vebt @ sa @ deg ).

% "2.prems"(1)
thf(fact_40_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y3: list_VEBT_VEBT,Z: list_VEBT_VEBT] : Y3 = Z )
    = ( ^ [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
          ( ( ( size_s6755466524823107622T_VEBT @ Xs )
            = ( size_s6755466524823107622T_VEBT @ Ys ) )
          & ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
             => ( ( nth_VEBT_VEBT @ Xs @ I )
                = ( nth_VEBT_VEBT @ Ys @ I ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_41_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y3: list_o,Z: list_o] : Y3 = Z )
    = ( ^ [Xs: list_o,Ys: list_o] :
          ( ( ( size_size_list_o @ Xs )
            = ( size_size_list_o @ Ys ) )
          & ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
             => ( ( nth_o @ Xs @ I )
                = ( nth_o @ Ys @ I ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_42_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y3: list_nat,Z: list_nat] : Y3 = Z )
    = ( ^ [Xs: list_nat,Ys: list_nat] :
          ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys ) )
          & ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
             => ( ( nth_nat @ Xs @ I )
                = ( nth_nat @ Ys @ I ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_43_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y3: list_int,Z: list_int] : Y3 = Z )
    = ( ^ [Xs: list_int,Ys: list_int] :
          ( ( ( size_size_list_int @ Xs )
            = ( size_size_list_int @ Ys ) )
          & ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
             => ( ( nth_int @ Xs @ I )
                = ( nth_int @ Ys @ I ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_44_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vEBT_VEBT > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ K )
           => ? [X4: vEBT_VEBT] : ( P @ I @ X4 ) ) )
      = ( ? [Xs: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ Xs )
              = K )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ K )
               => ( P @ I @ ( nth_VEBT_VEBT @ Xs @ I ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_45_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > $o > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ K )
           => ? [X4: $o] : ( P @ I @ X4 ) ) )
      = ( ? [Xs: list_o] :
            ( ( ( size_size_list_o @ Xs )
              = K )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ K )
               => ( P @ I @ ( nth_o @ Xs @ I ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_46_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ K )
           => ? [X4: nat] : ( P @ I @ X4 ) ) )
      = ( ? [Xs: list_nat] :
            ( ( ( size_size_list_nat @ Xs )
              = K )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ K )
               => ( P @ I @ ( nth_nat @ Xs @ I ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_47_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > int > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ K )
           => ? [X4: int] : ( P @ I @ X4 ) ) )
      = ( ? [Xs: list_int] :
            ( ( ( size_size_list_int @ Xs )
              = K )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ K )
               => ( P @ I @ ( nth_int @ Xs @ I ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_48_nth__equalityI,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys2 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
              = ( nth_VEBT_VEBT @ Ys2 @ I2 ) ) )
       => ( Xs2 = Ys2 ) ) ) ).

% nth_equalityI
thf(fact_49_nth__equalityI,axiom,
    ! [Xs2: list_o,Ys2: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_o @ Ys2 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
           => ( ( nth_o @ Xs2 @ I2 )
              = ( nth_o @ Ys2 @ I2 ) ) )
       => ( Xs2 = Ys2 ) ) ) ).

% nth_equalityI
thf(fact_50_nth__equalityI,axiom,
    ! [Xs2: list_nat,Ys2: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
           => ( ( nth_nat @ Xs2 @ I2 )
              = ( nth_nat @ Ys2 @ I2 ) ) )
       => ( Xs2 = Ys2 ) ) ) ).

% nth_equalityI
thf(fact_51_nth__equalityI,axiom,
    ! [Xs2: list_int,Ys2: list_int] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_int @ Ys2 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
           => ( ( nth_int @ Xs2 @ I2 )
              = ( nth_int @ Ys2 @ I2 ) ) )
       => ( Xs2 = Ys2 ) ) ) ).

% nth_equalityI
thf(fact_52_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_53_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_power_complex @ ( numera6690914467698888265omplex @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numera6690914467698888265omplex @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_54_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_55_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_56_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_57_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_power_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numeral_numeral_rat @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_58__C2_OIH_C_I1_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ treeList2 ) )
     => ( ( vEBT_invar_vebt @ X3 @ na )
        & ! [Xa: vEBT_VEBT] :
            ( ( vEBT_invar_vebt @ Xa @ na )
           => ( ( ( vEBT_VEBT_set_vebt @ X3 )
                = ( vEBT_VEBT_set_vebt @ Xa ) )
             => ( Xa = X3 ) ) ) ) ) ).

% "2.IH"(1)
thf(fact_59__C2_Ohyps_C_I4_J,axiom,
    ( deg
    = ( plus_plus_nat @ na @ m ) ) ).

% "2.hyps"(4)
thf(fact_60_valid__eq,axiom,
    vEBT_VEBT_valid = vEBT_invar_vebt ).

% valid_eq
thf(fact_61_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_VEBT_valid @ T @ D )
     => ( vEBT_invar_vebt @ T @ D ) ) ).

% valid_eq2
thf(fact_62_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_invar_vebt @ T @ D )
     => ( vEBT_VEBT_valid @ T @ D ) ) ).

% valid_eq1
thf(fact_63_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_set_vebt @ T )
        = ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_set_vebt'_valid
thf(fact_64_inthall,axiom,
    ! [Xs2: list_complex,P: complex > $o,N: nat] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ ( set_complex2 @ Xs2 ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
       => ( P @ ( nth_complex @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_65_inthall,axiom,
    ! [Xs2: list_real,P: real > $o,N: nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ ( set_real2 @ Xs2 ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
       => ( P @ ( nth_real @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_66_inthall,axiom,
    ! [Xs2: list_set_nat,P: set_nat > $o,N: nat] :
      ( ! [X5: set_nat] :
          ( ( member_set_nat @ X5 @ ( set_set_nat2 @ Xs2 ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs2 ) )
       => ( P @ ( nth_set_nat @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_67_inthall,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_68_inthall,axiom,
    ! [Xs2: list_o,P: $o > $o,N: nat] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ ( set_o2 @ Xs2 ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
       => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_69_inthall,axiom,
    ! [Xs2: list_nat,P: nat > $o,N: nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
       => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_70_inthall,axiom,
    ! [Xs2: list_int,P: int > $o,N: nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ ( set_int2 @ Xs2 ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
       => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_71__C2_OIH_C_I2_J,axiom,
    ! [S: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ S @ m )
     => ( ( ( vEBT_VEBT_set_vebt @ summary2 )
          = ( vEBT_VEBT_set_vebt @ S ) )
       => ( S = summary2 ) ) ) ).

% "2.IH"(2)
thf(fact_72_add__numeral__left,axiom,
    ! [V: num,W: num,Z2: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ W ) @ Z2 ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_73_add__numeral__left,axiom,
    ! [V: num,W: num,Z2: complex] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z2 ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_74_add__numeral__left,axiom,
    ! [V: num,W: num,Z2: real] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z2 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_75_add__numeral__left,axiom,
    ! [V: num,W: num,Z2: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_76_add__numeral__left,axiom,
    ! [V: num,W: num,Z2: int] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_77_add__numeral__left,axiom,
    ! [V: num,W: num,Z2: rat] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z2 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_78_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( numera1916890842035813515d_enat @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_79_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_80_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_81_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_82_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_83_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_84__C2_Ohyps_C_I5_J,axiom,
    ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ summary2 @ X_1 ) ).

% "2.hyps"(5)
thf(fact_85__C2_Ohyps_C_I6_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ treeList2 ) )
     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) ).

% "2.hyps"(6)
thf(fact_86_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_87_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_88_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_89_mem__Collect__eq,axiom,
    ! [A: complex,P: complex > $o] :
      ( ( member_complex @ A @ ( collect_complex @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_90_mem__Collect__eq,axiom,
    ! [A: real,P: real > $o] :
      ( ( member_real @ A @ ( collect_real @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_91_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_92_mem__Collect__eq,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_93_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_94_mem__Collect__eq,axiom,
    ! [A: int,P: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_95_Collect__mem__eq,axiom,
    ! [A2: set_complex] :
      ( ( collect_complex
        @ ^ [X6: complex] : ( member_complex @ X6 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_96_Collect__mem__eq,axiom,
    ! [A2: set_real] :
      ( ( collect_real
        @ ^ [X6: real] : ( member_real @ X6 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_97_Collect__mem__eq,axiom,
    ! [A2: set_list_nat] :
      ( ( collect_list_nat
        @ ^ [X6: list_nat] : ( member_list_nat @ X6 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_98_Collect__mem__eq,axiom,
    ! [A2: set_set_nat] :
      ( ( collect_set_nat
        @ ^ [X6: set_nat] : ( member_set_nat @ X6 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_99_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X6: nat] : ( member_nat @ X6 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_100_Collect__mem__eq,axiom,
    ! [A2: set_int] :
      ( ( collect_int
        @ ^ [X6: int] : ( member_int @ X6 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_101_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_102_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_103_Collect__cong,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X5: set_nat] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_set_nat @ P )
        = ( collect_set_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_104_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_105_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_106_enat__less__induct,axiom,
    ! [P: extended_enat > $o,N: extended_enat] :
      ( ! [N2: extended_enat] :
          ( ! [M2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ M2 @ N2 )
             => ( P @ M2 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% enat_less_induct
thf(fact_107_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numera1916890842035813515d_enat @ ( bit0 @ N ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) ) ).

% numeral_Bit0
thf(fact_108_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_Bit0
thf(fact_109_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).

% numeral_Bit0
thf(fact_110_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).

% numeral_Bit0
thf(fact_111_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).

% numeral_Bit0
thf(fact_112_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).

% numeral_Bit0
thf(fact_113_nth__mem,axiom,
    ! [N: nat,Xs2: list_complex] :
      ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
     => ( member_complex @ ( nth_complex @ Xs2 @ N ) @ ( set_complex2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_114_nth__mem,axiom,
    ! [N: nat,Xs2: list_real] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
     => ( member_real @ ( nth_real @ Xs2 @ N ) @ ( set_real2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_115_nth__mem,axiom,
    ! [N: nat,Xs2: list_set_nat] :
      ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs2 ) )
     => ( member_set_nat @ ( nth_set_nat @ Xs2 @ N ) @ ( set_set_nat2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_116_nth__mem,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ N ) @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_117_nth__mem,axiom,
    ! [N: nat,Xs2: list_o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
     => ( member_o @ ( nth_o @ Xs2 @ N ) @ ( set_o2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_118_nth__mem,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( member_nat @ ( nth_nat @ Xs2 @ N ) @ ( set_nat2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_119_nth__mem,axiom,
    ! [N: nat,Xs2: list_int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
     => ( member_int @ ( nth_int @ Xs2 @ N ) @ ( set_int2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_120_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ! [X5: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( P @ X5 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_121_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_o,P: $o > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
     => ( ! [X5: $o] :
            ( ( member_o @ X5 @ ( set_o2 @ Xs2 ) )
           => ( P @ X5 ) )
       => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_122_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
           => ( P @ X5 ) )
       => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_123_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_int,P: int > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ ( set_int2 @ Xs2 ) )
           => ( P @ X5 ) )
       => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_124_in__set__conv__nth,axiom,
    ! [X: complex,Xs2: list_complex] :
      ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3451745648224563538omplex @ Xs2 ) )
            & ( ( nth_complex @ Xs2 @ I )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_125_in__set__conv__nth,axiom,
    ! [X: real,Xs2: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
            & ( ( nth_real @ Xs2 @ I )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_126_in__set__conv__nth,axiom,
    ! [X: set_nat,Xs2: list_set_nat] :
      ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3254054031482475050et_nat @ Xs2 ) )
            & ( ( nth_set_nat @ Xs2 @ I )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_127_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
            & ( ( nth_VEBT_VEBT @ Xs2 @ I )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_128_in__set__conv__nth,axiom,
    ! [X: $o,Xs2: list_o] :
      ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
            & ( ( nth_o @ Xs2 @ I )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_129_in__set__conv__nth,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
            & ( ( nth_nat @ Xs2 @ I )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_130_in__set__conv__nth,axiom,
    ! [X: int,Xs2: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
            & ( ( nth_int @ Xs2 @ I )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_131_all__nth__imp__all__set,axiom,
    ! [Xs2: list_complex,P: complex > $o,X: complex] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ Xs2 ) )
         => ( P @ ( nth_complex @ Xs2 @ I2 ) ) )
     => ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_132_all__nth__imp__all__set,axiom,
    ! [Xs2: list_real,P: real > $o,X: real] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs2 ) )
         => ( P @ ( nth_real @ Xs2 @ I2 ) ) )
     => ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_133_all__nth__imp__all__set,axiom,
    ! [Xs2: list_set_nat,P: set_nat > $o,X: set_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs2 ) )
         => ( P @ ( nth_set_nat @ Xs2 @ I2 ) ) )
     => ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_134_all__nth__imp__all__set,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
         => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_135_all__nth__imp__all__set,axiom,
    ! [Xs2: list_o,P: $o > $o,X: $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
         => ( P @ ( nth_o @ Xs2 @ I2 ) ) )
     => ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_136_all__nth__imp__all__set,axiom,
    ! [Xs2: list_nat,P: nat > $o,X: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
         => ( P @ ( nth_nat @ Xs2 @ I2 ) ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_137_all__nth__imp__all__set,axiom,
    ! [Xs2: list_int,P: int > $o,X: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
         => ( P @ ( nth_int @ Xs2 @ I2 ) ) )
     => ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_138_all__set__conv__all__nth,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X6: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( P @ X6 ) ) )
      = ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_139_all__set__conv__all__nth,axiom,
    ! [Xs2: list_o,P: $o > $o] :
      ( ( ! [X6: $o] :
            ( ( member_o @ X6 @ ( set_o2 @ Xs2 ) )
           => ( P @ X6 ) ) )
      = ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
           => ( P @ ( nth_o @ Xs2 @ I ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_140_all__set__conv__all__nth,axiom,
    ! [Xs2: list_nat,P: nat > $o] :
      ( ( ! [X6: nat] :
            ( ( member_nat @ X6 @ ( set_nat2 @ Xs2 ) )
           => ( P @ X6 ) ) )
      = ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
           => ( P @ ( nth_nat @ Xs2 @ I ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_141_all__set__conv__all__nth,axiom,
    ! [Xs2: list_int,P: int > $o] :
      ( ( ! [X6: int] :
            ( ( member_int @ X6 @ ( set_int2 @ Xs2 ) )
           => ( P @ X6 ) ) )
      = ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
           => ( P @ ( nth_int @ Xs2 @ I ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_142_pow_Osimps_I1_J,axiom,
    ! [X: num] :
      ( ( pow @ X @ one )
      = X ) ).

% pow.simps(1)
thf(fact_143_neq__if__length__neq,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
       != ( size_s6755466524823107622T_VEBT @ Ys2 ) )
     => ( Xs2 != Ys2 ) ) ).

% neq_if_length_neq
thf(fact_144_neq__if__length__neq,axiom,
    ! [Xs2: list_o,Ys2: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
       != ( size_size_list_o @ Ys2 ) )
     => ( Xs2 != Ys2 ) ) ).

% neq_if_length_neq
thf(fact_145_neq__if__length__neq,axiom,
    ! [Xs2: list_nat,Ys2: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
       != ( size_size_list_nat @ Ys2 ) )
     => ( Xs2 != Ys2 ) ) ).

% neq_if_length_neq
thf(fact_146_neq__if__length__neq,axiom,
    ! [Xs2: list_int,Ys2: list_int] :
      ( ( ( size_size_list_int @ Xs2 )
       != ( size_size_list_int @ Ys2 ) )
     => ( Xs2 != Ys2 ) ) ).

% neq_if_length_neq
thf(fact_147_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_148_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_o] :
      ( ( size_size_list_o @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_149_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_nat] :
      ( ( size_size_list_nat @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_150_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_int] :
      ( ( size_size_list_int @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_151_length__induct,axiom,
    ! [P: list_VEBT_VEBT > $o,Xs2: list_VEBT_VEBT] :
      ( ! [Xs3: list_VEBT_VEBT] :
          ( ! [Ys3: list_VEBT_VEBT] :
              ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys3 ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_152_length__induct,axiom,
    ! [P: list_o > $o,Xs2: list_o] :
      ( ! [Xs3: list_o] :
          ( ! [Ys3: list_o] :
              ( ( ord_less_nat @ ( size_size_list_o @ Ys3 ) @ ( size_size_list_o @ Xs3 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_153_length__induct,axiom,
    ! [P: list_nat > $o,Xs2: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys3: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs3 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_154_length__induct,axiom,
    ! [P: list_int > $o,Xs2: list_int] :
      ( ! [Xs3: list_int] :
          ( ! [Ys3: list_int] :
              ( ( ord_less_nat @ ( size_size_list_int @ Ys3 ) @ ( size_size_list_int @ Xs3 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_155_add__def,axiom,
    ( vEBT_VEBT_add
    = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).

% add_def
thf(fact_156_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_157_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_158_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_159_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_160_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_161_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_162_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_163_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_164_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_165_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X ) @ X ) ) ) ).

% valid_insert_both_member_options_add
thf(fact_166_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
         => ( ( vEBT_V8194947554948674370ptions @ T @ X )
           => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y ) @ X ) ) ) ) ) ).

% valid_insert_both_member_options_pres
thf(fact_167_ab,axiom,
    ! [X3: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ treeList2 ) )
     => ( ( vEBT_VEBT_set_vebt @ X3 )
        = bot_bot_set_nat ) ) ).

% ab
thf(fact_168_pow__sum,axiom,
    ! [A: nat,B: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).

% pow_sum
thf(fact_169__C2_Oprems_C_I2_J,axiom,
    ( ( vEBT_VEBT_set_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ deg @ treeList2 @ summary2 ) )
    = ( vEBT_VEBT_set_vebt @ sa ) ) ).

% "2.prems"(2)
thf(fact_170_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
     => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% high_bound_aux
thf(fact_171_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_172_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_173_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_174_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_175_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_176_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_177_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_178_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_179_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
     => ( Deg = N ) ) ).

% deg_deg_n
thf(fact_180_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ T )
     => ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_12 ) ) ).

% not_min_Null_member
thf(fact_181_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X )
       => ( vEBT_vebt_member @ T @ X ) ) ) ).

% valid_member_both_member_options
thf(fact_182_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X )
        = ( vEBT_vebt_member @ T @ X ) ) ) ).

% both_member_options_equiv_member
thf(fact_183_high__def,axiom,
    ( vEBT_VEBT_high
    = ( ^ [X6: nat,N3: nat] : ( divide_divide_nat @ X6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% high_def
thf(fact_184_semiring__norm_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).

% semiring_norm(6)
thf(fact_185_semiring__norm_I2_J,axiom,
    ( ( plus_plus_num @ one @ one )
    = ( bit0 @ one ) ) ).

% semiring_norm(2)
thf(fact_186_sprop,axiom,
    ( ( sa
      = ( vEBT_Node @ none_P5556105721700978146at_nat @ deg @ treeList @ summary ) )
    & ( deg
      = ( plus_plus_nat @ na @ m ) )
    & ( ( size_s6755466524823107622T_VEBT @ treeList )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
    & ( vEBT_invar_vebt @ summary @ m )
    & ! [X3: vEBT_VEBT] :
        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ treeList ) )
       => ( vEBT_invar_vebt @ X3 @ na ) )
    & ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ summary @ X_1 ) ) ).

% sprop
thf(fact_187__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062treeList_H_Asummary_H_O_As_A_061_ANode_ANone_Adeg_AtreeList_H_Asummary_H_A_092_060and_062_Adeg_A_061_An_A_L_Am_A_092_060and_062_Alength_AtreeList_H_A_061_A2_A_094_Am_A_092_060and_062_Ainvar__vebt_Asummary_H_Am_A_092_060and_062_A_I_092_060forall_062t_092_060in_062set_AtreeList_H_O_Ainvar__vebt_At_An_J_A_092_060and_062_A_I_092_060nexists_062i_O_Aboth__member__options_Asummary_H_Ai_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [TreeList2: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
        ~ ( ( sa
            = ( vEBT_Node @ none_P5556105721700978146at_nat @ deg @ TreeList2 @ Summary2 ) )
          & ( deg
            = ( plus_plus_nat @ na @ m ) )
          & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
          & ( vEBT_invar_vebt @ Summary2 @ m )
          & ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
             => ( vEBT_invar_vebt @ X3 @ na ) )
          & ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 ) ) ).

% \<open>\<And>thesis. (\<And>treeList' summary'. s = Node None deg treeList' summary' \<and> deg = n + m \<and> length treeList' = 2 ^ m \<and> invar_vebt summary' m \<and> (\<forall>t\<in>set treeList'. invar_vebt t n) \<and> (\<nexists>i. both_member_options summary' i) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_188_power__divide,axiom,
    ! [A: complex,B: complex,N: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N )
      = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).

% power_divide
thf(fact_189_power__divide,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
      = ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).

% power_divide
thf(fact_190_power__divide,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ A @ B ) @ N )
      = ( divide_divide_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).

% power_divide
thf(fact_191_add__One__commute,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ N )
      = ( plus_plus_num @ N @ one ) ) ).

% add_One_commute
thf(fact_192_divide__numeral__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_193_divide__numeral__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_194_divide__numeral__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_195_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_196_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_197_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_198_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_199_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I3: real,J: real,K: real,L: real] :
      ( ( ( I3 = J )
        & ( K = L ) )
     => ( ( plus_plus_real @ I3 @ K )
        = ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_200_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I3: rat,J: rat,K: rat,L: rat] :
      ( ( ( I3 = J )
        & ( K = L ) )
     => ( ( plus_plus_rat @ I3 @ K )
        = ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_201_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I3: nat,J: nat,K: nat,L: nat] :
      ( ( ( I3 = J )
        & ( K = L ) )
     => ( ( plus_plus_nat @ I3 @ K )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_202_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I3: int,J: int,K: int,L: int] :
      ( ( ( I3 = J )
        & ( K = L ) )
     => ( ( plus_plus_int @ I3 @ K )
        = ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_203_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_204_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_205_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_206_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_207_group__cancel_Oadd2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2
        = ( plus_plus_real @ K @ B ) )
     => ( ( plus_plus_real @ A @ B2 )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_208_group__cancel_Oadd2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2
        = ( plus_plus_rat @ K @ B ) )
     => ( ( plus_plus_rat @ A @ B2 )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_209_group__cancel_Oadd2,axiom,
    ! [B2: nat,K: nat,B: nat,A: nat] :
      ( ( B2
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A @ B2 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_210_group__cancel_Oadd2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2
        = ( plus_plus_int @ K @ B ) )
     => ( ( plus_plus_int @ A @ B2 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_211_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_212_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_213_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_214_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_215_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_216_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_217_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_218_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_219_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_220_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_221_add_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A3: real,B3: real] : ( plus_plus_real @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_222_add_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A3: rat,B3: rat] : ( plus_plus_rat @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_223_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_224_add_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_225_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_226_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_227_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_228_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_229_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_230_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_231_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_232_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_233_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_234_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_235_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_236_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_237_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE_nat
thf(fact_238_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ~ ( P @ N2 )
         => ? [M2: nat] :
              ( ( ord_less_nat @ M2 @ N2 )
              & ~ ( P @ M2 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_239_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N2 )
             => ( P @ M2 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_240_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_241_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_242_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_243_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_244_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
      = ( ( ord_less_nat @ M @ N )
        | ( ord_less_nat @ N @ M ) ) ) ).

% nat_neq_iff
thf(fact_245_size__neq__size__imp__neq,axiom,
    ! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ X )
       != ( size_s6755466524823107622T_VEBT @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_246_size__neq__size__imp__neq,axiom,
    ! [X: list_o,Y: list_o] :
      ( ( ( size_size_list_o @ X )
       != ( size_size_list_o @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_247_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( ( size_size_list_nat @ X )
       != ( size_size_list_nat @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_248_size__neq__size__imp__neq,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( ( size_size_list_int @ X )
       != ( size_size_list_int @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_249_size__neq__size__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( ( size_size_num @ X )
       != ( size_size_num @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_250_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_251_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_252_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_253_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_254_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_255_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_256_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_257_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_258_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_259_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_260_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_261_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_262_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_263_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_264_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_265_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_266_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_267_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_268_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_269_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_270_add__strict__mono,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_le72135733267957522d_enat @ C @ D )
       => ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ ( plus_p3455044024723400733d_enat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_271_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I3: real,J: real,K: real,L: real] :
      ( ( ( ord_less_real @ I3 @ J )
        & ( K = L ) )
     => ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_272_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I3: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I3 @ J )
        & ( K = L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I3 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_273_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I3: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I3 @ J )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_274_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I3: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I3 @ J )
        & ( K = L ) )
     => ( ord_less_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_275_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I3: real,J: real,K: real,L: real] :
      ( ( ( I3 = J )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_276_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I3: rat,J: rat,K: rat,L: rat] :
      ( ( ( I3 = J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I3 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_277_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I3: nat,J: nat,K: nat,L: nat] :
      ( ( ( I3 = J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_278_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I3: int,J: int,K: int,L: int] :
      ( ( ( I3 = J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_279_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I3: real,J: real,K: real,L: real] :
      ( ( ( ord_less_real @ I3 @ J )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_280_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I3: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I3 @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I3 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_281_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I3: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I3 @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_282_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I3: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I3 @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_283_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% less_add_eq_less
thf(fact_284_trans__less__add2,axiom,
    ! [I3: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I3 @ J )
     => ( ord_less_nat @ I3 @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_less_add2
thf(fact_285_trans__less__add1,axiom,
    ! [I3: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I3 @ J )
     => ( ord_less_nat @ I3 @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_less_add1
thf(fact_286_add__less__mono1,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I3 @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_287_not__add__less2,axiom,
    ! [J: nat,I3: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I3 ) @ I3 ) ).

% not_add_less2
thf(fact_288_not__add__less1,axiom,
    ! [I3: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I3 @ J ) @ I3 ) ).

% not_add_less1
thf(fact_289_add__less__mono,axiom,
    ! [I3: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I3 @ J )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_290_add__lessD1,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I3 @ J ) @ K )
     => ( ord_less_nat @ I3 @ K ) ) ).

% add_lessD1
thf(fact_291_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_292_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
               => ( ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
thf(fact_293_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).

% both_member_options_ding
thf(fact_294_div__exp__eq,axiom,
    ! [A: nat,M: nat,N: 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 ) ) @ N ) )
      = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).

% div_exp_eq
thf(fact_295_div__exp__eq,axiom,
    ! [A: int,M: nat,N: 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 ) ) @ N ) )
      = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).

% div_exp_eq
thf(fact_296_buildup__gives__empty,axiom,
    ! [N: nat] :
      ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
      = bot_bot_set_nat ) ).

% buildup_gives_empty
thf(fact_297_field__less__half__sum,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_real @ X @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_298_field__less__half__sum,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ord_less_rat @ X @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_299_high__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
        = Y ) ) ).

% high_inv
thf(fact_300_field__sum__of__halves,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = X ) ).

% field_sum_of_halves
thf(fact_301_field__sum__of__halves,axiom,
    ! [X: rat] :
      ( ( plus_plus_rat @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
      = X ) ).

% field_sum_of_halves
thf(fact_302_numeral__Bit0__div__2,axiom,
    ! [N: num] :
      ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( numeral_numeral_nat @ N ) ) ).

% numeral_Bit0_div_2
thf(fact_303_numeral__Bit0__div__2,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( numeral_numeral_int @ N ) ) ).

% numeral_Bit0_div_2
thf(fact_304_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).

% VEBT_internal.minNull.simps(4)
thf(fact_305_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X ) ).

% vebt_member.simps(2)
thf(fact_306_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_307_bit__split__inv,axiom,
    ! [X: nat,D: nat] :
      ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D ) @ ( vEBT_VEBT_low @ X @ D ) @ D )
      = X ) ).

% bit_split_inv
thf(fact_308_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z2: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ W ) @ Z2 ) )
      = ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_309_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z2: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z2 ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_310_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z2: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z2 ) )
      = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_311_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_312_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_313_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z2: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z2 ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_314_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( numera1916890842035813515d_enat @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_315_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_316_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_317_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_318_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_319_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_320_low__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
        = X ) ) ).

% low_inv
thf(fact_321_bit__concat__def,axiom,
    ( vEBT_VEBT_bit_concat
    = ( ^ [H: nat,L2: nat,D2: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D2 ) ) @ L2 ) ) ) ).

% bit_concat_def
thf(fact_322_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_323_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_324_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_325_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_326_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_327_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_328_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_329_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_330_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_331_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_332_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_333_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_334_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_335_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_336_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_337_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_338_power__add__numeral,axiom,
    ! [A: complex,M: num,N: num] :
      ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_339_power__add__numeral,axiom,
    ! [A: real,M: num,N: num] :
      ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_340_power__add__numeral,axiom,
    ! [A: rat,M: num,N: num] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_341_power__add__numeral,axiom,
    ! [A: nat,M: num,N: num] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_342_power__add__numeral,axiom,
    ! [A: int,M: num,N: num] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_343_power__add__numeral2,axiom,
    ! [A: complex,M: num,N: num,B: complex] :
      ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_344_power__add__numeral2,axiom,
    ! [A: real,M: num,N: num,B: real] :
      ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_345_power__add__numeral2,axiom,
    ! [A: rat,M: num,N: num,B: rat] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_346_power__add__numeral2,axiom,
    ! [A: nat,M: num,N: num,B: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_347_power__add__numeral2,axiom,
    ! [A: int,M: num,N: num,B: int] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_348_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_349_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_350_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_351_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_352_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_353_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_354_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_355_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_356_mult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A3: real,B3: real] : ( times_times_real @ B3 @ A3 ) ) ) ).

% mult.commute
thf(fact_357_mult_Ocommute,axiom,
    ( times_times_rat
    = ( ^ [A3: rat,B3: rat] : ( times_times_rat @ B3 @ A3 ) ) ) ).

% mult.commute
thf(fact_358_mult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).

% mult.commute
thf(fact_359_mult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A3: int,B3: int] : ( times_times_int @ B3 @ A3 ) ) ) ).

% mult.commute
thf(fact_360_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_361_mult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_362_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_363_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_364_div__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q2 ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) ).

% div_mult2_eq
thf(fact_365_power__commutes,axiom,
    ! [A: complex,N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
      = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).

% power_commutes
thf(fact_366_power__commutes,axiom,
    ! [A: real,N: nat] :
      ( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
      = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).

% power_commutes
thf(fact_367_power__commutes,axiom,
    ! [A: rat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ A @ N ) @ A )
      = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).

% power_commutes
thf(fact_368_power__commutes,axiom,
    ! [A: nat,N: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).

% power_commutes
thf(fact_369_power__commutes,axiom,
    ! [A: int,N: nat] :
      ( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
      = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).

% power_commutes
thf(fact_370_power__mult__distrib,axiom,
    ! [A: complex,B: complex,N: nat] :
      ( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N )
      = ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_371_power__mult__distrib,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
      = ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_372_power__mult__distrib,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N )
      = ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_373_power__mult__distrib,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
      = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_374_power__mult__distrib,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
      = ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_375_power__commuting__commutes,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( ( ( times_times_complex @ X @ Y )
        = ( times_times_complex @ Y @ X ) )
     => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ Y )
        = ( times_times_complex @ Y @ ( power_power_complex @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_376_power__commuting__commutes,axiom,
    ! [X: real,Y: real,N: nat] :
      ( ( ( times_times_real @ X @ Y )
        = ( times_times_real @ Y @ X ) )
     => ( ( times_times_real @ ( power_power_real @ X @ N ) @ Y )
        = ( times_times_real @ Y @ ( power_power_real @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_377_power__commuting__commutes,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = ( times_times_rat @ Y @ X ) )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ Y )
        = ( times_times_rat @ Y @ ( power_power_rat @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_378_power__commuting__commutes,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = ( times_times_nat @ Y @ X ) )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
        = ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_379_power__commuting__commutes,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( ( times_times_int @ X @ Y )
        = ( times_times_int @ Y @ X ) )
     => ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y )
        = ( times_times_int @ Y @ ( power_power_int @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_380_power__mult,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_381_power__mult,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_382_power__mult,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_383_power__mult,axiom,
    ! [A: complex,M: nat,N: nat] :
      ( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_384_left__add__mult__distrib,axiom,
    ! [I3: nat,U: nat,J: nat,K: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I3 @ J ) @ U ) @ K ) ) ).

% left_add_mult_distrib
thf(fact_385_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% add_mult_distrib2
thf(fact_386_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
      = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% add_mult_distrib
thf(fact_387_less__mult__imp__div__less,axiom,
    ! [M: nat,I3: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( times_times_nat @ I3 @ N ) )
     => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I3 ) ) ).

% less_mult_imp_div_less
thf(fact_388_mult__numeral__1,axiom,
    ! [A: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_389_mult__numeral__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_390_mult__numeral__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_391_mult__numeral__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_392_mult__numeral__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_393_mult__numeral__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_394_mult__numeral__1__right,axiom,
    ! [A: extended_enat] :
      ( ( times_7803423173614009249d_enat @ A @ ( numera1916890842035813515d_enat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_395_mult__numeral__1__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_396_mult__numeral__1__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_397_mult__numeral__1__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_398_mult__numeral__1__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_399_mult__numeral__1__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_400_power__add,axiom,
    ! [A: complex,M: nat,N: nat] :
      ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).

% power_add
thf(fact_401_power__add,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).

% power_add
thf(fact_402_power__add,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).

% power_add
thf(fact_403_power__add,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).

% power_add
thf(fact_404_power__add,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).

% power_add
thf(fact_405_mult__2,axiom,
    ! [Z2: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_p3455044024723400733d_enat @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_406_mult__2,axiom,
    ! [Z2: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_plus_complex @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_407_mult__2,axiom,
    ! [Z2: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_plus_real @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_408_mult__2,axiom,
    ! [Z2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_plus_nat @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_409_mult__2,axiom,
    ! [Z2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_plus_int @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_410_mult__2,axiom,
    ! [Z2: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_plus_rat @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_411_mult__2__right,axiom,
    ! [Z2: extended_enat] :
      ( ( times_7803423173614009249d_enat @ Z2 @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) )
      = ( plus_p3455044024723400733d_enat @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_412_mult__2__right,axiom,
    ! [Z2: complex] :
      ( ( times_times_complex @ Z2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_413_mult__2__right,axiom,
    ! [Z2: real] :
      ( ( times_times_real @ Z2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_414_mult__2__right,axiom,
    ! [Z2: nat] :
      ( ( times_times_nat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_415_mult__2__right,axiom,
    ! [Z2: int] :
      ( ( times_times_int @ Z2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_416_mult__2__right,axiom,
    ! [Z2: rat] :
      ( ( times_times_rat @ Z2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_417_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_418_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_419_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_420_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_421_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_422_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_423_power4__eq__xxxx,axiom,
    ! [X: complex] :
      ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_424_power4__eq__xxxx,axiom,
    ! [X: real] :
      ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_real @ ( times_times_real @ ( times_times_real @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_425_power4__eq__xxxx,axiom,
    ! [X: rat] :
      ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_426_power4__eq__xxxx,axiom,
    ! [X: nat] :
      ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_427_power4__eq__xxxx,axiom,
    ! [X: int] :
      ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_428_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_429_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_430_power2__eq__square,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_rat @ A @ A ) ) ).

% power2_eq_square
thf(fact_431_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_432_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_433_power__even__eq,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_434_power__even__eq,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_435_power__even__eq,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_436_power__even__eq,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_437_power2__sum,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( power_8040749407984259932d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( power_8040749407984259932d_enat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8040749407984259932d_enat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_438_power2__sum,axiom,
    ! [X: complex,Y: complex] :
      ( ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_439_power2__sum,axiom,
    ! [X: real,Y: real] :
      ( ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_440_power2__sum,axiom,
    ! [X: nat,Y: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_441_power2__sum,axiom,
    ! [X: int,Y: int] :
      ( ( power_power_int @ ( plus_plus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_442_power2__sum,axiom,
    ! [X: rat,Y: rat] :
      ( ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( 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 ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_443_in__children__def,axiom,
    ( vEBT_V5917875025757280293ildren
    = ( ^ [N3: nat,TreeList3: list_VEBT_VEBT,X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ X6 @ N3 ) ) @ ( vEBT_VEBT_low @ X6 @ N3 ) ) ) ) ).

% in_children_def
thf(fact_444_mul__def,axiom,
    ( vEBT_VEBT_mul
    = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).

% mul_def
thf(fact_445_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_446_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_447_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).

% times_divide_eq_right
thf(fact_448_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_449_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_450_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).

% divide_divide_eq_right
thf(fact_451_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_452_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_453_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
      = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_454_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_455_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_456_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_457_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).

% buildup_nothing_in_leaf
thf(fact_458_low__def,axiom,
    ( vEBT_VEBT_low
    = ( ^ [X6: nat,N3: nat] : ( modulo_modulo_nat @ X6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% low_def
thf(fact_459_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N ) )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
               => ( ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
thf(fact_460_buildup__gives__valid,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).

% buildup_gives_valid
thf(fact_461_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ Uu @ none_P5556105721700978146at_nat @ Uv )
      = none_P5556105721700978146at_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_462_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: num > num > num,Uv: option_num] :
      ( ( vEBT_V819420779217536731ft_num @ Uu @ none_num @ Uv )
      = none_num ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_463_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: nat > nat > nat,Uv: option_nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ Uu @ none_nat @ Uv )
      = none_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_464_empty__Collect__eq,axiom,
    ! [P: list_nat > $o] :
      ( ( bot_bot_set_list_nat
        = ( collect_list_nat @ P ) )
      = ( ! [X6: list_nat] :
            ~ ( P @ X6 ) ) ) ).

% empty_Collect_eq
thf(fact_465_empty__Collect__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( bot_bot_set_set_nat
        = ( collect_set_nat @ P ) )
      = ( ! [X6: set_nat] :
            ~ ( P @ X6 ) ) ) ).

% empty_Collect_eq
thf(fact_466_empty__Collect__eq,axiom,
    ! [P: real > $o] :
      ( ( bot_bot_set_real
        = ( collect_real @ P ) )
      = ( ! [X6: real] :
            ~ ( P @ X6 ) ) ) ).

% empty_Collect_eq
thf(fact_467_empty__Collect__eq,axiom,
    ! [P: $o > $o] :
      ( ( bot_bot_set_o
        = ( collect_o @ P ) )
      = ( ! [X6: $o] :
            ~ ( P @ X6 ) ) ) ).

% empty_Collect_eq
thf(fact_468_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X6: nat] :
            ~ ( P @ X6 ) ) ) ).

% empty_Collect_eq
thf(fact_469_empty__Collect__eq,axiom,
    ! [P: int > $o] :
      ( ( bot_bot_set_int
        = ( collect_int @ P ) )
      = ( ! [X6: int] :
            ~ ( P @ X6 ) ) ) ).

% empty_Collect_eq
thf(fact_470_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N2: nat] :
          ( X
         != ( plus_plus_nat @ N2 @ N2 ) )
     => ~ ! [N2: nat] :
            ( X
           != ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) ) ) ).

% even_odd_cases
thf(fact_471_valid__0__not,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_0_not
thf(fact_472_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_tree_deg_neq_0
thf(fact_473_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% deg_not_0
thf(fact_474_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
     => ? [Info2: option4927543243414619207at_nat,TreeList4: list_VEBT_VEBT,S2: vEBT_VEBT] :
          ( Tree
          = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList4 @ S2 ) ) ) ).

% deg_SUcn_Node
thf(fact_475_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_476_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% nat.inject
thf(fact_477_empty__iff,axiom,
    ! [C: complex] :
      ~ ( member_complex @ C @ bot_bot_set_complex ) ).

% empty_iff
thf(fact_478_empty__iff,axiom,
    ! [C: set_nat] :
      ~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).

% empty_iff
thf(fact_479_empty__iff,axiom,
    ! [C: real] :
      ~ ( member_real @ C @ bot_bot_set_real ) ).

% empty_iff
thf(fact_480_empty__iff,axiom,
    ! [C: $o] :
      ~ ( member_o @ C @ bot_bot_set_o ) ).

% empty_iff
thf(fact_481_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_482_empty__iff,axiom,
    ! [C: int] :
      ~ ( member_int @ C @ bot_bot_set_int ) ).

% empty_iff
thf(fact_483_all__not__in__conv,axiom,
    ! [A2: set_complex] :
      ( ( ! [X6: complex] :
            ~ ( member_complex @ X6 @ A2 ) )
      = ( A2 = bot_bot_set_complex ) ) ).

% all_not_in_conv
thf(fact_484_all__not__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ( ! [X6: set_nat] :
            ~ ( member_set_nat @ X6 @ A2 ) )
      = ( A2 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
thf(fact_485_all__not__in__conv,axiom,
    ! [A2: set_real] :
      ( ( ! [X6: real] :
            ~ ( member_real @ X6 @ A2 ) )
      = ( A2 = bot_bot_set_real ) ) ).

% all_not_in_conv
thf(fact_486_all__not__in__conv,axiom,
    ! [A2: set_o] :
      ( ( ! [X6: $o] :
            ~ ( member_o @ X6 @ A2 ) )
      = ( A2 = bot_bot_set_o ) ) ).

% all_not_in_conv
thf(fact_487_all__not__in__conv,axiom,
    ! [A2: set_nat] :
      ( ( ! [X6: nat] :
            ~ ( member_nat @ X6 @ A2 ) )
      = ( A2 = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_488_all__not__in__conv,axiom,
    ! [A2: set_int] :
      ( ( ! [X6: int] :
            ~ ( member_int @ X6 @ A2 ) )
      = ( A2 = bot_bot_set_int ) ) ).

% all_not_in_conv
thf(fact_489_Collect__empty__eq,axiom,
    ! [P: list_nat > $o] :
      ( ( ( collect_list_nat @ P )
        = bot_bot_set_list_nat )
      = ( ! [X6: list_nat] :
            ~ ( P @ X6 ) ) ) ).

% Collect_empty_eq
thf(fact_490_Collect__empty__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( ( collect_set_nat @ P )
        = bot_bot_set_set_nat )
      = ( ! [X6: set_nat] :
            ~ ( P @ X6 ) ) ) ).

% Collect_empty_eq
thf(fact_491_Collect__empty__eq,axiom,
    ! [P: real > $o] :
      ( ( ( collect_real @ P )
        = bot_bot_set_real )
      = ( ! [X6: real] :
            ~ ( P @ X6 ) ) ) ).

% Collect_empty_eq
thf(fact_492_Collect__empty__eq,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( ! [X6: $o] :
            ~ ( P @ X6 ) ) ) ).

% Collect_empty_eq
thf(fact_493_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X6: nat] :
            ~ ( P @ X6 ) ) ) ).

% Collect_empty_eq
thf(fact_494_Collect__empty__eq,axiom,
    ! [P: int > $o] :
      ( ( ( collect_int @ P )
        = bot_bot_set_int )
      = ( ! [X6: int] :
            ~ ( P @ X6 ) ) ) ).

% Collect_empty_eq
thf(fact_495_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_496_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_497_mod__mod__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mod_trivial
thf(fact_498_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_499_not__gr__zero,axiom,
    ! [N: extended_enat] :
      ( ( ~ ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) )
      = ( N = zero_z5237406670263579293d_enat ) ) ).

% not_gr_zero
thf(fact_500_add_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.right_neutral
thf(fact_501_add_Oright__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.right_neutral
thf(fact_502_add_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.right_neutral
thf(fact_503_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_504_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_505_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A @ A ) )
      = ( A = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_506_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A @ A ) )
      = ( A = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_507_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_508_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_509_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_510_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_511_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_512_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_513_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_514_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_515_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_516_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_517_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_518_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_519_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_520_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_521_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_522_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_523_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_524_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_525_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_526_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_527_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_528_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = zero_zero_nat )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_529_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X @ Y ) )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_530_add__0,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add_0
thf(fact_531_add__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add_0
thf(fact_532_add__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add_0
thf(fact_533_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_534_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_535_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_536_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_537_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_538_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_539_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_540_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_541_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_542_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_543_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_544_division__ring__divide__zero,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% division_ring_divide_zero
thf(fact_545_division__ring__divide__zero,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% division_ring_divide_zero
thf(fact_546_division__ring__divide__zero,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% division_ring_divide_zero
thf(fact_547_bits__div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_div_0
thf(fact_548_bits__div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_div_0
thf(fact_549_bits__div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% bits_div_by_0
thf(fact_550_bits__div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% bits_div_by_0
thf(fact_551_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_less_eq
thf(fact_552_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_553_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_554_bits__mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_mod_0
thf(fact_555_bits__mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_mod_0
thf(fact_556_bits__mod__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_0
thf(fact_557_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_558_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_559_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_560_add__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc_right
thf(fact_561_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_562_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_563_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_add_self1
thf(fact_564_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_565_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_566_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_add_self2
thf(fact_567_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_568_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_569_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ( times_times_nat @ M @ K )
        = ( times_times_nat @ N @ K ) )
      = ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_570_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N ) )
      = ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_571_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_572_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_573_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( modulo_modulo_nat @ M @ N )
        = M ) ) ).

% mod_less
thf(fact_574_semiring__norm_I13_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).

% semiring_norm(13)
thf(fact_575_semiring__norm_I11_J,axiom,
    ! [M: num] :
      ( ( times_times_num @ M @ one )
      = M ) ).

% semiring_norm(11)
thf(fact_576_semiring__norm_I12_J,axiom,
    ! [N: num] :
      ( ( times_times_num @ one @ N )
      = N ) ).

% semiring_norm(12)
thf(fact_577_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_578_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_579_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_580_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_581_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_582_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_583_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_584_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_585_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_586_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_587_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_588_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_589_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_590_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_591_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_592_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_593_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_594_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_595_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_596_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_597_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_598_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_599_sum__squares__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
        = zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_600_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
        = zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_601_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
        = zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_602_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_603_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_604_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_605_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_606_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_607_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_608_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_609_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_610_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_611_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_612_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_613_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_614_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_615_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_616_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_617_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_618_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_619_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_620_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_621_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_622_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_623_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
      = zero_zero_rat ) ).

% power_0_Suc
thf(fact_624_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_625_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
      = zero_zero_real ) ).

% power_0_Suc
thf(fact_626_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_627_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
      = zero_zero_complex ) ).

% power_0_Suc
thf(fact_628_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_rat ) ).

% power_zero_numeral
thf(fact_629_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_nat ) ).

% power_zero_numeral
thf(fact_630_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
      = zero_zero_real ) ).

% power_zero_numeral
thf(fact_631_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
      = zero_zero_int ) ).

% power_zero_numeral
thf(fact_632_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
      = zero_zero_complex ) ).

% power_zero_numeral
thf(fact_633_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_634_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_635_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_636_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_637_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_638_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_639_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_640_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_641_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_div_trivial
thf(fact_642_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_643_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_644_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_div_trivial
thf(fact_645_power__Suc0__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_646_power__Suc0__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_647_power__Suc0__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_648_power__Suc0__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_649_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_650_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_651_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_652_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_653_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_654_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_655_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_656_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_657_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_658_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_659_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_660_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_661_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_662_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_663_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_664_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( times_times_nat @ M @ N ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% one_eq_mult_iff
thf(fact_665_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% mult_eq_1_iff
thf(fact_666_div__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
      = M ) ).

% div_by_Suc_0
thf(fact_667_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N ) ) ) ).

% mult_less_cancel2
thf(fact_668_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% nat_0_less_mult_iff
thf(fact_669_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_670_div__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( divide_divide_nat @ M @ N )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_671_power__Suc__0,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_672_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M: nat] :
      ( ( ( power_power_nat @ X @ M )
        = ( suc @ zero_zero_nat ) )
      = ( ( M = zero_zero_nat )
        | ( X
          = ( suc @ zero_zero_nat ) ) ) ) ).

% nat_power_eq_Suc_0_iff
thf(fact_673_mult__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( times_times_nat @ M @ ( suc @ N ) )
      = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).

% mult_Suc_right
thf(fact_674_nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_675_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( K = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
          = zero_zero_nat ) )
      & ( ( K != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
          = ( divide_divide_nat @ M @ N ) ) ) ) ).

% nat_mult_div_cancel_disj
thf(fact_676_mod__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_677_num__double,axiom,
    ! [N: num] :
      ( ( times_times_num @ ( bit0 @ one ) @ N )
      = ( bit0 @ N ) ) ).

% num_double
thf(fact_678_power__mult__numeral,axiom,
    ! [A: nat,M: num,N: num] :
      ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_679_power__mult__numeral,axiom,
    ! [A: real,M: num,N: num] :
      ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_680_power__mult__numeral,axiom,
    ! [A: int,M: num,N: num] :
      ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_681_power__mult__numeral,axiom,
    ! [A: complex,M: num,N: num] :
      ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_682_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_683_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_684_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_685_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_686_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_687_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_688_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_689_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_690_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_691_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_692_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_693_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_694_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_695_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_696_power__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( ( power_power_rat @ A @ N )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_697_power__eq__0__iff,axiom,
    ! [A: nat,N: nat] :
      ( ( ( power_power_nat @ A @ N )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_698_power__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( ( power_power_real @ A @ N )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_699_power__eq__0__iff,axiom,
    ! [A: int,N: nat] :
      ( ( ( power_power_int @ A @ N )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_700_power__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( ( power_power_complex @ A @ N )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_701_div__mult__self__is__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
        = M ) ) ).

% div_mult_self_is_m
thf(fact_702_div__mult__self1__is__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
        = M ) ) ).

% div_mult_self1_is_m
thf(fact_703_Suc__numeral,axiom,
    ! [N: num] :
      ( ( suc @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).

% Suc_numeral
thf(fact_704_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% Suc_mod_mult_self1
thf(fact_705_Suc__mod__mult__self2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% Suc_mod_mult_self2
thf(fact_706_Suc__mod__mult__self3,axiom,
    ! [K: nat,N: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% Suc_mod_mult_self3
thf(fact_707_Suc__mod__mult__self4,axiom,
    ! [N: nat,K: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% Suc_mod_mult_self4
thf(fact_708_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_709_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_710_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_711_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_712_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_713_add__2__eq__Suc_H,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( suc @ N ) ) ) ).

% add_2_eq_Suc'
thf(fact_714_add__2__eq__Suc,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
      = ( suc @ ( suc @ N ) ) ) ).

% add_2_eq_Suc
thf(fact_715_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_716_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_717_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_718_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_719_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_720_sum__power2__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_721_sum__power2__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_722_sum__power2__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_723_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_724_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).

% mod_Suc_Suc_eq
thf(fact_725_mod__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% mod_Suc_eq
thf(fact_726_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
          = N )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
          = zero_zero_nat ) )
      & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
         != N )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
          = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).

% mod_Suc
thf(fact_727_vebt__buildup_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ( ( X
         != ( suc @ zero_zero_nat ) )
       => ~ ! [Va: nat] :
              ( X
             != ( suc @ ( suc @ Va ) ) ) ) ) ).

% vebt_buildup.cases
thf(fact_728_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M3: nat] :
          ( N
          = ( suc @ M3 ) ) ) ).

% not0_implies_Suc
thf(fact_729_zero__reorient,axiom,
    ! [X: complex] :
      ( ( zero_zero_complex = X )
      = ( X = zero_zero_complex ) ) ).

% zero_reorient
thf(fact_730_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
      = ( X = zero_zero_real ) ) ).

% zero_reorient
thf(fact_731_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
      = ( X = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_732_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_733_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
      = ( X = zero_zero_int ) ) ).

% zero_reorient
thf(fact_734_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_735_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_736_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_737_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_738_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_739_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [X5: nat] : ( P @ X5 @ zero_zero_nat )
     => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
       => ( ! [X5: nat,Y4: nat] :
              ( ( P @ X5 @ Y4 )
             => ( P @ ( suc @ X5 ) @ ( suc @ Y4 ) ) )
         => ( P @ M @ N ) ) ) ) ).

% diff_induct
thf(fact_740_nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P @ N2 )
           => ( P @ ( suc @ N2 ) ) )
       => ( P @ N ) ) ) ).

% nat_induct
thf(fact_741_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y ) )
     => ( X = Y ) ) ).

% Suc_inject
thf(fact_742_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_743_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat
        = ( suc @ X2 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_744_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_745_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_746_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] :
      ( zero_zero_nat
     != ( suc @ X2 ) ) ).

% nat.distinct(1)
thf(fact_747_mod__induct,axiom,
    ! [P: nat > $o,N: nat,P2: nat,M: nat] :
      ( ( P @ N )
     => ( ( ord_less_nat @ N @ P2 )
       => ( ( ord_less_nat @ M @ P2 )
         => ( ! [N2: nat] :
                ( ( ord_less_nat @ N2 @ P2 )
               => ( ( P @ N2 )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P2 ) ) ) )
           => ( P @ M ) ) ) ) ) ).

% mod_induct
thf(fact_748_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_749_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_750_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_751_mod__less__divisor,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).

% mod_less_divisor
thf(fact_752_Ex__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N ) )
            & ( P @ I ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I: nat] :
            ( ( ord_less_nat @ I @ N )
            & ( P @ ( suc @ I ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_753_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M4: nat] :
            ( N
            = ( suc @ M4 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_754_All__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N ) )
           => ( P @ I ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ N )
           => ( P @ ( suc @ I ) ) ) ) ) ).

% All_less_Suc2
thf(fact_755_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M3: nat] :
          ( N
          = ( suc @ M3 ) ) ) ).

% gr0_implies_Suc
thf(fact_756_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( M = zero_zero_nat )
        | ? [J2: nat] :
            ( ( M
              = ( suc @ J2 ) )
            & ( ord_less_nat @ J2 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_757_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_758_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_759_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_760_div__less__mono,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( modulo_modulo_nat @ A2 @ N )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B2 @ N )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N ) @ ( divide_divide_nat @ B2 @ N ) ) ) ) ) ) ).

% div_less_mono
thf(fact_761_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_762_num_Osize_I5_J,axiom,
    ! [X2: num] :
      ( ( size_size_num @ ( bit0 @ X2 ) )
      = ( plus_plus_nat @ ( size_size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(5)
thf(fact_763_n__less__n__mult__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).

% n_less_n_mult_m
thf(fact_764_n__less__m__mult__n,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).

% n_less_m_mult_n
thf(fact_765_one__less__mult,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).

% one_less_mult
thf(fact_766_power__gt__expt,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).

% power_gt_expt
thf(fact_767_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_rat @ zero_zero_rat @ N )
        = zero_zero_rat ) ) ).

% zero_power
thf(fact_768_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_nat @ zero_zero_nat @ N )
        = zero_zero_nat ) ) ).

% zero_power
thf(fact_769_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_real @ zero_zero_real @ N )
        = zero_zero_real ) ) ).

% zero_power
thf(fact_770_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_int @ zero_zero_int @ N )
        = zero_zero_int ) ) ).

% zero_power
thf(fact_771_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_complex @ zero_zero_complex @ N )
        = zero_zero_complex ) ) ).

% zero_power
thf(fact_772_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_773_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_774_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_775_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_776_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_777_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_778_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_779_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_780_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_781_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_782_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_783_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_784_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_785_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_786_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_787_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_788_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_789_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_790_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_791_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_792_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_793_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_794_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_795_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_796_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_797_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_798_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_799_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_800_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_801_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_802_power__mod,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N ) @ B )
      = ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B ) ) ).

% power_mod
thf(fact_803_power__mod,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N ) @ B )
      = ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B ) ) ).

% power_mod
thf(fact_804_power__mod,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B ) @ N ) @ B )
      = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N ) @ B ) ) ).

% power_mod
thf(fact_805_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_806_strict__inc__induct,axiom,
    ! [I3: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I3 @ J )
     => ( ! [I2: nat] :
            ( ( J
              = ( suc @ I2 ) )
           => ( P @ I2 ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ J )
             => ( ( P @ ( suc @ I2 ) )
               => ( P @ I2 ) ) )
         => ( P @ I3 ) ) ) ) ).

% strict_inc_induct
thf(fact_807_less__Suc__induct,axiom,
    ! [I3: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I3 @ J )
     => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
       => ( ! [I2: nat,J3: nat,K2: nat] :
              ( ( ord_less_nat @ I2 @ J3 )
             => ( ( ord_less_nat @ J3 @ K2 )
               => ( ( P @ I2 @ J3 )
                 => ( ( P @ J3 @ K2 )
                   => ( P @ I2 @ K2 ) ) ) ) )
         => ( P @ I3 @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_808_less__trans__Suc,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I3 @ J )
     => ( ( ord_less_nat @ J @ K )
       => ( ord_less_nat @ ( suc @ I3 ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_809_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_less_SucD
thf(fact_810_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
       => ( M = N ) ) ) ).

% less_antisym
thf(fact_811_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M )
      = ( ? [M5: nat] :
            ( ( M
              = ( suc @ M5 ) )
            & ( ord_less_nat @ N @ M5 ) ) ) ) ).

% Suc_less_eq2
thf(fact_812_All__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N ) )
           => ( P @ I ) ) )
      = ( ( P @ N )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ N )
           => ( P @ I ) ) ) ) ).

% All_less_Suc
thf(fact_813_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_814_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( M = N ) ) ) ).

% less_Suc_eq
thf(fact_815_Ex__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N ) )
            & ( P @ I ) ) )
      = ( ( P @ N )
        | ? [I: nat] :
            ( ( ord_less_nat @ I @ N )
            & ( P @ I ) ) ) ) ).

% Ex_less_Suc
thf(fact_816_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_817_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( M = N ) ) ) ).

% less_SucE
thf(fact_818_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ( suc @ M )
         != N )
       => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).

% Suc_lessI
thf(fact_819_Suc__lessE,axiom,
    ! [I3: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I3 ) @ K )
     => ~ ! [J3: nat] :
            ( ( ord_less_nat @ I3 @ J3 )
           => ( K
             != ( suc @ J3 ) ) ) ) ).

% Suc_lessE
thf(fact_820_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_lessD
thf(fact_821_Nat_OlessE,axiom,
    ! [I3: nat,K: nat] :
      ( ( ord_less_nat @ I3 @ K )
     => ( ( K
         != ( suc @ I3 ) )
       => ~ ! [J3: nat] :
              ( ( ord_less_nat @ I3 @ J3 )
             => ( K
               != ( suc @ J3 ) ) ) ) ) ).

% Nat.lessE
thf(fact_822_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_823_add__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc
thf(fact_824_add__Suc__shift,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).

% add_Suc_shift
thf(fact_825_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ ( suc @ K ) @ M )
        = ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( M = N ) ) ).

% Suc_mult_cancel1
thf(fact_826_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_827_zero__less__iff__neq__zero,axiom,
    ! [N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
      = ( N != zero_z5237406670263579293d_enat ) ) ).

% zero_less_iff_neq_zero
thf(fact_828_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_829_gr__implies__not__zero,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ M @ N )
     => ( N != zero_z5237406670263579293d_enat ) ) ).

% gr_implies_not_zero
thf(fact_830_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_831_not__less__zero,axiom,
    ! [N: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).

% not_less_zero
thf(fact_832_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_833_gr__zeroI,axiom,
    ! [N: extended_enat] :
      ( ( N != zero_z5237406670263579293d_enat )
     => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ).

% gr_zeroI
thf(fact_834_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).

% less_numeral_extra(3)
thf(fact_835_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_836_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_837_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_838_less__numeral__extra_I3_J,axiom,
    ~ ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ zero_z5237406670263579293d_enat ) ).

% less_numeral_extra(3)
thf(fact_839_field__lbound__gt__zero,axiom,
    ! [D1: real,D22: real] :
      ( ( ord_less_real @ zero_zero_real @ D1 )
     => ( ( ord_less_real @ zero_zero_real @ D22 )
       => ? [E: real] :
            ( ( ord_less_real @ zero_zero_real @ E )
            & ( ord_less_real @ E @ D1 )
            & ( ord_less_real @ E @ D22 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_840_field__lbound__gt__zero,axiom,
    ! [D1: rat,D22: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ D1 )
     => ( ( ord_less_rat @ zero_zero_rat @ D22 )
       => ? [E: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ E )
            & ( ord_less_rat @ E @ D1 )
            & ( ord_less_rat @ E @ D22 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_841_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_z5237406670263579293d_enat
     != ( numera1916890842035813515d_enat @ N ) ) ).

% zero_neq_numeral
thf(fact_842_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_complex
     != ( numera6690914467698888265omplex @ N ) ) ).

% zero_neq_numeral
thf(fact_843_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_real
     != ( numeral_numeral_real @ N ) ) ).

% zero_neq_numeral
thf(fact_844_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N ) ) ).

% zero_neq_numeral
thf(fact_845_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_int
     != ( numeral_numeral_int @ N ) ) ).

% zero_neq_numeral
thf(fact_846_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_rat
     != ( numeral_numeral_rat @ N ) ) ).

% zero_neq_numeral
thf(fact_847_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_848_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_849_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_850_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_851_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_852_add_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.comm_neutral
thf(fact_853_add_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.comm_neutral
thf(fact_854_add_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.comm_neutral
thf(fact_855_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_856_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_857_add_Ogroup__left__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_858_add_Ogroup__left__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_859_add_Ogroup__left__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_860_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_861_verit__sum__simplify,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% verit_sum_simplify
thf(fact_862_verit__sum__simplify,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% verit_sum_simplify
thf(fact_863_verit__sum__simplify,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% verit_sum_simplify
thf(fact_864_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_865_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_866_power__not__zero,axiom,
    ! [A: rat,N: nat] :
      ( ( A != zero_zero_rat )
     => ( ( power_power_rat @ A @ N )
       != zero_zero_rat ) ) ).

% power_not_zero
thf(fact_867_power__not__zero,axiom,
    ! [A: nat,N: nat] :
      ( ( A != zero_zero_nat )
     => ( ( power_power_nat @ A @ N )
       != zero_zero_nat ) ) ).

% power_not_zero
thf(fact_868_power__not__zero,axiom,
    ! [A: real,N: nat] :
      ( ( A != zero_zero_real )
     => ( ( power_power_real @ A @ N )
       != zero_zero_real ) ) ).

% power_not_zero
thf(fact_869_power__not__zero,axiom,
    ! [A: int,N: nat] :
      ( ( A != zero_zero_int )
     => ( ( power_power_int @ A @ N )
       != zero_zero_int ) ) ).

% power_not_zero
thf(fact_870_power__not__zero,axiom,
    ! [A: complex,N: nat] :
      ( ( A != zero_zero_complex )
     => ( ( power_power_complex @ A @ N )
       != zero_zero_complex ) ) ).

% power_not_zero
thf(fact_871_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_872_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ~ ( P @ N2 )
             => ? [M2: nat] :
                  ( ( ord_less_nat @ M2 @ N2 )
                  & ~ ( P @ M2 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_873_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_874_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_875_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_876_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_877_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_878_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_879_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_880_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_881_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N ) )
      = ( ( K = zero_zero_nat )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_882_mult__0,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% mult_0
thf(fact_883_divide__neg__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_neg_neg
thf(fact_884_divide__neg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_neg_neg
thf(fact_885_divide__neg__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_neg_pos
thf(fact_886_divide__neg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_neg_pos
thf(fact_887_divide__pos__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_pos_neg
thf(fact_888_divide__pos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_pos_neg
thf(fact_889_divide__pos__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_pos_pos
thf(fact_890_divide__pos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_pos_pos
thf(fact_891_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_892_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_893_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_894_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_895_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_896_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_897_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_898_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_899_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_900_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_901_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_902_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_903_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_904_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_905_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_906_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_907_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_908_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_909_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_910_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_911_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_912_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_913_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_914_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_915_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_916_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_917_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_918_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_919_frac__eq__eq,axiom,
    ! [Y: complex,Z2: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z2 != zero_zero_complex )
       => ( ( ( divide1717551699836669952omplex @ X @ Y )
            = ( divide1717551699836669952omplex @ W @ Z2 ) )
          = ( ( times_times_complex @ X @ Z2 )
            = ( times_times_complex @ W @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_920_frac__eq__eq,axiom,
    ! [Y: real,Z2: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( ( ( divide_divide_real @ X @ Y )
            = ( divide_divide_real @ W @ Z2 ) )
          = ( ( times_times_real @ X @ Z2 )
            = ( times_times_real @ W @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_921_frac__eq__eq,axiom,
    ! [Y: rat,Z2: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( ( divide_divide_rat @ X @ Y )
            = ( divide_divide_rat @ W @ Z2 ) )
          = ( ( times_times_rat @ X @ Z2 )
            = ( times_times_rat @ W @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_922_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_923_split__mod,axiom,
    ! [P: nat > $o,M: nat,N: nat] :
      ( ( P @ ( modulo_modulo_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ M ) )
        & ( ( N != zero_zero_nat )
         => ! [I: nat,J2: nat] :
              ( ( ord_less_nat @ J2 @ N )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I ) @ J2 ) )
               => ( P @ J2 ) ) ) ) ) ) ).

% split_mod
thf(fact_924_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_925_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_926_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_927_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_928_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_929_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ( N = zero_zero_nat )
        | ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases_iff
thf(fact_930_less__2__cases,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
     => ( ( N = zero_zero_nat )
        | ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases
thf(fact_931_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_932_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_933_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > real,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_934_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > rat,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_935_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > num,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_936_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_937_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > int,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_938_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > extended_enat,N: nat,M: nat] :
      ( ! [N2: nat] : ( ord_le72135733267957522d_enat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_le72135733267957522d_enat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_939_lift__Suc__mono__less,axiom,
    ! [F: nat > real,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_940_lift__Suc__mono__less,axiom,
    ! [F: nat > rat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_rat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_941_lift__Suc__mono__less,axiom,
    ! [F: nat > num,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_942_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_943_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_944_lift__Suc__mono__less,axiom,
    ! [F: nat > extended_enat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_le72135733267957522d_enat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_le72135733267957522d_enat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_945_power__Suc,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ A @ ( suc @ N ) )
      = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).

% power_Suc
thf(fact_946_power__Suc,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ A @ ( suc @ N ) )
      = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).

% power_Suc
thf(fact_947_power__Suc,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ A @ ( suc @ N ) )
      = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).

% power_Suc
thf(fact_948_power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).

% power_Suc
thf(fact_949_power__Suc,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( suc @ N ) )
      = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).

% power_Suc
thf(fact_950_power__Suc2,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ A @ ( suc @ N ) )
      = ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_951_power__Suc2,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ A @ ( suc @ N ) )
      = ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_952_power__Suc2,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ A @ ( suc @ N ) )
      = ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_953_power__Suc2,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_954_power__Suc2,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( suc @ N ) )
      = ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_955_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_956_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_957_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_958_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_959_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_960_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_961_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_962_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_963_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_964_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_965_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_966_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_967_mult__imp__div__pos__less,axiom,
    ! [Y: real,X: real,Z2: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ X @ ( times_times_real @ Z2 @ Y ) )
       => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z2 ) ) ) ).

% mult_imp_div_pos_less
thf(fact_968_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X: rat,Z2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ X @ ( times_times_rat @ Z2 @ Y ) )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z2 ) ) ) ).

% mult_imp_div_pos_less
thf(fact_969_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z2: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ ( times_times_real @ Z2 @ Y ) @ X )
       => ( ord_less_real @ Z2 @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_970_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z2: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ ( times_times_rat @ Z2 @ Y ) @ X )
       => ( ord_less_rat @ Z2 @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_971_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_972_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_973_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_974_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_975_divide__add__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z2 ) @ Y )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% divide_add_eq_iff
thf(fact_976_divide__add__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X @ Z2 ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% divide_add_eq_iff
thf(fact_977_divide__add__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z2 ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% divide_add_eq_iff
thf(fact_978_add__divide__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z2 ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).

% add_divide_eq_iff
thf(fact_979_add__divide__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z2 ) )
        = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).

% add_divide_eq_iff
thf(fact_980_add__divide__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z2 ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).

% add_divide_eq_iff
thf(fact_981_add__num__frac,axiom,
    ! [Y: complex,Z2: complex,X: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ X @ Y ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z2 @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_982_add__num__frac,axiom,
    ! [Y: real,Z2: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ Z2 @ ( divide_divide_real @ X @ Y ) )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z2 @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_983_add__num__frac,axiom,
    ! [Y: rat,Z2: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ X @ Y ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z2 @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_984_add__frac__num,axiom,
    ! [Y: complex,X: complex,Z2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ Z2 )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z2 @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_985_add__frac__num,axiom,
    ! [Y: real,X: real,Z2: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z2 )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z2 @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_986_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z2 )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z2 @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_987_add__frac__eq,axiom,
    ! [Y: complex,Z2: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z2 != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z2 ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z2 ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z2 ) ) ) ) ) ).

% add_frac_eq
thf(fact_988_add__frac__eq,axiom,
    ! [Y: real,Z2: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z2 ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) ) ) ) ).

% add_frac_eq
thf(fact_989_add__frac__eq,axiom,
    ! [Y: rat,Z2: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z2 ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) ) ) ) ).

% add_frac_eq
thf(fact_990_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
          = A ) )
      & ( ( Z2 != zero_zero_complex )
       => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_991_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
          = A ) )
      & ( ( Z2 != zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_992_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
          = A ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_993_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_994_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
          = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_995_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_996_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ? [K2: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_997_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M4: nat,N3: nat] :
        ? [K3: nat] :
          ( N3
          = ( suc @ ( plus_plus_nat @ M4 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_998_less__add__Suc2,axiom,
    ! [I3: nat,M: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ M @ I3 ) ) ) ).

% less_add_Suc2
thf(fact_999_less__add__Suc1,axiom,
    ! [I3: nat,M: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ I3 @ M ) ) ) ).

% less_add_Suc1
thf(fact_1000_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ~ ! [Q3: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).

% less_natE
thf(fact_1001_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_mult_less_cancel1
thf(fact_1002_mult__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).

% mult_Suc
thf(fact_1003_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).

% not_numeral_less_zero
thf(fact_1004_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).

% not_numeral_less_zero
thf(fact_1005_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_1006_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_1007_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).

% not_numeral_less_zero
thf(fact_1008_zero__less__numeral,axiom,
    ! [N: num] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).

% zero_less_numeral
thf(fact_1009_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).

% zero_less_numeral
thf(fact_1010_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_less_numeral
thf(fact_1011_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).

% zero_less_numeral
thf(fact_1012_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).

% zero_less_numeral
thf(fact_1013_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_1014_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_1015_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_1016_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_1017_pos__add__strict,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ A )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ord_le72135733267957522d_enat @ B @ ( plus_p3455044024723400733d_enat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_1018_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C2: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C2 ) )
           => ( C2 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_1019_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ~ ! [C2: extended_enat] :
            ( ( B
              = ( plus_p3455044024723400733d_enat @ A @ C2 ) )
           => ( C2 = zero_z5237406670263579293d_enat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_1020_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_1021_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_1022_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_1023_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_1024_add__pos__pos,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ A )
     => ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ B )
       => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_1025_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_1026_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_1027_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_1028_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_1029_add__neg__neg,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ zero_z5237406670263579293d_enat )
     => ( ( ord_le72135733267957522d_enat @ B @ zero_z5237406670263579293d_enat )
       => ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ zero_z5237406670263579293d_enat ) ) ) ).

% add_neg_neg
thf(fact_1030_zero__less__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).

% zero_less_power
thf(fact_1031_zero__less__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).

% zero_less_power
thf(fact_1032_zero__less__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).

% zero_less_power
thf(fact_1033_zero__less__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).

% zero_less_power
thf(fact_1034_less__imp__add__positive,axiom,
    ! [I3: nat,J: nat] :
      ( ( ord_less_nat @ I3 @ J )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I3 @ K2 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_1035_mult__less__mono1,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I3 @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).

% mult_less_mono1
thf(fact_1036_mult__less__mono2,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I3 @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J ) ) ) ) ).

% mult_less_mono2
thf(fact_1037_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ( times_times_nat @ K @ M )
          = ( times_times_nat @ K @ N ) )
        = ( M = N ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_1038_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( ord_less_nat @ M @ N ) ) ) ).

% nat_mult_less_cancel1
thf(fact_1039_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( divide_divide_nat @ M @ N )
        = zero_zero_nat )
      = ( ( ord_less_nat @ M @ N )
        | ( N = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
thf(fact_1040_nat__power__less__imp__less,axiom,
    ! [I3: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ I3 )
     => ( ( ord_less_nat @ ( power_power_nat @ I3 @ M ) @ ( power_power_nat @ I3 @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% nat_power_less_imp_less
thf(fact_1041_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_1042_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_1043_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_1044_nat__bit__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P @ N2 )
           => ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
       => ( ! [N2: nat] :
              ( ( P @ N2 )
             => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_bit_induct
thf(fact_1045_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% Suc_n_div_2_gt_zero
thf(fact_1046_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% div_2_gt_zero
thf(fact_1047_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_1048_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_1049_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_1050_odd__power__less__zero,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).

% odd_power_less_zero
thf(fact_1051_odd__power__less__zero,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).

% odd_power_less_zero
thf(fact_1052_odd__power__less__zero,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).

% odd_power_less_zero
thf(fact_1053_div__mod__decomp,axiom,
    ! [A2: nat,N: nat] :
      ( A2
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N ) @ N ) @ ( modulo_modulo_nat @ A2 @ N ) ) ) ).

% div_mod_decomp
thf(fact_1054_mod__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).

% mod_mult2_eq
thf(fact_1055_linordered__field__no__ub,axiom,
    ! [X3: real] :
    ? [X_12: real] : ( ord_less_real @ X3 @ X_12 ) ).

% linordered_field_no_ub
thf(fact_1056_linordered__field__no__ub,axiom,
    ! [X3: rat] :
    ? [X_12: rat] : ( ord_less_rat @ X3 @ X_12 ) ).

% linordered_field_no_ub
thf(fact_1057_linordered__field__no__lb,axiom,
    ! [X3: real] :
    ? [Y4: real] : ( ord_less_real @ Y4 @ X3 ) ).

% linordered_field_no_lb
thf(fact_1058_linordered__field__no__lb,axiom,
    ! [X3: rat] :
    ? [Y4: rat] : ( ord_less_rat @ Y4 @ X3 ) ).

% linordered_field_no_lb
thf(fact_1059_sum__squares__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
      = ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_1060_sum__squares__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
      = ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_1061_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
      = ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_1062_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_1063_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_1064_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_1065_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_1066_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_1067_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_1068_not__psubset__empty,axiom,
    ! [A2: set_real] :
      ~ ( ord_less_set_real @ A2 @ bot_bot_set_real ) ).

% not_psubset_empty
thf(fact_1069_not__psubset__empty,axiom,
    ! [A2: set_o] :
      ~ ( ord_less_set_o @ A2 @ bot_bot_set_o ) ).

% not_psubset_empty
thf(fact_1070_not__psubset__empty,axiom,
    ! [A2: set_nat] :
      ~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).

% not_psubset_empty
thf(fact_1071_not__psubset__empty,axiom,
    ! [A2: set_int] :
      ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).

% not_psubset_empty
thf(fact_1072_emptyE,axiom,
    ! [A: complex] :
      ~ ( member_complex @ A @ bot_bot_set_complex ) ).

% emptyE
thf(fact_1073_emptyE,axiom,
    ! [A: set_nat] :
      ~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).

% emptyE
thf(fact_1074_emptyE,axiom,
    ! [A: real] :
      ~ ( member_real @ A @ bot_bot_set_real ) ).

% emptyE
thf(fact_1075_emptyE,axiom,
    ! [A: $o] :
      ~ ( member_o @ A @ bot_bot_set_o ) ).

% emptyE
thf(fact_1076_emptyE,axiom,
    ! [A: nat] :
      ~ ( member_nat @ A @ bot_bot_set_nat ) ).

% emptyE
thf(fact_1077_emptyE,axiom,
    ! [A: int] :
      ~ ( member_int @ A @ bot_bot_set_int ) ).

% emptyE
thf(fact_1078_equals0D,axiom,
    ! [A2: set_complex,A: complex] :
      ( ( A2 = bot_bot_set_complex )
     => ~ ( member_complex @ A @ A2 ) ) ).

% equals0D
thf(fact_1079_equals0D,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( A2 = bot_bot_set_set_nat )
     => ~ ( member_set_nat @ A @ A2 ) ) ).

% equals0D
thf(fact_1080_equals0D,axiom,
    ! [A2: set_real,A: real] :
      ( ( A2 = bot_bot_set_real )
     => ~ ( member_real @ A @ A2 ) ) ).

% equals0D
thf(fact_1081_equals0D,axiom,
    ! [A2: set_o,A: $o] :
      ( ( A2 = bot_bot_set_o )
     => ~ ( member_o @ A @ A2 ) ) ).

% equals0D
thf(fact_1082_equals0D,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( A2 = bot_bot_set_nat )
     => ~ ( member_nat @ A @ A2 ) ) ).

% equals0D
thf(fact_1083_equals0D,axiom,
    ! [A2: set_int,A: int] :
      ( ( A2 = bot_bot_set_int )
     => ~ ( member_int @ A @ A2 ) ) ).

% equals0D
thf(fact_1084_equals0I,axiom,
    ! [A2: set_complex] :
      ( ! [Y4: complex] :
          ~ ( member_complex @ Y4 @ A2 )
     => ( A2 = bot_bot_set_complex ) ) ).

% equals0I
thf(fact_1085_equals0I,axiom,
    ! [A2: set_set_nat] :
      ( ! [Y4: set_nat] :
          ~ ( member_set_nat @ Y4 @ A2 )
     => ( A2 = bot_bot_set_set_nat ) ) ).

% equals0I
thf(fact_1086_equals0I,axiom,
    ! [A2: set_real] :
      ( ! [Y4: real] :
          ~ ( member_real @ Y4 @ A2 )
     => ( A2 = bot_bot_set_real ) ) ).

% equals0I
thf(fact_1087_equals0I,axiom,
    ! [A2: set_o] :
      ( ! [Y4: $o] :
          ~ ( member_o @ Y4 @ A2 )
     => ( A2 = bot_bot_set_o ) ) ).

% equals0I
thf(fact_1088_equals0I,axiom,
    ! [A2: set_nat] :
      ( ! [Y4: nat] :
          ~ ( member_nat @ Y4 @ A2 )
     => ( A2 = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_1089_equals0I,axiom,
    ! [A2: set_int] :
      ( ! [Y4: int] :
          ~ ( member_int @ Y4 @ A2 )
     => ( A2 = bot_bot_set_int ) ) ).

% equals0I
thf(fact_1090_ex__in__conv,axiom,
    ! [A2: set_complex] :
      ( ( ? [X6: complex] : ( member_complex @ X6 @ A2 ) )
      = ( A2 != bot_bot_set_complex ) ) ).

% ex_in_conv
thf(fact_1091_ex__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ( ? [X6: set_nat] : ( member_set_nat @ X6 @ A2 ) )
      = ( A2 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
thf(fact_1092_ex__in__conv,axiom,
    ! [A2: set_real] :
      ( ( ? [X6: real] : ( member_real @ X6 @ A2 ) )
      = ( A2 != bot_bot_set_real ) ) ).

% ex_in_conv
thf(fact_1093_ex__in__conv,axiom,
    ! [A2: set_o] :
      ( ( ? [X6: $o] : ( member_o @ X6 @ A2 ) )
      = ( A2 != bot_bot_set_o ) ) ).

% ex_in_conv
thf(fact_1094_ex__in__conv,axiom,
    ! [A2: set_nat] :
      ( ( ? [X6: nat] : ( member_nat @ X6 @ A2 ) )
      = ( A2 != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_1095_ex__in__conv,axiom,
    ! [A2: set_int] :
      ( ( ? [X6: int] : ( member_int @ X6 @ A2 ) )
      = ( A2 != bot_bot_set_int ) ) ).

% ex_in_conv
thf(fact_1096_length__pos__if__in__set,axiom,
    ! [X: complex,Xs2: list_complex] :
      ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_1097_length__pos__if__in__set,axiom,
    ! [X: real,Xs2: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_1098_length__pos__if__in__set,axiom,
    ! [X: set_nat,Xs2: list_set_nat] :
      ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_1099_length__pos__if__in__set,axiom,
    ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_1100_length__pos__if__in__set,axiom,
    ! [X: $o,Xs2: list_o] :
      ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_1101_length__pos__if__in__set,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_1102_length__pos__if__in__set,axiom,
    ! [X: int,Xs2: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_1103_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( divide_divide_nat @ M @ N ) ) ) ).

% nat_mult_div_cancel1
thf(fact_1104_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q2 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N )
        = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q2 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_1105_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_1106_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_1107_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_1108_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_1109_Suc__nat__number__of__add,axiom,
    ! [V: num,N: nat] :
      ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).

% Suc_nat_number_of_add
thf(fact_1110_zero__power2,axiom,
    ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_rat ) ).

% zero_power2
thf(fact_1111_zero__power2,axiom,
    ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% zero_power2
thf(fact_1112_zero__power2,axiom,
    ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_real ) ).

% zero_power2
thf(fact_1113_zero__power2,axiom,
    ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% zero_power2
thf(fact_1114_zero__power2,axiom,
    ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_complex ) ).

% zero_power2
thf(fact_1115_dividend__less__times__div,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).

% dividend_less_times_div
thf(fact_1116_dividend__less__div__times,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).

% dividend_less_div_times
thf(fact_1117_split__div,axiom,
    ! [P: nat > $o,M: nat,N: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ zero_zero_nat ) )
        & ( ( N != zero_zero_nat )
         => ! [I: nat,J2: nat] :
              ( ( ord_less_nat @ J2 @ N )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I ) @ J2 ) )
               => ( P @ I ) ) ) ) ) ) ).

% split_div
thf(fact_1118_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_1119_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_1120_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_1121_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_1122_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_1123_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_1124_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_1125_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
thf(fact_1126_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
thf(fact_1127_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
thf(fact_1128_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
thf(fact_1129_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( 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 @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_1130_div__exp__mod__exp__eq,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( 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 @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_1131_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,N: nat,M: nat] :
      ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
      = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_1132_power__odd__eq,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_1133_power__odd__eq,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_1134_power__odd__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_1135_power__odd__eq,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_1136_power__odd__eq,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_1137_sum__power2__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_1138_sum__power2__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_1139_sum__power2__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_1140_not__sum__power2__lt__zero,axiom,
    ! [X: real,Y: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).

% not_sum_power2_lt_zero
thf(fact_1141_not__sum__power2__lt__zero,axiom,
    ! [X: rat,Y: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).

% not_sum_power2_lt_zero
thf(fact_1142_not__sum__power2__lt__zero,axiom,
    ! [X: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).

% not_sum_power2_lt_zero
thf(fact_1143_times__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z2: complex,W: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z2 @ W ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z2 ) @ ( times_times_complex @ Y @ W ) ) ) ).

% times_divide_times_eq
thf(fact_1144_times__divide__times__eq,axiom,
    ! [X: real,Y: real,Z2: real,W: real] :
      ( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z2 @ W ) )
      = ( divide_divide_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ W ) ) ) ).

% times_divide_times_eq
thf(fact_1145_times__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z2: rat,W: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z2 @ W ) )
      = ( divide_divide_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ Y @ W ) ) ) ).

% times_divide_times_eq
thf(fact_1146_divide__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z2: complex,W: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z2 @ W ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W ) @ ( times_times_complex @ Y @ Z2 ) ) ) ).

% divide_divide_times_eq
thf(fact_1147_divide__divide__times__eq,axiom,
    ! [X: real,Y: real,Z2: real,W: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z2 @ W ) )
      = ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y @ Z2 ) ) ) ).

% divide_divide_times_eq
thf(fact_1148_divide__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z2: rat,W: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z2 @ W ) )
      = ( divide_divide_rat @ ( times_times_rat @ X @ W ) @ ( times_times_rat @ Y @ Z2 ) ) ) ).

% divide_divide_times_eq
thf(fact_1149_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_1150_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_1151_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
      = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).

% divide_divide_eq_left'
thf(fact_1152_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_1153_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_1154_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_1155_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
thf(fact_1156_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
thf(fact_1157_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != ( suc @ zero_zero_nat ) )
      = ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
thf(fact_1158_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_1159_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_1160_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_1161_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_1162_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_1163_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_1164_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_1165_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_1166_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_1167_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_1168_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_1169_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_1170_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_1171_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_1172_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_1173_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_1174_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ Tree @ N )
     => ( ( vEBT_vebt_member @ Tree @ X )
       => ( ( vEBT_V5719532721284313246member @ Tree @ X )
          | ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).

% member_valid_both_member_options
thf(fact_1175_mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mod_0
thf(fact_1176_mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mod_0
thf(fact_1177_mod__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% mod_0
thf(fact_1178_mod__by__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ zero_zero_nat )
      = A ) ).

% mod_by_0
thf(fact_1179_mod__by__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ zero_zero_int )
      = A ) ).

% mod_by_0
thf(fact_1180_mod__by__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% mod_by_0
thf(fact_1181_mod__self,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ A )
      = zero_zero_nat ) ).

% mod_self
thf(fact_1182_mod__self,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ A )
      = zero_zero_int ) ).

% mod_self
thf(fact_1183_mod__self,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ A )
      = zero_z3403309356797280102nteger ) ).

% mod_self
thf(fact_1184_double__not__eq__Suc__double,axiom,
    ! [M: nat,N: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
     != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% double_not_eq_Suc_double
thf(fact_1185_Suc__double__not__eq__double,axiom,
    ! [M: nat,N: nat] :
      ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
     != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% Suc_double_not_eq_double
thf(fact_1186_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X ) ).

% buildup_nothing_in_min_max
thf(fact_1187_both__member__options__def,axiom,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [T2: vEBT_VEBT,X6: nat] :
          ( ( vEBT_V5719532721284313246member @ T2 @ X6 )
          | ( vEBT_VEBT_membermima @ T2 @ X6 ) ) ) ) ).

% both_member_options_def
thf(fact_1188_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_1189_i0__less,axiom,
    ! [N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
      = ( N != zero_z5237406670263579293d_enat ) ) ).

% i0_less
thf(fact_1190_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_1191_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_1192_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_1193_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_1194_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_1195_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_1196_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_1197_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_1198_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_1199_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_1200_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_1201_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_1202_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_1203_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_1204_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_1205_mult__zero__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% mult_zero_right
thf(fact_1206_mult__zero__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% mult_zero_right
thf(fact_1207_mult__zero__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_1208_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_1209_mult__zero__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_1210_mult__zero__left,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% mult_zero_left
thf(fact_1211_mult__zero__left,axiom,
    ! [A: real] :
      ( ( times_times_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% mult_zero_left
thf(fact_1212_mult__zero__left,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_1213_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_1214_mult__zero__left,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_1215_div__by__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% div_by_0
thf(fact_1216_div__by__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% div_by_0
thf(fact_1217_div__by__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% div_by_0
thf(fact_1218_div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% div_by_0
thf(fact_1219_div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% div_by_0
thf(fact_1220_div__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% div_0
thf(fact_1221_div__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% div_0
thf(fact_1222_div__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% div_0
thf(fact_1223_div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% div_0
thf(fact_1224_div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% div_0
thf(fact_1225_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_1226_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_1227_psubsetD,axiom,
    ! [A2: set_complex,B2: set_complex,C: complex] :
      ( ( ord_less_set_complex @ A2 @ B2 )
     => ( ( member_complex @ C @ A2 )
       => ( member_complex @ C @ B2 ) ) ) ).

% psubsetD
thf(fact_1228_psubsetD,axiom,
    ! [A2: set_real,B2: set_real,C: real] :
      ( ( ord_less_set_real @ A2 @ B2 )
     => ( ( member_real @ C @ A2 )
       => ( member_real @ C @ B2 ) ) ) ).

% psubsetD
thf(fact_1229_psubsetD,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
      ( ( ord_less_set_set_nat @ A2 @ B2 )
     => ( ( member_set_nat @ C @ A2 )
       => ( member_set_nat @ C @ B2 ) ) ) ).

% psubsetD
thf(fact_1230_psubsetD,axiom,
    ! [A2: set_nat,B2: set_nat,C: nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ( ( member_nat @ C @ A2 )
       => ( member_nat @ C @ B2 ) ) ) ).

% psubsetD
thf(fact_1231_psubsetD,axiom,
    ! [A2: set_int,B2: set_int,C: int] :
      ( ( ord_less_set_int @ A2 @ B2 )
     => ( ( member_int @ C @ A2 )
       => ( member_int @ C @ B2 ) ) ) ).

% psubsetD
thf(fact_1232_not__iless0,axiom,
    ! [N: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).

% not_iless0
thf(fact_1233_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
      = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
        & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).

% enat_0_less_mult_iff
thf(fact_1234_linorder__neqE__linordered__idom,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ ( ord_less_real @ X @ Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_1235_linorder__neqE__linordered__idom,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ ( ord_less_rat @ X @ Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_1236_linorder__neqE__linordered__idom,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_1237_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_1238_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_1239_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_1240_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_1241_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_1242_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_1243_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_1244_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_1245_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_1246_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_1247_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_1248_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_1249_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_1250_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_1251_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_1252_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_1253_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_1254_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_1255_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_1256_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_1257_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_1258_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_1259_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_1260_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_1261_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_1262_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_1263_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_1264_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_1265_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_1266_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_1267_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_1268_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_1269_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_1270_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_1271_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_1272_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_1273_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_1274_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_1275_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_1276_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_1277_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_1278_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_1279_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_1280_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_1281_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_1282_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_1283_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_1284_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_1285_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_1286_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_1287_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_1288_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_1289_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_1290_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_1291_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_1292_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_1293_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_1294_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_1295_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_1296_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_1297_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_1298_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_1299_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_1300_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_1301_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_1302_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_1303_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_1304_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_1305_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_1306_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_1307_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_1308_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_1309_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_1310_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_1311_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_1312_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_1313_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_1314_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_1315_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_1316_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_1317_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_1318_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_1319_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_1320_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_1321_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_1322_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_1323_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_1324_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_1325_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_1326_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_1327_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_1328_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_1329_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_1330_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_1331_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_1332_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_1333_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_1334_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_1335_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_1336_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_1337_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_1338_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_1339_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_1340_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_1341_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_1342_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_1343_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_1344_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_1345_not__square__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).

% not_square_less_zero
thf(fact_1346_not__square__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_1347_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_1348_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_1349_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_1350_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_1351_add__less__zeroD,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
     => ( ( ord_less_real @ X @ zero_zero_real )
        | ( ord_less_real @ Y @ zero_zero_real ) ) ) ).

% add_less_zeroD
thf(fact_1352_add__less__zeroD,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ X @ Y ) @ zero_zero_rat )
     => ( ( ord_less_rat @ X @ zero_zero_rat )
        | ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).

% add_less_zeroD
thf(fact_1353_add__less__zeroD,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
     => ( ( ord_less_int @ X @ zero_zero_int )
        | ( ord_less_int @ Y @ zero_zero_int ) ) ) ).

% add_less_zeroD
thf(fact_1354_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_1355_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_1356_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_1357_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_1358_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_1359_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_1360_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
      = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_1361_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
      = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_1362_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
      = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_1363_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( ( modulo_modulo_nat @ M @ D )
        = zero_zero_nat )
     => ? [Q3: nat] :
          ( M
          = ( times_times_nat @ D @ Q3 ) ) ) ).

% mod_eq_0D
thf(fact_1364_nat__mod__eq__iff,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N )
        = ( modulo_modulo_nat @ Y @ N ) )
      = ( ? [Q1: nat,Q22: nat] :
            ( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).

% nat_mod_eq_iff
thf(fact_1365_not__sum__squares__lt__zero,axiom,
    ! [X: real,Y: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).

% not_sum_squares_lt_zero
thf(fact_1366_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).

% not_sum_squares_lt_zero
thf(fact_1367_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).

% not_sum_squares_lt_zero
thf(fact_1368_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = zero_zero_nat )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
thf(fact_1369_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = zero_zero_int )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
thf(fact_1370_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
thf(fact_1371_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
      = zero_zero_nat ) ).

% cong_exp_iff_simps(1)
thf(fact_1372_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
      = zero_zero_int ) ).

% cong_exp_iff_simps(1)
thf(fact_1373_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
      = zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(1)
thf(fact_1374_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_1375_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_1376_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_1377_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_1378_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_1379_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_1380_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_1381_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_1382_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_1383_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_1384_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_1385_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_1386_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_1387_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_1388_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_1389_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_1390_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_1391_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_1392_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_1393_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_1394_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_1395_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_1396_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_1397_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_1398_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_1399_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_1400_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_1401_pos2,axiom,
    ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).

% pos2
thf(fact_1402_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_1403_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_1404_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_1405_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_1406_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_1407_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X )
      = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) ) ).

% vebt_insert.simps(3)
thf(fact_1408_set__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_1409_set__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_1410_set__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_1411_flip__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_1412_flip__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_1413_flip__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_1414_unset__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_1415_unset__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_1416_unset__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_1417_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) @ X )
      = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) ) ).

% vebt_insert.simps(2)
thf(fact_1418_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [M3: nat] : ( P @ M3 @ zero_zero_nat )
     => ( ! [M3: nat,N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ( P @ N2 @ ( modulo_modulo_nat @ M3 @ N2 ) )
             => ( P @ M3 @ N2 ) ) )
       => ( P @ M @ N ) ) ) ).

% gcd_nat_induct
thf(fact_1419_Leaf__0__not,axiom,
    ! [A: $o,B: $o] :
      ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).

% Leaf_0_not
thf(fact_1420_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
      ( ( ( vEBT_Leaf @ X21 @ X22 )
        = ( vEBT_Leaf @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% VEBT.inject(2)
thf(fact_1421_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% unset_bit_negative_int_iff
thf(fact_1422_set__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% set_bit_negative_int_iff
thf(fact_1423_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% flip_bit_negative_int_iff
thf(fact_1424_zdiv__mono__strict,axiom,
    ! [A2: int,B2: int,N: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ( ( modulo_modulo_int @ A2 @ N )
            = zero_zero_int )
         => ( ( ( modulo_modulo_int @ B2 @ N )
              = zero_zero_int )
           => ( ord_less_int @ ( divide_divide_int @ A2 @ N ) @ ( divide_divide_int @ B2 @ N ) ) ) ) ) ) ).

% zdiv_mono_strict
thf(fact_1425_div__mod__decomp__int,axiom,
    ! [A2: int,N: int] :
      ( A2
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ N ) @ N ) @ ( modulo_modulo_int @ A2 @ N ) ) ) ).

% div_mod_decomp_int
thf(fact_1426_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_1427_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_1428_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_1429_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( ( modulo_modulo_int @ M @ D )
        = zero_zero_int )
      = ( ? [Q4: int] :
            ( M
            = ( times_times_int @ D @ Q4 ) ) ) ) ).

% zmod_eq_0_iff
thf(fact_1430_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( ( modulo_modulo_int @ M @ D )
        = zero_zero_int )
     => ? [Q3: int] :
          ( M
          = ( times_times_int @ D @ Q3 ) ) ) ).

% zmod_eq_0D
thf(fact_1431_times__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( times_times_int @ K @ zero_zero_int )
      = zero_zero_int ) ).

% times_int_code(1)
thf(fact_1432_times__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( times_times_int @ zero_zero_int @ L )
      = zero_zero_int ) ).

% times_int_code(2)
thf(fact_1433_imult__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( ( times_7803423173614009249d_enat @ M @ N )
        = zero_z5237406670263579293d_enat )
      = ( ( M = zero_z5237406670263579293d_enat )
        | ( N = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
thf(fact_1434_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_1435_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_1436_zmult__zless__mono2,axiom,
    ! [I3: int,J: int,K: int] :
      ( ( ord_less_int @ I3 @ J )
     => ( ( ord_less_int @ zero_zero_int @ K )
       => ( ord_less_int @ ( times_times_int @ K @ I3 ) @ ( times_times_int @ K @ J ) ) ) ) ).

% zmult_zless_mono2
thf(fact_1437_less__int__code_I1_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_int_code(1)
thf(fact_1438_plus__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( plus_plus_int @ zero_zero_int @ L )
      = L ) ).

% plus_int_code(2)
thf(fact_1439_plus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( plus_plus_int @ K @ zero_zero_int )
      = K ) ).

% plus_int_code(1)
thf(fact_1440_iadd__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( ( plus_p3455044024723400733d_enat @ M @ N )
        = zero_z5237406670263579293d_enat )
      = ( ( M = zero_z5237406670263579293d_enat )
        & ( N = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
thf(fact_1441_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
      = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).

% int_distrib(2)
thf(fact_1442_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] :
      ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
      = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).

% int_distrib(1)
thf(fact_1443_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X22: $o] :
      ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X22 ) )
      = zero_zero_nat ) ).

% VEBT.size(4)
thf(fact_1444_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X22: $o] :
      ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
     != ( vEBT_Leaf @ X21 @ X22 ) ) ).

% VEBT.distinct(1)
thf(fact_1445_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
          ( Y
         != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
     => ~ ! [X212: $o,X222: $o] :
            ( Y
           != ( vEBT_Leaf @ X212 @ X222 ) ) ) ).

% VEBT.exhaust
thf(fact_1446_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).

% VEBT_internal.minNull.simps(3)
thf(fact_1447_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).

% VEBT_internal.minNull.simps(2)
thf(fact_1448_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).

% VEBT_internal.minNull.simps(1)
thf(fact_1449_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_1450_vebt__buildup_Osimps_I1_J,axiom,
    ( ( vEBT_vebt_buildup @ zero_zero_nat )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(1)
thf(fact_1451_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_1452_vebt__buildup_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(2)
thf(fact_1453_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X )
     => ( ( X
         != ( vEBT_Leaf @ $false @ $false ) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).

% VEBT_internal.minNull.elims(2)
thf(fact_1454_realpow__pos__nth2,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ? [R: real] :
          ( ( ord_less_real @ zero_zero_real @ R )
          & ( ( power_power_real @ R @ ( suc @ N ) )
            = A ) ) ) ).

% realpow_pos_nth2
thf(fact_1455_realpow__pos__nth,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [R: real] :
            ( ( ord_less_real @ zero_zero_real @ R )
            & ( ( power_power_real @ R @ N )
              = A ) ) ) ) ).

% realpow_pos_nth
thf(fact_1456_realpow__pos__nth__unique,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [X5: real] :
            ( ( ord_less_real @ zero_zero_real @ X5 )
            & ( ( power_power_real @ X5 @ N )
              = A )
            & ! [Y5: real] :
                ( ( ( ord_less_real @ zero_zero_real @ Y5 )
                  & ( ( power_power_real @ Y5 @ N )
                    = A ) )
               => ( Y5 = X5 ) ) ) ) ) ).

% realpow_pos_nth_unique
thf(fact_1457_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_1458_four__x__squared,axiom,
    ! [X: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% four_x_squared
thf(fact_1459_VEBT__internal_Oinsert_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_VEBT_insert @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X ) ) ).

% VEBT_internal.insert'.simps(1)
thf(fact_1460_option_Osize_I3_J,axiom,
    ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_1461_option_Osize_I3_J,axiom,
    ( ( size_size_option_nat @ none_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_1462_option_Osize_I3_J,axiom,
    ( ( size_size_option_num @ none_num )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_1463_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_1464_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_1465_signed__take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_1466_signed__take__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_1467_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
thf(fact_1468_add__scale__eq__noteq,axiom,
    ! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
      ( ( R2 != zero_zero_complex )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_complex @ A @ ( times_times_complex @ R2 @ C ) )
         != ( plus_plus_complex @ B @ ( times_times_complex @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_1469_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
         != ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_1470_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
         != ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_1471_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
         != ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_1472_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B: int,C: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
         != ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_1473_mult__less__iff1,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z2 )
     => ( ( ord_less_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
        = ( ord_less_real @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_1474_mult__less__iff1,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z2 )
     => ( ( ord_less_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ Y @ Z2 ) )
        = ( ord_less_rat @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_1475_mult__less__iff1,axiom,
    ! [Z2: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
        = ( ord_less_int @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_1476_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
thf(fact_1477_num_Osize__gen_I2_J,axiom,
    ! [X2: num] :
      ( ( size_num @ ( bit0 @ X2 ) )
      = ( plus_plus_nat @ ( size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(2)
thf(fact_1478_concat__bit__Suc,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
      = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).

% concat_bit_Suc
thf(fact_1479_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( N = one_one_nat )
       => ? [A5: $o,B5: $o] :
            ( T
            = ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ).

% deg_1_Leafy
thf(fact_1480_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
     => ? [A5: $o,B5: $o] :
          ( T
          = ( vEBT_Leaf @ A5 @ B5 ) ) ) ).

% deg_1_Leaf
thf(fact_1481_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
      = ( ? [A3: $o,B3: $o] :
            ( T
            = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ).

% deg1Leaf
thf(fact_1482_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( ( vEBT_vebt_mint @ T )
        = none_nat )
     => ( vEBT_VEBT_minNull @ T ) ) ).

% minminNull
thf(fact_1483_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ T )
     => ( ( vEBT_vebt_mint @ T )
        = none_nat ) ) ).

% minNullmin
thf(fact_1484_power__one__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_1485_power__one__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_1486_power__one__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_1487_power__one__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_1488_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M @ N ) )
      = ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_1489_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = one_one_nat )
      = ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_1490_real__divide__square__eq,axiom,
    ! [R2: real,A: real] :
      ( ( divide_divide_real @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ R2 ) )
      = ( divide_divide_real @ A @ R2 ) ) ).

% real_divide_square_eq
thf(fact_1491_mult_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.right_neutral
thf(fact_1492_mult_Oright__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.right_neutral
thf(fact_1493_mult_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.right_neutral
thf(fact_1494_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_1495_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_1496_mult__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ one_one_complex @ A )
      = A ) ).

% mult_1
thf(fact_1497_mult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% mult_1
thf(fact_1498_mult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% mult_1
thf(fact_1499_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_1500_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_1501_div__by__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ one_one_complex )
      = A ) ).

% div_by_1
thf(fact_1502_div__by__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ one_one_real )
      = A ) ).

% div_by_1
thf(fact_1503_div__by__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ one_one_rat )
      = A ) ).

% div_by_1
thf(fact_1504_div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% div_by_1
thf(fact_1505_div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% div_by_1
thf(fact_1506_bits__div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% bits_div_by_1
thf(fact_1507_bits__div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% bits_div_by_1
thf(fact_1508_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ one_one_rat @ N )
      = one_one_rat ) ).

% power_one
thf(fact_1509_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ one_one_nat @ N )
      = one_one_nat ) ).

% power_one
thf(fact_1510_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_real @ one_one_real @ N )
      = one_one_real ) ).

% power_one
thf(fact_1511_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_int @ one_one_int @ N )
      = one_one_int ) ).

% power_one
thf(fact_1512_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ one_one_complex @ N )
      = one_one_complex ) ).

% power_one
thf(fact_1513_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_1514_signed__take__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
      = zero_zero_int ) ).

% signed_take_bit_of_0
thf(fact_1515_concat__bit__0,axiom,
    ! [K: int,L: int] :
      ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
      = L ) ).

% concat_bit_0
thf(fact_1516_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_1517_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_1518_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_1519_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_1520_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_1521_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_1522_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_1523_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_1524_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_1525_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_1526_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_1527_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_1528_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_1529_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_1530_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_1531_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_1532_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% div_self
thf(fact_1533_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% div_self
thf(fact_1534_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% div_self
thf(fact_1535_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ A @ A )
        = one_one_nat ) ) ).

% div_self
thf(fact_1536_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ A @ A )
        = one_one_int ) ) ).

% div_self
thf(fact_1537_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_1538_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_1539_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_1540_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_1541_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_1542_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_1543_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_1544_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_1545_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_1546_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_1547_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_1548_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% divide_self
thf(fact_1549_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% divide_self
thf(fact_1550_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% divide_self
thf(fact_1551_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_1552_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_1553_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_1554_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_1555_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_1556_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_1557_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numera1916890842035813515d_enat @ N )
        = one_on7984719198319812577d_enat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_1558_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numera6690914467698888265omplex @ N )
        = one_one_complex )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_1559_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_real @ N )
        = one_one_real )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_1560_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_nat @ N )
        = one_one_nat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_1561_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_int @ N )
        = one_one_int )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_1562_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_rat @ N )
        = one_one_rat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_1563_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_on7984719198319812577d_enat
        = ( numera1916890842035813515d_enat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_1564_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_complex
        = ( numera6690914467698888265omplex @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_1565_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_real
        = ( numeral_numeral_real @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_1566_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat
        = ( numeral_numeral_nat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_1567_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_1568_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_rat
        = ( numeral_numeral_rat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_1569_power__inject__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( power_power_real @ A @ M )
          = ( power_power_real @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_1570_power__inject__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ( power_power_rat @ A @ M )
          = ( power_power_rat @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_1571_power__inject__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ( power_power_nat @ A @ M )
          = ( power_power_nat @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_1572_power__inject__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( power_power_int @ A @ M )
          = ( power_power_int @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_1573_mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_1574_mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_1575_mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% mod_by_1
thf(fact_1576_bits__mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_1577_bits__mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_1578_bits__mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_by_1
thf(fact_1579_not__real__square__gt__zero,axiom,
    ! [X: real] :
      ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
      = ( X = zero_zero_real ) ) ).

% not_real_square_gt_zero
thf(fact_1580_signed__take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
      = one_one_int ) ).

% signed_take_bit_Suc_1
thf(fact_1581_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_1582_concat__bit__negative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L ) @ zero_zero_int )
      = ( ord_less_int @ L @ zero_zero_int ) ) ).

% concat_bit_negative_iff
thf(fact_1583_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_1584_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_1585_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_1586_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_1587_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_1588_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_1589_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_1590_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_1591_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_1592_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_1593_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_1594_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_1595_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_1596_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_1597_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_1598_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_1599_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_1600_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_1601_power__strict__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_1602_power__strict__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_1603_power__strict__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_1604_power__strict__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_1605_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_1606_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N: nat] :
      ( ( ( numeral_numeral_nat @ K )
       != one_one_nat )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
        = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
thf(fact_1607_one__add__one,axiom,
    ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat )
    = ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_1608_one__add__one,axiom,
    ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_1609_one__add__one,axiom,
    ( ( plus_plus_real @ one_one_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_1610_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_1611_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_1612_one__add__one,axiom,
    ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_1613_power__strict__decreasing__iff,axiom,
    ! [B: real,M: nat,N: 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 @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1614_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1615_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1616_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1617_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_1618_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_1619_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_1620_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_1621_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_1622_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_1623_signed__take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_bit0
thf(fact_1624_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
      = ( numera1916890842035813515d_enat @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_1625_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_1626_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
      = ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_1627_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_1628_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
      = ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_1629_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
      = ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_1630_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
      = ( numera1916890842035813515d_enat @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_1631_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_1632_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_1633_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_1634_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_1635_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_1636_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_1637_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_1638_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_1639_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_1640_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_1641_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_1642_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_1643_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_1644_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_1645_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_1646_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_1647_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_1648_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_1649_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_1650_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_1651_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_1652_real__arch__pow__inv,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ X @ one_one_real )
       => ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X @ N2 ) @ Y ) ) ) ).

% real_arch_pow_inv
thf(fact_1653_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ? [N2: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N2 ) ) ) ).

% real_arch_pow
thf(fact_1654_one__reorient,axiom,
    ! [X: complex] :
      ( ( one_one_complex = X )
      = ( X = one_one_complex ) ) ).

% one_reorient
thf(fact_1655_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
      = ( X = one_one_real ) ) ).

% one_reorient
thf(fact_1656_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
      = ( X = one_one_rat ) ) ).

% one_reorient
thf(fact_1657_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_1658_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
      = ( X = one_one_int ) ) ).

% one_reorient
thf(fact_1659_signed__take__bit__add,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K @ L ) ) ) ).

% signed_take_bit_add
thf(fact_1660_signed__take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ K @ L ) ) ) ).

% signed_take_bit_mult
thf(fact_1661_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
thf(fact_1662_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_1663_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_1664_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_1665_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_1666_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ one_one_real ) ).

% less_numeral_extra(4)
thf(fact_1667_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_1668_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_1669_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_1670_less__numeral__extra_I4_J,axiom,
    ~ ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat ) ).

% less_numeral_extra(4)
thf(fact_1671_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_1672_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_1673_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_1674_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_1675_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_1676_mult_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.comm_neutral
thf(fact_1677_mult_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.comm_neutral
thf(fact_1678_mult_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.comm_neutral
thf(fact_1679_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_1680_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_1681_odd__nonzero,axiom,
    ! [Z2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
     != zero_zero_int ) ).

% odd_nonzero
thf(fact_1682_nat__mult__1,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ one_one_nat @ N )
      = N ) ).

% nat_mult_1
thf(fact_1683_nat__mult__1__right,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ N @ one_one_nat )
      = N ) ).

% nat_mult_1_right
thf(fact_1684_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
thf(fact_1685_concat__bit__assoc,axiom,
    ! [N: nat,K: int,M: nat,L: int,R2: int] :
      ( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L @ R2 ) )
      = ( bit_concat_bit @ ( plus_plus_nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L ) @ R2 ) ) ).

% concat_bit_assoc
thf(fact_1686_not__one__less__zero,axiom,
    ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).

% not_one_less_zero
thf(fact_1687_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_1688_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_1689_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_1690_not__one__less__zero,axiom,
    ~ ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat ) ).

% not_one_less_zero
thf(fact_1691_zero__less__one,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% zero_less_one
thf(fact_1692_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_1693_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_1694_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_1695_zero__less__one,axiom,
    ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).

% zero_less_one
thf(fact_1696_less__numeral__extra_I1_J,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% less_numeral_extra(1)
thf(fact_1697_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_1698_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_1699_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_1700_less__numeral__extra_I1_J,axiom,
    ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).

% less_numeral_extra(1)
thf(fact_1701_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat ) ).

% not_numeral_less_one
thf(fact_1702_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).

% not_numeral_less_one
thf(fact_1703_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_1704_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_1705_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_1706_less__1__mult,axiom,
    ! [M: real,N: real] :
      ( ( ord_less_real @ one_one_real @ M )
     => ( ( ord_less_real @ one_one_real @ N )
       => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_1707_less__1__mult,axiom,
    ! [M: rat,N: rat] :
      ( ( ord_less_rat @ one_one_rat @ M )
     => ( ( ord_less_rat @ one_one_rat @ N )
       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_1708_less__1__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ M )
     => ( ( ord_less_nat @ one_one_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_1709_less__1__mult,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ one_one_int @ M )
     => ( ( ord_less_int @ one_one_int @ N )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_1710_less__add__one,axiom,
    ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).

% less_add_one
thf(fact_1711_less__add__one,axiom,
    ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).

% less_add_one
thf(fact_1712_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_1713_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_1714_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_1715_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_1716_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_1717_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_1718_add__mono1,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ one_on7984719198319812577d_enat ) @ ( plus_p3455044024723400733d_enat @ B @ one_on7984719198319812577d_enat ) ) ) ).

% add_mono1
thf(fact_1719_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_1720_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_1721_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_1722_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat ) ) ).

% one_plus_numeral_commute
thf(fact_1723_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).

% one_plus_numeral_commute
thf(fact_1724_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).

% one_plus_numeral_commute
thf(fact_1725_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).

% one_plus_numeral_commute
thf(fact_1726_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).

% one_plus_numeral_commute
thf(fact_1727_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).

% one_plus_numeral_commute
thf(fact_1728_numeral__One,axiom,
    ( ( numera1916890842035813515d_enat @ one )
    = one_on7984719198319812577d_enat ) ).

% numeral_One
thf(fact_1729_numeral__One,axiom,
    ( ( numera6690914467698888265omplex @ one )
    = one_one_complex ) ).

% numeral_One
thf(fact_1730_numeral__One,axiom,
    ( ( numeral_numeral_real @ one )
    = one_one_real ) ).

% numeral_One
thf(fact_1731_numeral__One,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numeral_One
thf(fact_1732_numeral__One,axiom,
    ( ( numeral_numeral_int @ one )
    = one_one_int ) ).

% numeral_One
thf(fact_1733_numeral__One,axiom,
    ( ( numeral_numeral_rat @ one )
    = one_one_rat ) ).

% numeral_One
thf(fact_1734_left__right__inverse__power,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( ( ( times_times_complex @ X @ Y )
        = one_one_complex )
     => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) )
        = one_one_complex ) ) ).

% left_right_inverse_power
thf(fact_1735_left__right__inverse__power,axiom,
    ! [X: real,Y: real,N: nat] :
      ( ( ( times_times_real @ X @ Y )
        = one_one_real )
     => ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) )
        = one_one_real ) ) ).

% left_right_inverse_power
thf(fact_1736_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = one_one_rat )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y @ N ) )
        = one_one_rat ) ) ).

% left_right_inverse_power
thf(fact_1737_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_1738_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( ( times_times_int @ X @ Y )
        = one_one_int )
     => ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
        = one_one_int ) ) ).

% left_right_inverse_power
thf(fact_1739_power__one__over,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
      = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).

% power_one_over
thf(fact_1740_power__one__over,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
      = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).

% power_one_over
thf(fact_1741_power__one__over,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
      = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).

% power_one_over
thf(fact_1742_power__0,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_1743_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_1744_power__0,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% power_0
thf(fact_1745_power__0,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_1746_power__0,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% power_0
thf(fact_1747_numerals_I1_J,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numerals(1)
thf(fact_1748_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_1749_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_1750_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_1751_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_1752_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
      = none_nat ) ).

% vebt_mint.simps(2)
thf(fact_1753_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M
        = ( times_times_nat @ M @ N ) )
     => ( ( N = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_1754_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
      = none_nat ) ).

% vebt_maxt.simps(2)
thf(fact_1755_odd__less__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
      = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).

% odd_less_0_iff
thf(fact_1756_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_1757_zero__less__two,axiom,
    ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).

% zero_less_two
thf(fact_1758_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_1759_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_1760_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_1761_zero__less__two,axiom,
    ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat ) ).

% zero_less_two
thf(fact_1762_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_1763_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_1764_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_1765_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_1766_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_1767_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_1768_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_1769_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_1770_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_1771_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_1772_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_1773_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_1774_power__gt1__lemma,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_1775_power__gt1__lemma,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_1776_power__gt1__lemma,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_1777_power__gt1__lemma,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_1778_power__less__power__Suc,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_1779_power__less__power__Suc,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_1780_power__less__power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_1781_power__less__power__Suc,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_1782_power__gt1,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_1783_power__gt1,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_1784_power__gt1,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_1785_power__gt1,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_1786_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N )
          = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N )
          = zero_zero_rat ) ) ) ).

% power_0_left
thf(fact_1787_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N )
          = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N )
          = zero_zero_nat ) ) ) ).

% power_0_left
thf(fact_1788_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N )
          = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N )
          = zero_zero_real ) ) ) ).

% power_0_left
thf(fact_1789_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N )
          = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N )
          = zero_zero_int ) ) ) ).

% power_0_left
thf(fact_1790_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N )
          = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N )
          = zero_zero_complex ) ) ) ).

% power_0_left
thf(fact_1791_power__less__imp__less__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_1792_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_1793_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_1794_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_1795_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_real @ one_one_real @ A )
       => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).

% power_strict_increasing
thf(fact_1796_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_rat @ one_one_rat @ A )
       => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N5 ) ) ) ) ).

% power_strict_increasing
thf(fact_1797_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_nat @ one_one_nat @ A )
       => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).

% power_strict_increasing
thf(fact_1798_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_int @ one_one_int @ A )
       => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).

% power_strict_increasing
thf(fact_1799_nat__induct__non__zero,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( P @ one_one_nat )
       => ( ! [N2: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_non_zero
thf(fact_1800_div__eq__dividend__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ( divide_divide_nat @ M @ N )
          = M )
        = ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
thf(fact_1801_div__less__dividend,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).

% div_less_dividend
thf(fact_1802_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ( times_times_int @ M @ N )
          = one_one_int )
        = ( ( M = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
thf(fact_1803_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ one_one_int @ K )
       => ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).

% int_div_less_self
thf(fact_1804_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( ( ( X = zero_zero_nat )
         => A )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => B )
            & ( X = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
thf(fact_1805_vebt__insert_Osimps_I1_J,axiom,
    ! [X: nat,A: $o,B: $o] :
      ( ( ( X = zero_zero_nat )
       => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
          = ( vEBT_Leaf @ $true @ B ) ) )
      & ( ( X != zero_zero_nat )
       => ( ( ( X = one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
              = ( vEBT_Leaf @ A @ $true ) ) )
          & ( ( X != one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
              = ( vEBT_Leaf @ A @ B ) ) ) ) ) ) ).

% vebt_insert.simps(1)
thf(fact_1806_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( ( ( X = zero_zero_nat )
         => A )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => B )
            & ( X = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
thf(fact_1807_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% signed_take_bit_int_less_exp
thf(fact_1808_power__Suc__less,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_1809_power__Suc__less,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_1810_power__Suc__less,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_1811_power__Suc__less,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_1812_power__Suc__less__one,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).

% power_Suc_less_one
thf(fact_1813_power__Suc__less__one,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).

% power_Suc_less_one
thf(fact_1814_power__Suc__less__one,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).

% power_Suc_less_one
thf(fact_1815_power__Suc__less__one,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).

% power_Suc_less_one
thf(fact_1816_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( ord_less_real @ A @ one_one_real )
         => ( ord_less_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_1817_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ( ord_less_rat @ A @ one_one_rat )
         => ( ord_less_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_1818_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ A @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_1819_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ( ord_less_int @ A @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_1820_one__power2,axiom,
    ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_rat ) ).

% one_power2
thf(fact_1821_one__power2,axiom,
    ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_power2
thf(fact_1822_one__power2,axiom,
    ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_real ) ).

% one_power2
thf(fact_1823_one__power2,axiom,
    ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_power2
thf(fact_1824_one__power2,axiom,
    ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_complex ) ).

% one_power2
thf(fact_1825_one__less__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_1826_one__less__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_1827_one__less__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_1828_one__less__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_1829_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_1830_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_1831_nat__induct2,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ( P @ one_one_nat )
       => ( ! [N2: nat] :
              ( ( P @ N2 )
             => ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct2
thf(fact_1832_Suc__times__mod__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
        = one_one_nat ) ) ).

% Suc_times_mod_eq
thf(fact_1833_add__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( B
        = ( plus_plus_complex @ B @ A ) )
      = ( A = zero_zero_complex ) ) ).

% add_0_iff
thf(fact_1834_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B
        = ( plus_plus_real @ B @ A ) )
      = ( A = zero_zero_real ) ) ).

% add_0_iff
thf(fact_1835_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B
        = ( plus_plus_rat @ B @ A ) )
      = ( A = zero_zero_rat ) ) ).

% add_0_iff
thf(fact_1836_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_1837_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B
        = ( plus_plus_int @ B @ A ) )
      = ( A = zero_zero_int ) ) ).

% add_0_iff
thf(fact_1838_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_1839_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_1840_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_1841_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_1842_crossproduct__eq,axiom,
    ! [W: real,Y: real,X: real,Z2: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X @ Z2 ) )
        = ( plus_plus_real @ ( times_times_real @ W @ Z2 ) @ ( times_times_real @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z2 ) ) ) ).

% crossproduct_eq
thf(fact_1843_crossproduct__eq,axiom,
    ! [W: rat,Y: rat,X: rat,Z2: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y ) @ ( times_times_rat @ X @ Z2 ) )
        = ( plus_plus_rat @ ( times_times_rat @ W @ Z2 ) @ ( times_times_rat @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z2 ) ) ) ).

% crossproduct_eq
thf(fact_1844_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,X: nat,Z2: nat] :
      ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z2 ) )
        = ( plus_plus_nat @ ( times_times_nat @ W @ Z2 ) @ ( times_times_nat @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z2 ) ) ) ).

% crossproduct_eq
thf(fact_1845_crossproduct__eq,axiom,
    ! [W: int,Y: int,X: int,Z2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z2 ) )
        = ( plus_plus_int @ ( times_times_int @ W @ Z2 ) @ ( times_times_int @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z2 ) ) ) ).

% crossproduct_eq
thf(fact_1846_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,M: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
       => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).

% set_n_deg_not_0
thf(fact_1847_misiz,axiom,
    ! [T: vEBT_VEBT,N: nat,M: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( some_nat @ M )
          = ( vEBT_vebt_mint @ T ) )
       => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% misiz
thf(fact_1848_dbl__simps_I3_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_1849_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_1850_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_1851_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_1852_even__succ__mod__exp,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
          = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_1853_even__succ__mod__exp,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
          = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_1854_even__succ__mod__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
          = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_1855_even__succ__div__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
          = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_1856_even__succ__div__exp,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
          = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_1857_even__succ__div__exp,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
          = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_1858_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_1859_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_1860_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_1861_VEBT__internal_Oinsert_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_VEBT_insert @ X @ Xa2 )
        = Y )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ( Y
             != ( vEBT_vebt_insert @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList4 @ Summary3 ) )
             => ~ ( ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
                   => ( Y
                      = ( vEBT_Node @ Info2 @ Deg2 @ TreeList4 @ Summary3 ) ) )
                  & ( ~ ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
                   => ( Y
                      = ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList4 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ).

% VEBT_internal.insert'.elims
thf(fact_1862_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_1863_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_1864_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_1865_arith__geo__mean,axiom,
    ! [U: real,X: real,Y: real] :
      ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_real @ X @ Y ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_1866_arith__geo__mean,axiom,
    ! [U: rat,X: rat,Y: rat] :
      ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_rat @ X @ Y ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
         => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_1867_maxt__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% maxt_member
thf(fact_1868_maxbmo,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( ( vEBT_vebt_maxt @ T )
        = ( some_nat @ X ) )
     => ( vEBT_V8194947554948674370ptions @ T @ X ) ) ).

% maxbmo
thf(fact_1869_add__shift,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = Z2 )
      = ( ( vEBT_VEBT_add @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z2 ) ) ) ).

% add_shift
thf(fact_1870_mul__shift,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = Z2 )
      = ( ( vEBT_VEBT_mul @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z2 ) ) ) ).

% mul_shift
thf(fact_1871_max__in__set__def,axiom,
    ( vEBT_VEBT_max_in_set
    = ( ^ [Xs: set_nat,X6: nat] :
          ( ( member_nat @ X6 @ Xs )
          & ! [Y6: nat] :
              ( ( member_nat @ Y6 @ Xs )
             => ( ord_less_eq_nat @ Y6 @ X6 ) ) ) ) ) ).

% max_in_set_def
thf(fact_1872_min__in__set__def,axiom,
    ( vEBT_VEBT_min_in_set
    = ( ^ [Xs: set_nat,X6: nat] :
          ( ( member_nat @ X6 @ Xs )
          & ! [Y6: nat] :
              ( ( member_nat @ Y6 @ Xs )
             => ( ord_less_eq_nat @ X6 @ Y6 ) ) ) ) ) ).

% min_in_set_def
thf(fact_1873_power__shift,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ( power_power_nat @ X @ Y )
        = Z2 )
      = ( ( vEBT_VEBT_power @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z2 ) ) ) ).

% power_shift
thf(fact_1874_mint__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% mint_member
thf(fact_1875_semiring__norm_I71_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(71)
thf(fact_1876_semiring__norm_I68_J,axiom,
    ! [N: num] : ( ord_less_eq_num @ one @ N ) ).

% semiring_norm(68)
thf(fact_1877_subset__empty,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
      = ( A2 = bot_bot_set_real ) ) ).

% subset_empty
thf(fact_1878_subset__empty,axiom,
    ! [A2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
      = ( A2 = bot_bot_set_o ) ) ).

% subset_empty
thf(fact_1879_subset__empty,axiom,
    ! [A2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
      = ( A2 = bot_bot_set_int ) ) ).

% subset_empty
thf(fact_1880_subset__empty,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
      = ( A2 = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_1881_empty__subsetI,axiom,
    ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).

% empty_subsetI
thf(fact_1882_empty__subsetI,axiom,
    ! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).

% empty_subsetI
thf(fact_1883_empty__subsetI,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).

% empty_subsetI
thf(fact_1884_empty__subsetI,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).

% empty_subsetI
thf(fact_1885_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_1886_psubsetI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_set_nat @ A2 @ B2 ) ) ) ).

% psubsetI
thf(fact_1887_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Mini ) )
       => ( ( vEBT_vebt_member @ T @ X )
         => ( ord_less_eq_nat @ Mini @ X ) ) ) ) ).

% mint_corr_help
thf(fact_1888_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( ( vEBT_vebt_member @ T @ X )
         => ( ord_less_eq_nat @ X @ Maxi ) ) ) ) ).

% maxt_corr_help
thf(fact_1889_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_1890_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_1891_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_1892_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_1893_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_1894_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_1895_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_1896_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_1897_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_1898_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_1899_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_1900_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_1901_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_1902_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_1903_semiring__norm_I69_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).

% semiring_norm(69)
thf(fact_1904_dvd__0__right,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).

% dvd_0_right
thf(fact_1905_dvd__0__right,axiom,
    ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).

% dvd_0_right
thf(fact_1906_dvd__0__right,axiom,
    ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).

% dvd_0_right
thf(fact_1907_dvd__0__right,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).

% dvd_0_right
thf(fact_1908_dvd__0__right,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% dvd_0_right
thf(fact_1909_dvd__0__right,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).

% dvd_0_right
thf(fact_1910_dvd__0__left__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
thf(fact_1911_dvd__0__left__iff,axiom,
    ! [A: complex] :
      ( ( dvd_dvd_complex @ zero_zero_complex @ A )
      = ( A = zero_zero_complex ) ) ).

% dvd_0_left_iff
thf(fact_1912_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
      = ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
thf(fact_1913_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
      = ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
thf(fact_1914_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
thf(fact_1915_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
      = ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
thf(fact_1916_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_1917_dvd__1__left,axiom,
    ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).

% dvd_1_left
thf(fact_1918_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_1919_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_1920_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_1921_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_1922_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_1923_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_1924_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_1925_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_1926_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_1927_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_1928_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_1929_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_1930_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_1931_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N @ M ) ) ).

% Suc_le_mono
thf(fact_1932_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_1933_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_1934_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( K = zero_zero_nat )
        | ( dvd_dvd_nat @ M @ N ) ) ) ).

% nat_mult_dvd_cancel_disj
thf(fact_1935_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_1936_not__None__eq,axiom,
    ! [X: option_nat] :
      ( ( X != none_nat )
      = ( ? [Y6: nat] :
            ( X
            = ( some_nat @ Y6 ) ) ) ) ).

% not_None_eq
thf(fact_1937_not__None__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( X != none_P5556105721700978146at_nat )
      = ( ? [Y6: product_prod_nat_nat] :
            ( X
            = ( some_P7363390416028606310at_nat @ Y6 ) ) ) ) ).

% not_None_eq
thf(fact_1938_not__None__eq,axiom,
    ! [X: option_num] :
      ( ( X != none_num )
      = ( ? [Y6: num] :
            ( X
            = ( some_num @ Y6 ) ) ) ) ).

% not_None_eq
thf(fact_1939_not__Some__eq,axiom,
    ! [X: option_nat] :
      ( ( ! [Y6: nat] :
            ( X
           != ( some_nat @ Y6 ) ) )
      = ( X = none_nat ) ) ).

% not_Some_eq
thf(fact_1940_not__Some__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( ! [Y6: product_prod_nat_nat] :
            ( X
           != ( some_P7363390416028606310at_nat @ Y6 ) ) )
      = ( X = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
thf(fact_1941_not__Some__eq,axiom,
    ! [X: option_num] :
      ( ( ! [Y6: num] :
            ( X
           != ( some_num @ Y6 ) ) )
      = ( X = none_num ) ) ).

% not_Some_eq
thf(fact_1942_of__bool__eq_I1_J,axiom,
    ( ( zero_n1201886186963655149omplex @ $false )
    = zero_zero_complex ) ).

% of_bool_eq(1)
thf(fact_1943_of__bool__eq_I1_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $false )
    = zero_zero_real ) ).

% of_bool_eq(1)
thf(fact_1944_of__bool__eq_I1_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $false )
    = zero_zero_rat ) ).

% of_bool_eq(1)
thf(fact_1945_of__bool__eq_I1_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $false )
    = zero_zero_nat ) ).

% of_bool_eq(1)
thf(fact_1946_of__bool__eq_I1_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $false )
    = zero_zero_int ) ).

% of_bool_eq(1)
thf(fact_1947_of__bool__eq_I1_J,axiom,
    ( ( zero_n356916108424825756nteger @ $false )
    = zero_z3403309356797280102nteger ) ).

% of_bool_eq(1)
thf(fact_1948_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n1201886186963655149omplex @ P )
        = zero_zero_complex )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_1949_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P )
        = zero_zero_real )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_1950_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P )
        = zero_zero_rat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_1951_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = zero_zero_nat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_1952_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = zero_zero_int )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_1953_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n356916108424825756nteger @ P )
        = zero_z3403309356797280102nteger )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_1954_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_1955_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_1956_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_le72135733267957522d_enat @ ( zero_n1046097342994218471d_enat @ P ) @ ( zero_n1046097342994218471d_enat @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_1957_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_1958_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_1959_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_1960_concat__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L ) )
      = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).

% concat_bit_nonnegative_iff
thf(fact_1961_unset__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% unset_bit_nonnegative_int_iff
thf(fact_1962_dbl__simps_I2_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% dbl_simps(2)
thf(fact_1963_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_real @ zero_zero_real )
    = zero_zero_real ) ).

% dbl_simps(2)
thf(fact_1964_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% dbl_simps(2)
thf(fact_1965_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_int @ zero_zero_int )
    = zero_zero_int ) ).

% dbl_simps(2)
thf(fact_1966_set__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% set_bit_nonnegative_int_iff
thf(fact_1967_flip__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% flip_bit_nonnegative_int_iff
thf(fact_1968_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_1969_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_1970_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_1971_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_1972_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_1973_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_1974_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_1975_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_1976_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_1977_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_1978_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_1979_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_1980_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_1981_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_1982_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_1983_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_1984_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_1985_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_1986_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_1987_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_1988_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_1989_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_1990_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_1991_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_1992_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_1993_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_1994_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_1995_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_1996_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_1997_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_1998_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_1999_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_2000_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_2001_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_2002_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_2003_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_2004_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_2005_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_2006_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_2007_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_2008_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_2009_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_2010_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_2011_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_2012_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_2013_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_2014_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_2015_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_2016_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_2017_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_2018_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_2019_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_2020_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_2021_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_2022_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_2023_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_2024_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_2025_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_2026_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_2027_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_2028_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_2029_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_2030_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_2031_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_2032_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_2033_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_2034_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_2035_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_2036_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_2037_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_2038_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_2039_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_2040_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_2041_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_2042_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_2043_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_2044_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_2045_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_2046_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_2047_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_2048_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_2049_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_2050_Suc__0__mod__eq,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( zero_n2687167440665602831ol_nat
        @ ( N
         != ( suc @ zero_zero_nat ) ) ) ) ).

% Suc_0_mod_eq
thf(fact_2051_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_2052_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_2053_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_2054_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_2055_mint__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
       => ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X ) ) ) ) ).

% mint_sound
thf(fact_2056_mint__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X ) )
       => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).

% mint_corr
thf(fact_2057_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
       => ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X ) ) ) ) ).

% maxt_sound
thf(fact_2058_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X ) )
       => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).

% maxt_corr
thf(fact_2059_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
      = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_2060_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_2061_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_2062_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) )
      = ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_2063_enat__ord__number_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).

% enat_ord_number(1)
thf(fact_2064_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_2065_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_2066_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_2067_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_2068_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_2069_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_2070_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_2071_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_2072_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_2073_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_2074_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_2075_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_2076_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_2077_even__Suc__Suc__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% even_Suc_Suc_iff
thf(fact_2078_even__Suc,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% even_Suc
thf(fact_2079_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_2080_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_2081_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_2082_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_2083_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_2084_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_2085_pow__divides__pow__iff,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
        = ( dvd_dvd_nat @ A @ B ) ) ) ).

% pow_divides_pow_iff
thf(fact_2086_pow__divides__pow__iff,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
        = ( dvd_dvd_int @ A @ B ) ) ) ).

% pow_divides_pow_iff
thf(fact_2087_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
      = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
        & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).

% one_le_mult_iff
thf(fact_2088_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_2089_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% mult_le_cancel2
thf(fact_2090_lesseq__shift,axiom,
    ( ord_less_eq_nat
    = ( ^ [X6: nat,Y6: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X6 ) @ ( some_nat @ Y6 ) ) ) ) ).

% lesseq_shift
thf(fact_2091_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_2092_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_2093_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_2094_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_2095_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_2096_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_2097_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_2098_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_2099_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_2100_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_2101_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_2102_power__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2103_power__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2104_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2105_power__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2106_power__mono__iff,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
            = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_2107_power__mono__iff,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
            = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_2108_power__mono__iff,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
            = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_2109_power__mono__iff,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
            = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_2110_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_2111_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_2112_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_2113_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_2114_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_2115_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_2116_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_2117_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_2118_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_2119_even__Suc__div__two,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_Suc_div_two
thf(fact_2120_odd__Suc__div__two,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% odd_Suc_div_two
thf(fact_2121_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_2122_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_2123_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P2 ) ) )
      = P2 ) ).

% odd_of_bool_self
thf(fact_2124_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_2125_succ__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
      = ( ( vEBT_vebt_member @ T @ Y )
        & ( ord_less_nat @ X @ Y )
        & ! [Z3: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z3 )
              & ( ord_less_nat @ X @ Z3 ) )
           => ( ord_less_eq_nat @ Y @ Z3 ) ) ) ) ).

% succ_member
thf(fact_2126_pred__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
      = ( ( vEBT_vebt_member @ T @ Y )
        & ( ord_less_nat @ Y @ X )
        & ! [Z3: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z3 )
              & ( ord_less_nat @ Z3 @ X ) )
           => ( ord_less_eq_nat @ Z3 @ Y ) ) ) ) ).

% pred_member
thf(fact_2127_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_2128_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_2129_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_2130_power2__eq__iff__nonneg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_2131_power2__eq__iff__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_2132_power2__eq__iff__nonneg,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_2133_power2__eq__iff__nonneg,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_2134_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_2135_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_2136_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_2137_power__decreasing__iff,axiom,
    ! [B: real,M: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( ord_less_real @ B @ one_one_real )
       => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2138_power__decreasing__iff,axiom,
    ! [B: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2139_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2140_power__decreasing__iff,axiom,
    ! [B: int,M: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2141_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_2142_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_2143_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_2144_power__less__zero__eq,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        & ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% power_less_zero_eq
thf(fact_2145_power__less__zero__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% power_less_zero_eq
thf(fact_2146_power__less__zero__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        & ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% power_less_zero_eq
thf(fact_2147_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_2148_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_2149_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_2150_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_2151_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_2152_of__bool__half__eq__0,axiom,
    ! [B: $o] :
      ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = zero_z3403309356797280102nteger ) ).

% of_bool_half_eq_0
thf(fact_2153_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_2154_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_2155_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_2156_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_2157_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_2158_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_2159_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_2160_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_2161_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_2162_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_2163_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_2164_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_2165_even__power,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% even_power
thf(fact_2166_even__power,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% even_power
thf(fact_2167_even__power,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% even_power
thf(fact_2168_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_2169_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_2170_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_2171_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_2172_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_2173_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_2174_one__div__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_2175_one__div__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_2176_one__div__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_2177_bits__1__div__exp,axiom,
    ! [N: nat] :
      ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_2178_bits__1__div__exp,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_2179_bits__1__div__exp,axiom,
    ! [N: nat] :
      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_2180_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_2181_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_2182_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_2183_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_2184_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_2185_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_2186_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_2187_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_2188_dvd__power__le,axiom,
    ! [X: code_integer,Y: code_integer,N: nat,M: nat] :
      ( ( dvd_dvd_Code_integer @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_2189_dvd__power__le,axiom,
    ! [X: nat,Y: nat,N: nat,M: nat] :
      ( ( dvd_dvd_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_2190_dvd__power__le,axiom,
    ! [X: real,Y: real,N: nat,M: nat] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_2191_dvd__power__le,axiom,
    ! [X: int,Y: int,N: nat,M: nat] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_2192_dvd__power__le,axiom,
    ! [X: complex,Y: complex,N: nat,M: nat] :
      ( ( dvd_dvd_complex @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_2193_power__le__dvd,axiom,
    ! [A: code_integer,N: nat,B: code_integer,M: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_2194_power__le__dvd,axiom,
    ! [A: nat,N: nat,B: nat,M: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_2195_power__le__dvd,axiom,
    ! [A: real,N: nat,B: real,M: nat] :
      ( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_2196_power__le__dvd,axiom,
    ! [A: int,N: nat,B: int,M: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_2197_power__le__dvd,axiom,
    ! [A: complex,N: nat,B: complex,M: nat] :
      ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_2198_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_2199_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_2200_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_2201_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_2202_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: complex] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_2203_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_2204_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_2205_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_2206_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_2207_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_2208_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_2209_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_2210_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_2211_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_2212_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B ) )
       => ? [X5: nat] :
            ( ( P @ X5 )
            & ! [Y5: nat] :
                ( ( P @ Y5 )
               => ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_2213_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
      | ( ord_less_eq_nat @ N @ M ) ) ).

% nat_le_linear
thf(fact_2214_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( M = N ) ) ) ).

% le_antisym
thf(fact_2215_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_2216_le__trans,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I3 @ K ) ) ) ).

% le_trans
thf(fact_2217_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_2218_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_2219_lift__Suc__antimono__le,axiom,
    ! [F: nat > rat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_2220_lift__Suc__antimono__le,axiom,
    ! [F: nat > num,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_2221_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_2222_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_2223_lift__Suc__mono__le,axiom,
    ! [F: nat > set_nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_2224_lift__Suc__mono__le,axiom,
    ! [F: nat > rat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_2225_lift__Suc__mono__le,axiom,
    ! [F: nat > num,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_2226_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_2227_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_2228_zdvd__antisym__nonneg,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ N )
       => ( ( dvd_dvd_int @ M @ N )
         => ( ( dvd_dvd_int @ N @ M )
           => ( M = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
thf(fact_2229_dvd__imp__le,axiom,
    ! [K: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat @ K @ N ) ) ) ).

% dvd_imp_le
thf(fact_2230_power__increasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_real @ one_one_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).

% power_increasing
thf(fact_2231_power__increasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N5 ) ) ) ) ).

% power_increasing
thf(fact_2232_power__increasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).

% power_increasing
thf(fact_2233_power__increasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_int @ one_one_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).

% power_increasing
thf(fact_2234_zdvd__imp__le,axiom,
    ! [Z2: int,N: int] :
      ( ( dvd_dvd_int @ Z2 @ N )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int @ Z2 @ N ) ) ) ).

% zdvd_imp_le
thf(fact_2235_power__dvd__imp__le,axiom,
    ! [I3: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I3 @ M ) @ ( power_power_nat @ I3 @ N ) )
     => ( ( ord_less_nat @ one_one_nat @ I3 )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_dvd_imp_le
thf(fact_2236_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_2237_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2052037380579107095ol_rat
        @ ( P
          & Q ) )
      = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).

% of_bool_conj
thf(fact_2238_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_2239_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_2240_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_2241_dvd__field__iff,axiom,
    ( dvd_dvd_complex
    = ( ^ [A3: complex,B3: complex] :
          ( ( A3 = zero_zero_complex )
         => ( B3 = zero_zero_complex ) ) ) ) ).

% dvd_field_iff
thf(fact_2242_dvd__field__iff,axiom,
    ( dvd_dvd_real
    = ( ^ [A3: real,B3: real] :
          ( ( A3 = zero_zero_real )
         => ( B3 = zero_zero_real ) ) ) ) ).

% dvd_field_iff
thf(fact_2243_dvd__field__iff,axiom,
    ( dvd_dvd_rat
    = ( ^ [A3: rat,B3: rat] :
          ( ( A3 = zero_zero_rat )
         => ( B3 = zero_zero_rat ) ) ) ) ).

% dvd_field_iff
thf(fact_2244_dvd__0__left,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
     => ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
thf(fact_2245_dvd__0__left,axiom,
    ! [A: complex] :
      ( ( dvd_dvd_complex @ zero_zero_complex @ A )
     => ( A = zero_zero_complex ) ) ).

% dvd_0_left
thf(fact_2246_dvd__0__left,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
thf(fact_2247_dvd__0__left,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
thf(fact_2248_dvd__0__left,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
thf(fact_2249_dvd__0__left,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
thf(fact_2250_option_Odistinct_I1_J,axiom,
    ! [X2: nat] :
      ( none_nat
     != ( some_nat @ X2 ) ) ).

% option.distinct(1)
thf(fact_2251_option_Odistinct_I1_J,axiom,
    ! [X2: product_prod_nat_nat] :
      ( none_P5556105721700978146at_nat
     != ( some_P7363390416028606310at_nat @ X2 ) ) ).

% option.distinct(1)
thf(fact_2252_option_Odistinct_I1_J,axiom,
    ! [X2: num] :
      ( none_num
     != ( some_num @ X2 ) ) ).

% option.distinct(1)
thf(fact_2253_option_OdiscI,axiom,
    ! [Option: option_nat,X2: nat] :
      ( ( Option
        = ( some_nat @ X2 ) )
     => ( Option != none_nat ) ) ).

% option.discI
thf(fact_2254_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X2: product_prod_nat_nat] :
      ( ( Option
        = ( some_P7363390416028606310at_nat @ X2 ) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
thf(fact_2255_option_OdiscI,axiom,
    ! [Option: option_num,X2: num] :
      ( ( Option
        = ( some_num @ X2 ) )
     => ( Option != none_num ) ) ).

% option.discI
thf(fact_2256_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X23: nat] :
            ( Y
           != ( some_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_2257_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] :
            ( Y
           != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_2258_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X23: num] :
            ( Y
           != ( some_num @ X23 ) ) ) ).

% option.exhaust
thf(fact_2259_split__option__ex,axiom,
    ( ( ^ [P3: option_nat > $o] :
        ? [X7: option_nat] : ( P3 @ X7 ) )
    = ( ^ [P4: option_nat > $o] :
          ( ( P4 @ none_nat )
          | ? [X6: nat] : ( P4 @ ( some_nat @ X6 ) ) ) ) ) ).

% split_option_ex
thf(fact_2260_split__option__ex,axiom,
    ( ( ^ [P3: option4927543243414619207at_nat > $o] :
        ? [X7: option4927543243414619207at_nat] : ( P3 @ X7 ) )
    = ( ^ [P4: option4927543243414619207at_nat > $o] :
          ( ( P4 @ none_P5556105721700978146at_nat )
          | ? [X6: product_prod_nat_nat] : ( P4 @ ( some_P7363390416028606310at_nat @ X6 ) ) ) ) ) ).

% split_option_ex
thf(fact_2261_split__option__ex,axiom,
    ( ( ^ [P3: option_num > $o] :
        ? [X7: option_num] : ( P3 @ X7 ) )
    = ( ^ [P4: option_num > $o] :
          ( ( P4 @ none_num )
          | ? [X6: num] : ( P4 @ ( some_num @ X6 ) ) ) ) ) ).

% split_option_ex
thf(fact_2262_split__option__all,axiom,
    ( ( ^ [P3: option_nat > $o] :
        ! [X7: option_nat] : ( P3 @ X7 ) )
    = ( ^ [P4: option_nat > $o] :
          ( ( P4 @ none_nat )
          & ! [X6: nat] : ( P4 @ ( some_nat @ X6 ) ) ) ) ) ).

% split_option_all
thf(fact_2263_split__option__all,axiom,
    ( ( ^ [P3: option4927543243414619207at_nat > $o] :
        ! [X7: option4927543243414619207at_nat] : ( P3 @ X7 ) )
    = ( ^ [P4: option4927543243414619207at_nat > $o] :
          ( ( P4 @ none_P5556105721700978146at_nat )
          & ! [X6: product_prod_nat_nat] : ( P4 @ ( some_P7363390416028606310at_nat @ X6 ) ) ) ) ) ).

% split_option_all
thf(fact_2264_split__option__all,axiom,
    ( ( ^ [P3: option_num > $o] :
        ! [X7: option_num] : ( P3 @ X7 ) )
    = ( ^ [P4: option_num > $o] :
          ( ( P4 @ none_num )
          & ! [X6: num] : ( P4 @ ( some_num @ X6 ) ) ) ) ) ).

% split_option_all
thf(fact_2265_combine__options__cases,axiom,
    ! [X: option_nat,P: option_nat > option_nat > $o,Y: option_nat] :
      ( ( ( X = none_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P @ X @ Y ) )
       => ( ! [A5: nat,B5: nat] :
              ( ( X
                = ( some_nat @ A5 ) )
             => ( ( Y
                  = ( some_nat @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2266_combine__options__cases,axiom,
    ! [X: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P @ X @ Y ) )
       => ( ! [A5: nat,B5: product_prod_nat_nat] :
              ( ( X
                = ( some_nat @ A5 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2267_combine__options__cases,axiom,
    ! [X: option_nat,P: option_nat > option_num > $o,Y: option_num] :
      ( ( ( X = none_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X @ Y ) )
       => ( ! [A5: nat,B5: num] :
              ( ( X
                = ( some_nat @ A5 ) )
             => ( ( Y
                  = ( some_num @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2268_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y: option_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P @ X @ Y ) )
       => ( ! [A5: product_prod_nat_nat,B5: nat] :
              ( ( X
                = ( some_P7363390416028606310at_nat @ A5 ) )
             => ( ( Y
                  = ( some_nat @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2269_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P @ X @ Y ) )
       => ( ! [A5: product_prod_nat_nat,B5: product_prod_nat_nat] :
              ( ( X
                = ( some_P7363390416028606310at_nat @ A5 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2270_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y: option_num] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X @ Y ) )
       => ( ! [A5: product_prod_nat_nat,B5: num] :
              ( ( X
                = ( some_P7363390416028606310at_nat @ A5 ) )
             => ( ( Y
                  = ( some_num @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2271_combine__options__cases,axiom,
    ! [X: option_num,P: option_num > option_nat > $o,Y: option_nat] :
      ( ( ( X = none_num )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P @ X @ Y ) )
       => ( ! [A5: num,B5: nat] :
              ( ( X
                = ( some_num @ A5 ) )
             => ( ( Y
                  = ( some_nat @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2272_combine__options__cases,axiom,
    ! [X: option_num,P: option_num > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_num )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P @ X @ Y ) )
       => ( ! [A5: num,B5: product_prod_nat_nat] :
              ( ( X
                = ( some_num @ A5 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2273_combine__options__cases,axiom,
    ! [X: option_num,P: option_num > option_num > $o,Y: option_num] :
      ( ( ( X = none_num )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X @ Y ) )
       => ( ! [A5: num,B5: num] :
              ( ( X
                = ( some_num @ A5 ) )
             => ( ( Y
                  = ( some_num @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2274_dvd__productE,axiom,
    ! [P2: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ P2 @ ( times_times_nat @ A @ B ) )
     => ~ ! [X5: nat,Y4: nat] :
            ( ( P2
              = ( times_times_nat @ X5 @ Y4 ) )
           => ( ( dvd_dvd_nat @ X5 @ A )
             => ~ ( dvd_dvd_nat @ Y4 @ B ) ) ) ) ).

% dvd_productE
thf(fact_2275_dvd__productE,axiom,
    ! [P2: int,A: int,B: int] :
      ( ( dvd_dvd_int @ P2 @ ( times_times_int @ A @ B ) )
     => ~ ! [X5: int,Y4: int] :
            ( ( P2
              = ( times_times_int @ X5 @ Y4 ) )
           => ( ( dvd_dvd_int @ X5 @ A )
             => ~ ( dvd_dvd_int @ Y4 @ B ) ) ) ) ).

% dvd_productE
thf(fact_2276_division__decomp,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
     => ? [B6: nat,C3: nat] :
          ( ( A
            = ( times_times_nat @ B6 @ C3 ) )
          & ( dvd_dvd_nat @ B6 @ B )
          & ( dvd_dvd_nat @ C3 @ C ) ) ) ).

% division_decomp
thf(fact_2277_division__decomp,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
     => ? [B6: int,C3: int] :
          ( ( A
            = ( times_times_int @ B6 @ C3 ) )
          & ( dvd_dvd_int @ B6 @ B )
          & ( dvd_dvd_int @ C3 @ C ) ) ) ).

% division_decomp
thf(fact_2278_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_2279_dvdE,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ~ ! [K2: real] :
            ( A
           != ( times_times_real @ B @ K2 ) ) ) ).

% dvdE
thf(fact_2280_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ~ ! [K2: rat] :
            ( A
           != ( times_times_rat @ B @ K2 ) ) ) ).

% dvdE
thf(fact_2281_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ~ ! [K2: nat] :
            ( A
           != ( times_times_nat @ B @ K2 ) ) ) ).

% dvdE
thf(fact_2282_dvdE,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ~ ! [K2: int] :
            ( A
           != ( times_times_int @ B @ K2 ) ) ) ).

% dvdE
thf(fact_2283_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_2284_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A
        = ( times_times_real @ B @ K ) )
     => ( dvd_dvd_real @ B @ A ) ) ).

% dvdI
thf(fact_2285_dvdI,axiom,
    ! [A: rat,B: rat,K: rat] :
      ( ( A
        = ( times_times_rat @ B @ K ) )
     => ( dvd_dvd_rat @ B @ A ) ) ).

% dvdI
thf(fact_2286_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A
        = ( times_times_nat @ B @ K ) )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% dvdI
thf(fact_2287_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A
        = ( times_times_int @ B @ K ) )
     => ( dvd_dvd_int @ B @ A ) ) ).

% dvdI
thf(fact_2288_dvd__def,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [B3: code_integer,A3: code_integer] :
        ? [K3: code_integer] :
          ( A3
          = ( times_3573771949741848930nteger @ B3 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_2289_dvd__def,axiom,
    ( dvd_dvd_real
    = ( ^ [B3: real,A3: real] :
        ? [K3: real] :
          ( A3
          = ( times_times_real @ B3 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_2290_dvd__def,axiom,
    ( dvd_dvd_rat
    = ( ^ [B3: rat,A3: rat] :
        ? [K3: rat] :
          ( A3
          = ( times_times_rat @ B3 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_2291_dvd__def,axiom,
    ( dvd_dvd_nat
    = ( ^ [B3: nat,A3: nat] :
        ? [K3: nat] :
          ( A3
          = ( times_times_nat @ B3 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_2292_dvd__def,axiom,
    ( dvd_dvd_int
    = ( ^ [B3: int,A3: int] :
        ? [K3: int] :
          ( A3
          = ( times_times_int @ B3 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_2293_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_2294_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_2295_dvd__mult,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ C )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% dvd_mult
thf(fact_2296_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_2297_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_2298_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_2299_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_2300_dvd__mult2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_2301_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_2302_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_2303_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_2304_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_2305_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
     => ( dvd_dvd_rat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_2306_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_2307_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_2308_dvd__triv__left,axiom,
    ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).

% dvd_triv_left
thf(fact_2309_dvd__triv__left,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).

% dvd_triv_left
thf(fact_2310_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_2311_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_2312_dvd__triv__left,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).

% dvd_triv_left
thf(fact_2313_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_2314_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_2315_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ C @ D )
       => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_2316_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_2317_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_2318_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_2319_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_2320_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
     => ( dvd_dvd_rat @ B @ C ) ) ).

% dvd_mult_right
thf(fact_2321_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_2322_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_2323_dvd__triv__right,axiom,
    ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).

% dvd_triv_right
thf(fact_2324_dvd__triv__right,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).

% dvd_triv_right
thf(fact_2325_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_2326_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_2327_dvd__triv__right,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).

% dvd_triv_right
thf(fact_2328_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_2329_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_2330_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_2331_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_2332_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_2333_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_2334_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_2335_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_2336_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_2337_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_2338_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_2339_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_2340_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_2341_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_2342_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_2343_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_2344_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_2345_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_2346_dvd__div__eq__iff,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( dvd_dvd_rat @ C @ A )
     => ( ( dvd_dvd_rat @ C @ B )
       => ( ( ( divide_divide_rat @ A @ C )
            = ( divide_divide_rat @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_2347_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_2348_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_2349_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_2350_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_2351_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_2352_dvd__div__eq__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ C )
        = ( divide_divide_rat @ B @ C ) )
     => ( ( dvd_dvd_rat @ C @ A )
       => ( ( dvd_dvd_rat @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_2353_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_2354_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_2355_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_2356_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_2357_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_2358_dvd__power__same,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ X @ Y )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_2359_dvd__power__same,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( dvd_dvd_nat @ X @ Y )
     => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_2360_dvd__power__same,axiom,
    ! [X: real,Y: real,N: nat] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_2361_dvd__power__same,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_2362_dvd__power__same,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( ( dvd_dvd_complex @ X @ Y )
     => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_2363_gcd__nat_Oextremum,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% gcd_nat.extremum
thf(fact_2364_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
thf(fact_2365_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
thf(fact_2366_gcd__nat_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ( dvd_dvd_nat @ A @ zero_zero_nat )
        & ( A != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
thf(fact_2367_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
thf(fact_2368_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_2369_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_2370_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_2371_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_2372_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_2373_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_2374_verit__comp__simplify1_I3_J,axiom,
    ! [B4: extended_enat,A4: extended_enat] :
      ( ( ~ ( ord_le2932123472753598470d_enat @ B4 @ A4 ) )
      = ( ord_le72135733267957522d_enat @ A4 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2375_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_2376_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_2377_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_2378_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_2379_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_2380_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_2381_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_2382_dvd__mod,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N )
       => ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N ) ) ) ) ).

% dvd_mod
thf(fact_2383_dvd__mod,axiom,
    ! [K: int,M: int,N: int] :
      ( ( dvd_dvd_int @ K @ M )
     => ( ( dvd_dvd_int @ K @ N )
       => ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N ) ) ) ) ).

% dvd_mod
thf(fact_2384_dvd__mod,axiom,
    ! [K: code_integer,M: code_integer,N: code_integer] :
      ( ( dvd_dvd_Code_integer @ K @ M )
     => ( ( dvd_dvd_Code_integer @ K @ N )
       => ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N ) ) ) ) ).

% dvd_mod
thf(fact_2385_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_2386_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_2387_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_2388_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_2389_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_2390_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_2391_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I3: real,J: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I3 @ J )
        & ( K = L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2392_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I3: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I3 @ J )
        & ( K = L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I3 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2393_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I3: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I3 @ J )
        & ( K = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2394_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I3: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I3 @ J )
        & ( K = L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2395_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I3: real,J: real,K: real,L: real] :
      ( ( ( I3 = J )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2396_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I3: rat,J: rat,K: rat,L: rat] :
      ( ( ( I3 = J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I3 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2397_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I3: nat,J: nat,K: nat,L: nat] :
      ( ( ( I3 = J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2398_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I3: int,J: int,K: int,L: int] :
      ( ( ( I3 = J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2399_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I3: real,J: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I3 @ J )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2400_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I3: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I3 @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I3 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2401_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I3: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I3 @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2402_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I3: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I3 @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2403_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_2404_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_2405_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_2406_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_2407_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_2408_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_2409_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_2410_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_2411_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C2: nat] :
            ( B
           != ( plus_plus_nat @ A @ C2 ) ) ) ).

% less_eqE
thf(fact_2412_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_2413_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_2414_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_2415_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_2416_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
        ? [C4: nat] :
          ( B3
          = ( plus_plus_nat @ A3 @ C4 ) ) ) ) ).

% le_iff_add
thf(fact_2417_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_2418_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_2419_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_2420_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_2421_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_2422_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_2423_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_2424_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_2425_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_2426_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_2427_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_2428_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_2429_le__num__One__iff,axiom,
    ! [X: num] :
      ( ( ord_less_eq_num @ X @ one )
      = ( X = one ) ) ).

% le_num_One_iff
thf(fact_2430_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R3: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X5: nat] : ( R3 @ X5 @ X5 )
       => ( ! [X5: nat,Y4: nat,Z4: nat] :
              ( ( R3 @ X5 @ Y4 )
             => ( ( R3 @ Y4 @ Z4 )
               => ( R3 @ X5 @ Z4 ) ) )
         => ( ! [N2: nat] : ( R3 @ N2 @ ( suc @ N2 ) )
           => ( R3 @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_2431_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P @ M )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ M @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_2432_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
             => ( P @ M2 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_2433_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).

% not_less_eq_eq
thf(fact_2434_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_2435_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M @ N )
        | ( M
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_2436_Suc__le__D,axiom,
    ! [N: nat,M6: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
     => ? [M3: nat] :
          ( M6
          = ( suc @ M3 ) ) ) ).

% Suc_le_D
thf(fact_2437_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_2438_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M @ N )
       => ( M
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_2439_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_2440_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_2441_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_2442_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_2443_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_2444_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I3: nat,J: nat] :
      ( ! [I2: nat,J3: nat] :
          ( ( ord_less_nat @ I2 @ J3 )
         => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J3 ) ) )
     => ( ( ord_less_eq_nat @ I3 @ J )
       => ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_2445_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( M != N )
       => ( ord_less_nat @ M @ N ) ) ) ).

% le_neq_implies_less
thf(fact_2446_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_nat @ M @ N )
        | ( M = N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_or_eq_imp_le
thf(fact_2447_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M4: nat,N3: nat] :
          ( ( ord_less_nat @ M4 @ N3 )
          | ( M4 = N3 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_2448_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_imp_le_nat
thf(fact_2449_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M4: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M4 @ N3 )
          & ( M4 != N3 ) ) ) ) ).

% nat_less_le
thf(fact_2450_less__eq__real__def,axiom,
    ( ord_less_eq_real
    = ( ^ [X6: real,Y6: real] :
          ( ( ord_less_real @ X6 @ Y6 )
          | ( X6 = Y6 ) ) ) ) ).

% less_eq_real_def
thf(fact_2451_less__eq__int__code_I1_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% less_eq_int_code(1)
thf(fact_2452_subset__code_I1_J,axiom,
    ! [Xs2: list_complex,B2: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ B2 )
      = ( ! [X6: complex] :
            ( ( member_complex @ X6 @ ( set_complex2 @ Xs2 ) )
           => ( member_complex @ X6 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_2453_subset__code_I1_J,axiom,
    ! [Xs2: list_real,B2: set_real] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ B2 )
      = ( ! [X6: real] :
            ( ( member_real @ X6 @ ( set_real2 @ Xs2 ) )
           => ( member_real @ X6 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_2454_subset__code_I1_J,axiom,
    ! [Xs2: list_set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs2 ) @ B2 )
      = ( ! [X6: set_nat] :
            ( ( member_set_nat @ X6 @ ( set_set_nat2 @ Xs2 ) )
           => ( member_set_nat @ X6 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_2455_subset__code_I1_J,axiom,
    ! [Xs2: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ B2 )
      = ( ! [X6: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( member_VEBT_VEBT @ X6 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_2456_subset__code_I1_J,axiom,
    ! [Xs2: list_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ B2 )
      = ( ! [X6: int] :
            ( ( member_int @ X6 @ ( set_int2 @ Xs2 ) )
           => ( member_int @ X6 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_2457_subset__code_I1_J,axiom,
    ! [Xs2: list_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B2 )
      = ( ! [X6: nat] :
            ( ( member_nat @ X6 @ ( set_nat2 @ Xs2 ) )
           => ( member_nat @ X6 @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_2458_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ~ ( ( ord_less_eq_nat @ M @ N )
         => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).

% add_leE
thf(fact_2459_le__add1,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).

% le_add1
thf(fact_2460_le__add2,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).

% le_add2
thf(fact_2461_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% add_leD1
thf(fact_2462_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ K @ N ) ) ).

% add_leD2
thf(fact_2463_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N2: nat] :
          ( L
          = ( plus_plus_nat @ K @ N2 ) ) ) ).

% le_Suc_ex
thf(fact_2464_add__le__mono,axiom,
    ! [I3: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_le_mono
thf(fact_2465_add__le__mono1,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_le_mono1
thf(fact_2466_trans__le__add1,axiom,
    ! [I3: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_le_add1
thf(fact_2467_trans__le__add2,axiom,
    ! [I3: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_le_add2
thf(fact_2468_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M4: nat,N3: nat] :
        ? [K3: nat] :
          ( N3
          = ( plus_plus_nat @ M4 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_2469_mult__le__mono2,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ K @ I3 ) @ ( times_times_nat @ K @ J ) ) ) ).

% mult_le_mono2
thf(fact_2470_mult__le__mono1,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).

% mult_le_mono1
thf(fact_2471_mult__le__mono,axiom,
    ! [I3: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).

% mult_le_mono
thf(fact_2472_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_2473_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_2474_div__le__mono,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).

% div_le_mono
thf(fact_2475_div__le__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).

% div_le_dividend
thf(fact_2476_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( ord_less_set_nat @ A6 @ B7 )
          | ( A6 = B7 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_2477_subset__psubset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_set_nat @ B2 @ C5 )
       => ( ord_less_set_nat @ A2 @ C5 ) ) ) ).

% subset_psubset_trans
thf(fact_2478_subset__not__subset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ A6 @ B7 )
          & ~ ( ord_less_eq_set_nat @ B7 @ A6 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_2479_psubset__subset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C5 )
       => ( ord_less_set_nat @ A2 @ C5 ) ) ) ).

% psubset_subset_trans
thf(fact_2480_psubset__imp__subset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).

% psubset_imp_subset
thf(fact_2481_psubset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ A6 @ B7 )
          & ( A6 != B7 ) ) ) ) ).

% psubset_eq
thf(fact_2482_psubsetE,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
         => ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).

% psubsetE
thf(fact_2483_mod__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).

% mod_less_eq_dividend
thf(fact_2484_ile0__eq,axiom,
    ! [N: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
      = ( N = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
thf(fact_2485_i0__lb,axiom,
    ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).

% i0_lb
thf(fact_2486_dvd__power__iff,axiom,
    ! [X: code_integer,M: nat,N: nat] :
      ( ( X != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N ) )
        = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
          | ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_2487_dvd__power__iff,axiom,
    ! [X: nat,M: nat,N: nat] :
      ( ( X != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N ) )
        = ( ( dvd_dvd_nat @ X @ one_one_nat )
          | ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_2488_dvd__power__iff,axiom,
    ! [X: int,M: nat,N: nat] :
      ( ( X != zero_zero_int )
     => ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N ) )
        = ( ( dvd_dvd_int @ X @ one_one_int )
          | ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_2489_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ A @ one_one_real )
         => ( ord_less_eq_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2490_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ A @ one_one_rat )
         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2491_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ A @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2492_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ A @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2493_power__le__imp__le__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_2494_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_2495_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_2496_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_2497_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) )
      = ( some_P7363390416028606310at_nat @ ( F @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_2498_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: num > num > num,A: num,B: num] :
      ( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A ) @ ( some_num @ B ) )
      = ( some_num @ ( F @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_2499_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: nat > nat > nat,A: nat,B: nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A ) @ ( some_nat @ B ) )
      = ( some_nat @ ( F @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_2500_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_2501_unset__bit__less__eq,axiom,
    ! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).

% unset_bit_less_eq
thf(fact_2502_set__bit__greater__eq,axiom,
    ! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).

% set_bit_greater_eq
thf(fact_2503_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_2504_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_2505_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_2506_power__mono__odd,axiom,
    ! [N: nat,A: real,B: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).

% power_mono_odd
thf(fact_2507_power__mono__odd,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).

% power_mono_odd
thf(fact_2508_power__mono__odd,axiom,
    ! [N: nat,A: int,B: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).

% power_mono_odd
thf(fact_2509_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
        = ( ord_less_eq_nat @ M @ N ) ) ) ).

% dvd_power_iff_le
thf(fact_2510_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_2511_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_2512_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_2513_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_2514_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_2515_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_2516_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_2517_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_2518_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_2519_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_2520_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_2521_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_2522_of__bool__def,axiom,
    ( zero_n1201886186963655149omplex
    = ( ^ [P5: $o] : ( if_complex @ P5 @ one_one_complex @ zero_zero_complex ) ) ) ).

% of_bool_def
thf(fact_2523_of__bool__def,axiom,
    ( zero_n3304061248610475627l_real
    = ( ^ [P5: $o] : ( if_real @ P5 @ one_one_real @ zero_zero_real ) ) ) ).

% of_bool_def
thf(fact_2524_of__bool__def,axiom,
    ( zero_n2052037380579107095ol_rat
    = ( ^ [P5: $o] : ( if_rat @ P5 @ one_one_rat @ zero_zero_rat ) ) ) ).

% of_bool_def
thf(fact_2525_of__bool__def,axiom,
    ( zero_n2687167440665602831ol_nat
    = ( ^ [P5: $o] : ( if_nat @ P5 @ one_one_nat @ zero_zero_nat ) ) ) ).

% of_bool_def
thf(fact_2526_of__bool__def,axiom,
    ( zero_n2684676970156552555ol_int
    = ( ^ [P5: $o] : ( if_int @ P5 @ one_one_int @ zero_zero_int ) ) ) ).

% of_bool_def
thf(fact_2527_of__bool__def,axiom,
    ( zero_n356916108424825756nteger
    = ( ^ [P5: $o] : ( if_Code_integer @ P5 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).

% of_bool_def
thf(fact_2528_not__is__unit__0,axiom,
    ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).

% not_is_unit_0
thf(fact_2529_not__is__unit__0,axiom,
    ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).

% not_is_unit_0
thf(fact_2530_not__is__unit__0,axiom,
    ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).

% not_is_unit_0
thf(fact_2531_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_2532_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_2533_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_2534_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_2535_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_2536_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_2537_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_2538_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_2539_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_2540_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_2541_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_2542_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_2543_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_2544_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_2545_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_2546_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_2547_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_2548_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_2549_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_2550_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_2551_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_2552_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_2553_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_2554_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_2555_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_2556_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_2557_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_2558_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_2559_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_2560_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_2561_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_2562_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_2563_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_2564_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_2565_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_2566_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_2567_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_2568_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_2569_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_2570_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_2571_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_2572_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_2573_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_2574_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_2575_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_2576_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_2577_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_2578_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_2579_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_2580_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_2581_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_2582_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_2583_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_2584_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_2585_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_2586_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_2587_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_2588_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_2589_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_2590_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_2591_div__power,axiom,
    ! [B: code_integer,A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B ) @ N )
        = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).

% div_power
thf(fact_2592_div__power,axiom,
    ! [B: nat,A: nat,N: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N )
        = ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).

% div_power
thf(fact_2593_div__power,axiom,
    ! [B: int,A: int,N: nat] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N )
        = ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).

% div_power
thf(fact_2594_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_2595_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_2596_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_2597_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( modulo_modulo_nat @ B3 @ A3 )
          = zero_zero_nat ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_2598_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_int
    = ( ^ [A3: int,B3: int] :
          ( ( modulo_modulo_int @ B3 @ A3 )
          = zero_zero_int ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_2599_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [A3: code_integer,B3: code_integer] :
          ( ( modulo364778990260209775nteger @ B3 @ A3 )
          = zero_z3403309356797280102nteger ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_2600_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_2601_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_2602_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_2603_zero__le__numeral,axiom,
    ! [N: num] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).

% zero_le_numeral
thf(fact_2604_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).

% zero_le_numeral
thf(fact_2605_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).

% zero_le_numeral
thf(fact_2606_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_le_numeral
thf(fact_2607_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).

% zero_le_numeral
thf(fact_2608_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).

% not_numeral_le_zero
thf(fact_2609_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).

% not_numeral_le_zero
thf(fact_2610_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).

% not_numeral_le_zero
thf(fact_2611_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_2612_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_2613_dvd__pos__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ M @ N )
       => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).

% dvd_pos_nat
thf(fact_2614_nat__dvd__not__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).

% nat_dvd_not_less
thf(fact_2615_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_2616_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_2617_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_2618_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_2619_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_2620_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_2621_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_2622_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_2623_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_2624_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_2625_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_2626_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_2627_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_2628_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_2629_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_2630_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_2631_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_2632_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_2633_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_2634_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_2635_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_2636_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_2637_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_2638_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_2639_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_2640_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_2641_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_2642_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_2643_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_2644_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_2645_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_2646_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_2647_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_2648_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_2649_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_2650_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_2651_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_2652_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_2653_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_2654_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_2655_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_2656_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_2657_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_2658_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_2659_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_2660_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_2661_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_2662_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_2663_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_2664_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_2665_zero__le__square,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).

% zero_le_square
thf(fact_2666_zero__le__square,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).

% zero_le_square
thf(fact_2667_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_2668_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_2669_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_2670_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_2671_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_2672_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_2673_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_2674_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_2675_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_2676_add__nonpos__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ( plus_plus_real @ X @ Y )
            = zero_zero_real )
          = ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2677_add__nonpos__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ( ( plus_plus_rat @ X @ Y )
            = zero_zero_rat )
          = ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2678_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2679_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ zero_zero_int )
     => ( ( ord_less_eq_int @ Y @ zero_zero_int )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_2680_add__nonneg__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( plus_plus_real @ X @ Y )
            = zero_zero_real )
          = ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2681_add__nonneg__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ( plus_plus_rat @ X @ Y )
            = zero_zero_rat )
          = ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2682_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2683_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_2684_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_2685_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_2686_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_2687_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_2688_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_2689_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_2690_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_2691_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_2692_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_2693_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_2694_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_2695_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_2696_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_2697_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_2698_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_2699_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_2700_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_2701_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_2702_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_2703_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_2704_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_2705_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_2706_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_2707_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_2708_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_2709_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_2710_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_2711_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_2712_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_2713_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_2714_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_2715_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_2716_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_2717_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_2718_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_2719_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_2720_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_2721_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_2722_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_2723_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_2724_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_2725_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_2726_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_2727_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_2728_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I3: real,J: real,K: real,L: real] :
      ( ( ( ord_less_real @ I3 @ J )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2729_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I3: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I3 @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I3 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2730_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I3: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I3 @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2731_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I3: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I3 @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2732_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I3: real,J: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I3 @ J )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I3 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2733_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I3: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I3 @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I3 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2734_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I3: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I3 @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2735_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I3: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I3 @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2736_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_2737_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_2738_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_2739_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_2740_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_2741_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_2742_divide__nonneg__nonneg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_2743_divide__nonneg__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_2744_divide__nonneg__nonpos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonneg_nonpos
thf(fact_2745_divide__nonneg__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_nonpos
thf(fact_2746_divide__nonpos__nonneg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonpos_nonneg
thf(fact_2747_divide__nonpos__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_nonneg
thf(fact_2748_divide__nonpos__nonpos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_2749_divide__nonpos__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_2750_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_2751_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_2752_one__le__numeral,axiom,
    ! [N: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).

% one_le_numeral
thf(fact_2753_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).

% one_le_numeral
thf(fact_2754_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).

% one_le_numeral
thf(fact_2755_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).

% one_le_numeral
thf(fact_2756_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).

% one_le_numeral
thf(fact_2757_power__mono,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).

% power_mono
thf(fact_2758_power__mono,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).

% power_mono
thf(fact_2759_power__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).

% power_mono
thf(fact_2760_power__mono,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).

% power_mono
thf(fact_2761_zero__le__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).

% zero_le_power
thf(fact_2762_zero__le__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).

% zero_le_power
thf(fact_2763_zero__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).

% zero_le_power
thf(fact_2764_zero__le__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).

% zero_le_power
thf(fact_2765_one__le__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).

% one_le_power
thf(fact_2766_one__le__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).

% one_le_power
thf(fact_2767_one__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).

% one_le_power
thf(fact_2768_one__le__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).

% one_le_power
thf(fact_2769_zdvd__not__zless,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_int @ M @ N )
       => ~ ( dvd_dvd_int @ N @ M ) ) ) ).

% zdvd_not_zless
thf(fact_2770_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X5: nat,Y4: nat] :
      ( ( dvd_dvd_nat @ D3 @ A )
      & ( dvd_dvd_nat @ D3 @ B )
      & ( ( ( times_times_nat @ A @ X5 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D3 ) )
        | ( ( times_times_nat @ B @ X5 )
          = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D3 ) ) ) ) ).

% bezout_add_nat
thf(fact_2771_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X: nat,Y: nat] :
      ( ( dvd_dvd_nat @ D @ A )
     => ( ( dvd_dvd_nat @ D @ B )
       => ( ( ( ( times_times_nat @ A @ X )
              = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D ) )
            | ( ( times_times_nat @ B @ X )
              = ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
         => ? [X5: nat,Y4: 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 ) @ Y4 ) @ D ) )
                | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X5 )
                  = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D ) ) ) ) ) ) ) ).

% bezout_lemma_nat
thf(fact_2772_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_2773_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_2774_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_2775_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,N: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
     => ( ( K != zero_zero_int )
       => ( dvd_dvd_int @ M @ N ) ) ) ).

% zdvd_mult_cancel
thf(fact_2776_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_2777_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_le_eq
thf(fact_2778_dec__induct,axiom,
    ! [I3: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ( P @ I3 )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I3 @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ N2 )
                 => ( P @ ( suc @ N2 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_2779_inc__induct,axiom,
    ! [I3: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ( P @ J )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I3 @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ ( suc @ N2 ) )
                 => ( P @ N2 ) ) ) )
         => ( P @ I3 ) ) ) ) ).

% inc_induct
thf(fact_2780_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_le_lessD
thf(fact_2781_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% le_less_Suc_eq
thf(fact_2782_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% less_Suc_eq_le
thf(fact_2783_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_2784_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_2785_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K2 )
               => ~ ( P @ I4 ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_2786_dbl__def,axiom,
    ( neg_numeral_dbl_real
    = ( ^ [X6: real] : ( plus_plus_real @ X6 @ X6 ) ) ) ).

% dbl_def
thf(fact_2787_dbl__def,axiom,
    ( neg_numeral_dbl_rat
    = ( ^ [X6: rat] : ( plus_plus_rat @ X6 @ X6 ) ) ) ).

% dbl_def
thf(fact_2788_dbl__def,axiom,
    ( neg_numeral_dbl_int
    = ( ^ [X6: int] : ( plus_plus_int @ X6 @ X6 ) ) ) ).

% dbl_def
thf(fact_2789_zdvd__period,axiom,
    ! [A: int,D: int,X: int,T: int,C: int] :
      ( ( dvd_dvd_int @ A @ D )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
        = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).

% zdvd_period
thf(fact_2790_zdvd__reduce,axiom,
    ! [K: int,N: int,M: int] :
      ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
      = ( dvd_dvd_int @ K @ N ) ) ).

% zdvd_reduce
thf(fact_2791_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M3: nat,N2: nat] :
          ( ( ord_less_nat @ M3 @ N2 )
         => ( ord_less_nat @ ( F @ M3 ) @ ( F @ N2 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_2792_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_mult_le_cancel1
thf(fact_2793_Suc__div__le__mono,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).

% Suc_div_le_mono
thf(fact_2794_int__one__le__iff__zero__less,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ one_one_int @ Z2 )
      = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).

% int_one_le_iff_zero_less
thf(fact_2795_times__div__less__eq__dividend,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).

% times_div_less_eq_dividend
thf(fact_2796_div__times__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).

% div_times_less_eq_dividend
thf(fact_2797_mod__Suc__le__divisor,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).

% mod_Suc_le_divisor
thf(fact_2798_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_2799_zero__le__even__power,axiom,
    ! [N: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).

% zero_le_even_power
thf(fact_2800_zero__le__even__power,axiom,
    ! [N: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).

% zero_le_even_power
thf(fact_2801_zero__le__even__power,axiom,
    ! [N: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).

% zero_le_even_power
thf(fact_2802_zero__le__odd__power,axiom,
    ! [N: nat,A: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).

% zero_le_odd_power
thf(fact_2803_zero__le__odd__power,axiom,
    ! [N: nat,A: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).

% zero_le_odd_power
thf(fact_2804_zero__le__odd__power,axiom,
    ! [N: nat,A: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% zero_le_odd_power
thf(fact_2805_zero__le__power__eq,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_2806_zero__le__power__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_2807_zero__le__power__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_2808_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ Uw @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
      = none_P5556105721700978146at_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_2809_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: num > num > num,V: num] :
      ( ( vEBT_V819420779217536731ft_num @ Uw @ ( some_num @ V ) @ none_num )
      = none_num ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_2810_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: nat > nat > nat,V: nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ Uw @ ( some_nat @ V ) @ none_nat )
      = none_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_2811_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( ( vEBT_V1502963449132264192at_nat @ X @ Xa2 @ Xb )
        = Y )
     => ( ( ( Xa2 = none_P5556105721700978146at_nat )
         => ( Y != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] :
                ( Xa2
                = ( some_P7363390416028606310at_nat @ V2 ) )
           => ( ( Xb = none_P5556105721700978146at_nat )
             => ( Y != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A5: product_prod_nat_nat] :
                ( ( Xa2
                  = ( some_P7363390416028606310at_nat @ A5 ) )
               => ! [B5: product_prod_nat_nat] :
                    ( ( Xb
                      = ( some_P7363390416028606310at_nat @ B5 ) )
                   => ( Y
                     != ( some_P7363390416028606310at_nat @ ( X @ A5 @ B5 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_2812_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: num > num > num,Xa2: option_num,Xb: option_num,Y: option_num] :
      ( ( ( vEBT_V819420779217536731ft_num @ X @ Xa2 @ Xb )
        = Y )
     => ( ( ( Xa2 = none_num )
         => ( Y != none_num ) )
       => ( ( ? [V2: num] :
                ( Xa2
                = ( some_num @ V2 ) )
           => ( ( Xb = none_num )
             => ( Y != none_num ) ) )
         => ~ ! [A5: num] :
                ( ( Xa2
                  = ( some_num @ A5 ) )
               => ! [B5: num] :
                    ( ( Xb
                      = ( some_num @ B5 ) )
                   => ( Y
                     != ( some_num @ ( X @ A5 @ B5 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_2813_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: nat > nat > nat,Xa2: option_nat,Xb: option_nat,Y: option_nat] :
      ( ( ( vEBT_V4262088993061758097ft_nat @ X @ Xa2 @ Xb )
        = Y )
     => ( ( ( Xa2 = none_nat )
         => ( Y != none_nat ) )
       => ( ( ? [V2: nat] :
                ( Xa2
                = ( some_nat @ V2 ) )
           => ( ( Xb = none_nat )
             => ( Y != none_nat ) ) )
         => ~ ! [A5: nat] :
                ( ( Xa2
                  = ( some_nat @ A5 ) )
               => ! [B5: nat] :
                    ( ( Xb
                      = ( some_nat @ B5 ) )
                   => ( Y
                     != ( some_nat @ ( X @ A5 @ B5 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_2814_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C2: code_integer] :
              ( B
             != ( times_3573771949741848930nteger @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_2815_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [C2: nat] :
              ( B
             != ( times_times_nat @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_2816_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [C2: int] :
              ( B
             != ( times_times_int @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_2817_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_2818_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_2819_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_2820_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_2821_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_2822_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_2823_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_2824_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_2825_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_2826_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_2827_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_2828_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_2829_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_2830_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_2831_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_2832_even__numeral,axiom,
    ! [N: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_2833_even__numeral,axiom,
    ! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_2834_even__numeral,axiom,
    ! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_2835_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_2836_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_2837_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_2838_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_2839_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_2840_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_2841_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_2842_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_2843_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_2844_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_2845_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_2846_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_2847_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_2848_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_2849_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_2850_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_2851_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_2852_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_2853_is__unit__power__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_2854_is__unit__power__iff,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_2855_is__unit__power__iff,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_2856_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_2857_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_2858_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_2859_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_2860_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_2861_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_2862_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_2863_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_2864_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_2865_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_2866_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_2867_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_2868_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_2869_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_2870_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_2871_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_2872_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_2873_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_2874_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_2875_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_2876_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_2877_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_2878_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_2879_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_2880_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_2881_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_2882_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_2883_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_2884_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_2885_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_2886_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_2887_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_2888_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_2889_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_2890_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_2891_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_2892_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_2893_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_2894_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_2895_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_2896_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_2897_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_2898_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_2899_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_2900_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_2901_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_2902_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_2903_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_2904_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_2905_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_2906_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_2907_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_2908_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_2909_mult__le__cancel__iff1,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z2 )
     => ( ( ord_less_eq_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_2910_mult__le__cancel__iff1,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z2 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ Y @ Z2 ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_2911_mult__le__cancel__iff1,axiom,
    ! [Z2: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_2912_mult__le__cancel__iff2,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z2 )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ ( times_times_real @ Z2 @ Y ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_2913_mult__le__cancel__iff2,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z2 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X ) @ ( times_times_rat @ Z2 @ Y ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_2914_mult__le__cancel__iff2,axiom,
    ! [Z2: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z2 @ X ) @ ( times_times_int @ Z2 @ Y ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_2915_field__le__epsilon,axiom,
    ! [X: real,Y: real] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_epsilon
thf(fact_2916_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E )
         => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y @ E ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_epsilon
thf(fact_2917_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_2918_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_2919_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_2920_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_2921_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_2922_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_2923_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_2924_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_2925_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_2926_add__pos__nonneg,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ A )
     => ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ B )
       => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_2927_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_2928_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_2929_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_2930_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_2931_add__nonpos__neg,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ zero_z5237406670263579293d_enat )
     => ( ( ord_le72135733267957522d_enat @ B @ zero_z5237406670263579293d_enat )
       => ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ zero_z5237406670263579293d_enat ) ) ) ).

% add_nonpos_neg
thf(fact_2932_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_2933_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_2934_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_2935_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_2936_add__nonneg__pos,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ A )
     => ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ B )
       => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_2937_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_2938_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_2939_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_2940_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_2941_add__neg__nonpos,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ zero_z5237406670263579293d_enat )
     => ( ( ord_le2932123472753598470d_enat @ B @ zero_z5237406670263579293d_enat )
       => ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ zero_z5237406670263579293d_enat ) ) ) ).

% add_neg_nonpos
thf(fact_2942_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_2943_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_2944_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_2945_frac__le,axiom,
    ! [Y: real,X: real,W: real,Z2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_eq_real @ W @ Z2 )
           => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).

% frac_le
thf(fact_2946_frac__le,axiom,
    ! [Y: rat,X: rat,W: rat,Z2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_eq_rat @ W @ Z2 )
           => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).

% frac_le
thf(fact_2947_frac__less,axiom,
    ! [X: real,Y: real,W: real,Z2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_eq_real @ W @ Z2 )
           => ( ord_less_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).

% frac_less
thf(fact_2948_frac__less,axiom,
    ! [X: rat,Y: rat,W: rat,Z2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ X @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_eq_rat @ W @ Z2 )
           => ( ord_less_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).

% frac_less
thf(fact_2949_frac__less2,axiom,
    ! [X: real,Y: real,W: real,Z2: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_real @ W @ Z2 )
           => ( ord_less_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).

% frac_less2
thf(fact_2950_frac__less2,axiom,
    ! [X: rat,Y: rat,W: rat,Z2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ X @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_rat @ W @ Z2 )
           => ( ord_less_rat @ ( divide_divide_rat @ X @ Z2 ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).

% frac_less2
thf(fact_2951_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_2952_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_2953_divide__nonneg__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonneg_neg
thf(fact_2954_divide__nonneg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_neg
thf(fact_2955_divide__nonneg__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonneg_pos
thf(fact_2956_divide__nonneg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonneg_pos
thf(fact_2957_divide__nonpos__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonpos_neg
thf(fact_2958_divide__nonpos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonpos_neg
thf(fact_2959_divide__nonpos__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonpos_pos
thf(fact_2960_divide__nonpos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_pos
thf(fact_2961_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_2962_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_2963_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_2964_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_2965_sum__squares__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2966_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2967_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2968_sum__squares__ge__zero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_2969_sum__squares__ge__zero,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_2970_sum__squares__ge__zero,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_2971_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_2972_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_2973_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_2974_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_2975_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_2976_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_2977_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_2978_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_2979_mult__right__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_2980_mult__right__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_2981_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_2982_mult__left__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_2983_mult__left__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_2984_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_2985_power__less__imp__less__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_real @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2986_power__less__imp__less__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2987_power__less__imp__less__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2988_power__less__imp__less__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2989_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_2990_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_2991_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_2992_discrete,axiom,
    ( ord_less_int
    = ( ^ [A3: int] : ( ord_less_eq_int @ ( plus_plus_int @ A3 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_2993_power__le__one,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).

% power_le_one
thf(fact_2994_power__le__one,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).

% power_le_one
thf(fact_2995_power__le__one,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).

% power_le_one
thf(fact_2996_power__le__one,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ A @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).

% power_le_one
thf(fact_2997_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( dvd_dvd_nat @ M @ N ) ) ) ).

% nat_mult_dvd_cancel1
thf(fact_2998_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( dvd_dvd_nat @ M @ N ) ) ) ).

% dvd_mult_cancel
thf(fact_2999_dvd__mult__cancel2,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
        = ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
thf(fact_3000_dvd__mult__cancel1,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
        = ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
thf(fact_3001_power__inject__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( ( power_power_real @ A @ ( suc @ N ) )
        = ( power_power_real @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_3002_power__inject__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( ( power_power_rat @ A @ ( suc @ N ) )
        = ( power_power_rat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_3003_power__inject__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ( power_power_nat @ A @ ( suc @ N ) )
        = ( power_power_nat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_3004_power__inject__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( ( power_power_int @ A @ ( suc @ N ) )
        = ( power_power_int @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_3005_power__le__imp__le__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_3006_power__le__imp__le__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_3007_power__le__imp__le__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_3008_power__le__imp__le__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_3009_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_3010_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_3011_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_3012_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_3013_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_3014_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_3015_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D3: nat,X5: nat,Y4: nat] :
          ( ( dvd_dvd_nat @ D3 @ A )
          & ( dvd_dvd_nat @ D3 @ B )
          & ( ( times_times_nat @ A @ X5 )
            = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D3 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_3016_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
      = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).

% mod_greater_zero_iff_not_dvd
thf(fact_3017_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K2 )
               => ~ ( P @ I4 ) )
            & ( P @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_3018_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( ord_less_eq_nat @ M @ N ) ) ) ).

% nat_mult_le_cancel1
thf(fact_3019_div__le__mono2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).

% div_le_mono2
thf(fact_3020_div__greater__zero__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
      = ( ( ord_less_eq_nat @ N @ M )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% div_greater_zero_iff
thf(fact_3021_nat__one__le__power,axiom,
    ! [I3: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I3 )
     => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I3 @ N ) ) ) ).

% nat_one_le_power
thf(fact_3022_mod__le__divisor,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).

% mod_le_divisor
thf(fact_3023_le__imp__0__less,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).

% le_imp_0_less
thf(fact_3024_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_3025_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_3026_zdiv__eq__0__iff,axiom,
    ! [I3: int,K: int] :
      ( ( ( divide_divide_int @ I3 @ K )
        = zero_zero_int )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I3 )
          & ( ord_less_int @ I3 @ K ) )
        | ( ( ord_less_eq_int @ I3 @ zero_zero_int )
          & ( ord_less_int @ K @ I3 ) ) ) ) ).

% zdiv_eq_0_iff
thf(fact_3027_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_3028_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_3029_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_3030_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_3031_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_3032_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_3033_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I3: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I3 @ K ) )
        = ( ord_less_eq_int @ K @ I3 ) ) ) ).

% pos_imp_zdiv_pos_iff
thf(fact_3034_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_3035_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_3036_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_3037_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q2 )
        = ( modulo_modulo_nat @ N @ Q2 ) )
     => ( ( ord_less_eq_nat @ N @ M )
       => ~ ! [S2: nat] :
              ( M
             != ( plus_plus_nat @ N @ ( times_times_nat @ Q2 @ S2 ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_3038_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q2 )
        = ( modulo_modulo_nat @ N @ Q2 ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ~ ! [S2: nat] :
              ( N
             != ( plus_plus_nat @ M @ ( times_times_nat @ Q2 @ S2 ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_3039_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N )
        = ( modulo_modulo_nat @ Y @ N ) )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ? [Q3: nat] :
            ( X
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q3 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_3040_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_3041_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_3042_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_3043_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_3044_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_3045_zmod__trivial__iff,axiom,
    ! [I3: int,K: int] :
      ( ( ( modulo_modulo_int @ I3 @ K )
        = I3 )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I3 )
          & ( ord_less_int @ I3 @ K ) )
        | ( ( ord_less_eq_int @ I3 @ zero_zero_int )
          & ( ord_less_int @ K @ I3 ) ) ) ) ).

% zmod_trivial_iff
thf(fact_3046_power__le__zero__eq,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
      = ( ( ord_less_nat @ zero_zero_nat @ N )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( ord_less_eq_real @ A @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_3047_power__le__zero__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ N )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_3048_power__le__zero__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ N )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( ord_less_eq_int @ A @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_3049_option_Osize_I4_J,axiom,
    ! [X2: nat] :
      ( ( size_size_option_nat @ ( some_nat @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_3050_option_Osize_I4_J,axiom,
    ! [X2: product_prod_nat_nat] :
      ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_3051_option_Osize_I4_J,axiom,
    ! [X2: num] :
      ( ( size_size_option_num @ ( some_num @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_3052_even__zero,axiom,
    dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).

% even_zero
thf(fact_3053_even__zero,axiom,
    dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).

% even_zero
thf(fact_3054_even__zero,axiom,
    dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).

% even_zero
thf(fact_3055_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_3056_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_3057_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_3058_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_3059_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_3060_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_3061_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_3062_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_3063_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_3064_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_3065_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_3066_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_3067_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_3068_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_3069_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_3070_odd__one,axiom,
    ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).

% odd_one
thf(fact_3071_odd__one,axiom,
    ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).

% odd_one
thf(fact_3072_odd__one,axiom,
    ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).

% odd_one
thf(fact_3073_bit__eq__rec,axiom,
    ( ( ^ [Y3: code_integer,Z: code_integer] : Y3 = Z )
    = ( ^ [A3: code_integer,B3: code_integer] :
          ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
            = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) )
          & ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = ( divide6298287555418463151nteger @ B3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_3074_bit__eq__rec,axiom,
    ( ( ^ [Y3: nat,Z: nat] : Y3 = Z )
    = ( ^ [A3: nat,B3: nat] :
          ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
            = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) )
          & ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( divide_divide_nat @ B3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_3075_bit__eq__rec,axiom,
    ( ( ^ [Y3: int,Z: int] : Y3 = Z )
    = ( ^ [A3: int,B3: int] :
          ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
            = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) )
          & ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = ( divide_divide_int @ B3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_3076_dvd__power,axiom,
    ! [N: nat,X: code_integer] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_Code_integer ) )
     => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N ) ) ) ).

% dvd_power
thf(fact_3077_dvd__power,axiom,
    ! [N: nat,X: rat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_rat ) )
     => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N ) ) ) ).

% dvd_power
thf(fact_3078_dvd__power,axiom,
    ! [N: nat,X: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_nat ) )
     => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).

% dvd_power
thf(fact_3079_dvd__power,axiom,
    ! [N: nat,X: real] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_real ) )
     => ( dvd_dvd_real @ X @ ( power_power_real @ X @ N ) ) ) ).

% dvd_power
thf(fact_3080_dvd__power,axiom,
    ! [N: nat,X: int] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_int ) )
     => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).

% dvd_power
thf(fact_3081_dvd__power,axiom,
    ! [N: nat,X: complex] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_complex ) )
     => ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N ) ) ) ).

% dvd_power
thf(fact_3082_field__le__mult__one__interval,axiom,
    ! [X: real,Y: real] :
      ( ! [Z4: real] :
          ( ( ord_less_real @ zero_zero_real @ Z4 )
         => ( ( ord_less_real @ Z4 @ one_one_real )
           => ( ord_less_eq_real @ ( times_times_real @ Z4 @ X ) @ Y ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_3083_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z4: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ Z4 )
         => ( ( ord_less_rat @ Z4 @ one_one_rat )
           => ( ord_less_eq_rat @ ( times_times_rat @ Z4 @ X ) @ Y ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_3084_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_3085_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_3086_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_3087_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_3088_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_3089_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_3090_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_3091_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_3092_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_3093_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_3094_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_3095_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_3096_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_3097_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_3098_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_3099_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_3100_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_3101_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_3102_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_3103_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_3104_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_3105_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_3106_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_3107_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_3108_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_3109_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_3110_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_3111_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_3112_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_3113_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_3114_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_3115_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_3116_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_3117_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_3118_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_3119_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_3120_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_3121_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_3122_mult__imp__div__pos__le,axiom,
    ! [Y: real,X: real,Z2: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ X @ ( times_times_real @ Z2 @ Y ) )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z2 ) ) ) ).

% mult_imp_div_pos_le
thf(fact_3123_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X: rat,Z2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z2 @ Y ) )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z2 ) ) ) ).

% mult_imp_div_pos_le
thf(fact_3124_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z2: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ Y ) @ X )
       => ( ord_less_eq_real @ Z2 @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_3125_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z2: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ Y ) @ X )
       => ( ord_less_eq_rat @ Z2 @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_3126_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_3127_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_3128_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_3129_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_3130_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_3131_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_3132_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_3133_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_3134_convex__bound__le,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( ( ord_less_eq_real @ X @ A )
     => ( ( ord_less_eq_real @ Y @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3135_convex__bound__le,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( ( ord_less_eq_rat @ X @ A )
     => ( ( ord_less_eq_rat @ Y @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3136_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( ( ord_less_eq_int @ X @ A )
     => ( ( ord_less_eq_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3137_power__Suc__le__self,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_3138_power__Suc__le__self,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_3139_power__Suc__le__self,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_3140_power__Suc__le__self,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ A @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_3141_power__eq__imp__eq__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( ( power_power_real @ A @ N )
        = ( power_power_real @ B @ N ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_3142_power__eq__imp__eq__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( ( power_power_rat @ A @ N )
        = ( power_power_rat @ B @ N ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_3143_power__eq__imp__eq__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ( power_power_nat @ A @ N )
        = ( power_power_nat @ B @ N ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_3144_power__eq__imp__eq__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( ( power_power_int @ A @ N )
        = ( power_power_int @ B @ N ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_3145_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: real,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ( power_power_real @ A @ N )
              = ( power_power_real @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_3146_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ( power_power_rat @ A @ N )
              = ( power_power_rat @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_3147_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ( power_power_nat @ A @ N )
              = ( power_power_nat @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_3148_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ( power_power_int @ A @ N )
              = ( power_power_int @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_3149_self__le__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_3150_self__le__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_3151_self__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_3152_self__le__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_3153_not__exp__less__eq__0__int,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).

% not_exp_less_eq_0_int
thf(fact_3154_power2__nat__le__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% power2_nat_le_imp_le
thf(fact_3155_power2__nat__le__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% power2_nat_le_eq_le
thf(fact_3156_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_3157_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).

% signed_take_bit_int_less_self_iff
thf(fact_3158_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% signed_take_bit_int_greater_eq_self_iff
thf(fact_3159_vebt__mint_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( A
       => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A
       => ( ( B
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
              = ( some_nat @ one_one_nat ) ) )
          & ( ~ B
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
              = none_nat ) ) ) ) ) ).

% vebt_mint.simps(1)
thf(fact_3160_vebt__maxt_Osimps_I1_J,axiom,
    ! [B: $o,A: $o] :
      ( ( B
       => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( A
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
              = none_nat ) ) ) ) ) ).

% vebt_maxt.simps(1)
thf(fact_3161_two__realpow__ge__one,axiom,
    ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).

% two_realpow_ge_one
thf(fact_3162_div__nat__eqI,axiom,
    ! [N: nat,Q2: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M )
     => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
       => ( ( divide_divide_nat @ M @ N )
          = Q2 ) ) ) ).

% div_nat_eqI
thf(fact_3163_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q2 )
     => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q2 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_3164_q__pos__lemma,axiom,
    ! [B4: int,Q5: int,R4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q5 ) @ R4 ) )
     => ( ( ord_less_int @ R4 @ B4 )
       => ( ( ord_less_int @ zero_zero_int @ B4 )
         => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).

% q_pos_lemma
thf(fact_3165_zdiv__mono2__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B4: int,Q5: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 )
        = ( plus_plus_int @ ( times_times_int @ B4 @ Q5 ) @ R4 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q5 ) @ R4 ) )
       => ( ( ord_less_int @ R4 @ B4 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
           => ( ( ord_less_int @ zero_zero_int @ B4 )
             => ( ( ord_less_eq_int @ B4 @ B )
               => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_3166_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q2: int,R2: int,B4: int,Q5: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 )
        = ( plus_plus_int @ ( times_times_int @ B4 @ Q5 ) @ R4 ) )
     => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q5 ) @ R4 ) @ zero_zero_int )
       => ( ( ord_less_int @ R2 @ 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 @ Q5 @ Q2 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_3167_unique__quotient__lemma,axiom,
    ! [B: int,Q5: int,R4: int,Q2: int,R2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
       => ( ( ord_less_int @ R4 @ B )
         => ( ( ord_less_int @ R2 @ B )
           => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_3168_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q5: int,R4: int,Q2: int,R2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( ord_less_int @ B @ R4 )
           => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_3169_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_3170_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_3171_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_3172_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_3173_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_3174_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_3175_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_3176_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_3177_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_3178_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_3179_odd__pos,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% odd_pos
thf(fact_3180_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_3181_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_3182_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_3183_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_3184_convex__bound__lt,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( ( ord_less_real @ X @ A )
     => ( ( ord_less_real @ Y @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3185_convex__bound__lt,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( ( ord_less_rat @ X @ A )
     => ( ( ord_less_rat @ Y @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3186_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( ( ord_less_int @ X @ A )
     => ( ( ord_less_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3187_power2__le__imp__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_3188_power2__le__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_3189_power2__le__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_3190_power2__le__imp__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_3191_power2__eq__imp__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_3192_power2__eq__imp__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_3193_power2__eq__imp__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_3194_power2__eq__imp__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_3195_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_3196_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_3197_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_3198_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_3199_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_3200_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_3201_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_3202_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_3203_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_3204_power__strict__mono,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_3205_power__strict__mono,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_3206_power__strict__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_3207_power__strict__mono,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_3208_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_3209_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_3210_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_3211_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_3212_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_3213_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_3214_split__div_H,axiom,
    ! [P: nat > $o,M: nat,N: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
          & ( P @ zero_zero_nat ) )
        | ? [Q4: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M )
            & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
            & ( P @ Q4 ) ) ) ) ).

% split_div'
thf(fact_3215_verit__le__mono__div,axiom,
    ! [A2: nat,B2: nat,N: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B2 @ N )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B2 @ N ) ) ) ) ).

% verit_le_mono_div
thf(fact_3216_split__zdiv,axiom,
    ! [P: int > $o,N: int,K: int] :
      ( ( P @ ( divide_divide_int @ N @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ zero_zero_int ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I: int,J2: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
                & ( ord_less_int @ J2 @ K )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I ) @ J2 ) ) )
             => ( P @ I ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I: int,J2: int] :
              ( ( ( ord_less_int @ K @ J2 )
                & ( ord_less_eq_int @ J2 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I ) @ J2 ) ) )
             => ( P @ I ) ) ) ) ) ).

% split_zdiv
thf(fact_3217_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( divide_divide_int @ A @ B )
            = Q2 ) ) ) ) ).

% int_div_neg_eq
thf(fact_3218_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( divide_divide_int @ A @ B )
            = Q2 ) ) ) ) ).

% int_div_pos_eq
thf(fact_3219_split__zmod,axiom,
    ! [P: int > $o,N: int,K: int] :
      ( ( P @ ( modulo_modulo_int @ N @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ N ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I: int,J2: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
                & ( ord_less_int @ J2 @ K )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I ) @ J2 ) ) )
             => ( P @ J2 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I: int,J2: int] :
              ( ( ( ord_less_int @ K @ J2 )
                & ( ord_less_eq_int @ J2 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I ) @ J2 ) ) )
             => ( P @ J2 ) ) ) ) ) ).

% split_zmod
thf(fact_3220_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( modulo_modulo_int @ A @ B )
            = R2 ) ) ) ) ).

% int_mod_neg_eq
thf(fact_3221_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( modulo_modulo_int @ A @ B )
            = R2 ) ) ) ) ).

% int_mod_pos_eq
thf(fact_3222_verit__le__mono__div__int,axiom,
    ! [A2: int,B2: int,N: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N )
            @ ( if_int
              @ ( ( modulo_modulo_int @ B2 @ N )
                = zero_zero_int )
              @ one_one_int
              @ zero_zero_int ) )
          @ ( divide_divide_int @ B2 @ N ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_3223_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_3224_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_3225_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_3226_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_3227_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_3228_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_3229_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_3230_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_3231_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_3232_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_3233_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_3234_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_3235_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_3236_power2__less__imp__less,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_real @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_3237_power2__less__imp__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_3238_power2__less__imp__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ord_less_nat @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_3239_power2__less__imp__less,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_int @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_3240_sum__power2__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_3241_sum__power2__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_3242_sum__power2__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_3243_sum__power2__ge__zero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_3244_sum__power2__ge__zero,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_3245_sum__power2__ge__zero,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_3246_zero__le__even__power_H,axiom,
    ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_3247_zero__le__even__power_H,axiom,
    ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_3248_zero__le__even__power_H,axiom,
    ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_3249_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
       => ? [N2: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_3250_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ one_one_nat @ K )
       => ? [N2: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_3251_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_3252_split__neg__lemma,axiom,
    ! [K: int,P: int > int > $o,N: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
        = ( ! [I: int,J2: int] :
              ( ( ( ord_less_int @ K @ J2 )
                & ( ord_less_eq_int @ J2 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I ) @ J2 ) ) )
             => ( P @ I @ J2 ) ) ) ) ) ).

% split_neg_lemma
thf(fact_3253_split__pos__lemma,axiom,
    ! [K: int,P: int > int > $o,N: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
        = ( ! [I: int,J2: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J2 )
                & ( ord_less_int @ J2 @ K )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I ) @ J2 ) ) )
             => ( P @ I @ J2 ) ) ) ) ) ).

% split_pos_lemma
thf(fact_3254_exp__mod__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_3255_exp__mod__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_3256_exp__mod__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_3257_zero__less__power__eq,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
      = ( ( N = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( A != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_3258_zero__less__power__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
      = ( ( N = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( A != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_3259_zero__less__power__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
      = ( ( N = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( A != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_3260_sum__squares__bound,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_3261_sum__squares__bound,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_3262_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_3263_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_3264_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_3265_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_3266_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_3267_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_3268_mod__double__modulus,axiom,
    ! [M: code_integer,X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
       => ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( modulo364778990260209775nteger @ X @ M ) )
          | ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_3269_mod__double__modulus,axiom,
    ! [M: nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
       => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_nat @ X @ M ) )
          | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_3270_mod__double__modulus,axiom,
    ! [M: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_int @ X @ M ) )
          | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_3271_VEBT__internal_Oinsert_H_Osimps_I2_J,axiom,
    ! [Deg: nat,X: nat,Info: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) @ X )
       => ( ( vEBT_VEBT_insert @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) )
      & ( ~ ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) @ X )
       => ( ( vEBT_VEBT_insert @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X )
          = ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).

% VEBT_internal.insert'.simps(2)
thf(fact_3272_less__shift,axiom,
    ( ord_less_nat
    = ( ^ [X6: nat,Y6: nat] : ( vEBT_VEBT_less @ ( some_nat @ X6 ) @ ( some_nat @ Y6 ) ) ) ) ).

% less_shift
thf(fact_3273_greater__shift,axiom,
    ( ord_less_nat
    = ( ^ [Y6: nat,X6: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X6 ) @ ( some_nat @ Y6 ) ) ) ) ).

% greater_shift
thf(fact_3274_helpypredd,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Y ) )
       => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% helpypredd
thf(fact_3275_helpyd,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Y ) )
       => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% helpyd
thf(fact_3276_power__minus__is__div,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A @ B ) )
        = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% power_minus_is_div
thf(fact_3277_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 )
         => ! [X3: int] :
              ( ( P @ X3 )
             => ( P @ ( plus_plus_int @ X3 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_3278_even__even__mod__4__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).

% even_even_mod_4_iff
thf(fact_3279_div2__even__ext__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
          = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y ) )
       => ( X = Y ) ) ) ).

% div2_even_ext_nat
thf(fact_3280_unity__coeff__ex,axiom,
    ! [P: code_integer > $o,L: code_integer] :
      ( ( ? [X6: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X6 ) ) )
      = ( ? [X6: code_integer] :
            ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X6 @ zero_z3403309356797280102nteger ) )
            & ( P @ X6 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3281_unity__coeff__ex,axiom,
    ! [P: complex > $o,L: complex] :
      ( ( ? [X6: complex] : ( P @ ( times_times_complex @ L @ X6 ) ) )
      = ( ? [X6: complex] :
            ( ( dvd_dvd_complex @ L @ ( plus_plus_complex @ X6 @ zero_zero_complex ) )
            & ( P @ X6 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3282_unity__coeff__ex,axiom,
    ! [P: real > $o,L: real] :
      ( ( ? [X6: real] : ( P @ ( times_times_real @ L @ X6 ) ) )
      = ( ? [X6: real] :
            ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X6 @ zero_zero_real ) )
            & ( P @ X6 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3283_unity__coeff__ex,axiom,
    ! [P: rat > $o,L: rat] :
      ( ( ? [X6: rat] : ( P @ ( times_times_rat @ L @ X6 ) ) )
      = ( ? [X6: rat] :
            ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X6 @ zero_zero_rat ) )
            & ( P @ X6 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3284_unity__coeff__ex,axiom,
    ! [P: nat > $o,L: nat] :
      ( ( ? [X6: nat] : ( P @ ( times_times_nat @ L @ X6 ) ) )
      = ( ? [X6: nat] :
            ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X6 @ zero_zero_nat ) )
            & ( P @ X6 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3285_unity__coeff__ex,axiom,
    ! [P: int > $o,L: int] :
      ( ( ? [X6: int] : ( P @ ( times_times_int @ L @ X6 ) ) )
      = ( ? [X6: int] :
            ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X6 @ zero_zero_int ) )
            & ( P @ X6 ) ) ) ) ).

% unity_coeff_ex
thf(fact_3286_subset__antisym,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% subset_antisym
thf(fact_3287_subsetI,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A2 )
         => ( member_complex @ X5 @ B2 ) )
     => ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ).

% subsetI
thf(fact_3288_subsetI,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A2 )
         => ( member_real @ X5 @ B2 ) )
     => ( ord_less_eq_set_real @ A2 @ B2 ) ) ).

% subsetI
thf(fact_3289_subsetI,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ! [X5: set_nat] :
          ( ( member_set_nat @ X5 @ A2 )
         => ( member_set_nat @ X5 @ B2 ) )
     => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).

% subsetI
thf(fact_3290_subsetI,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( member_int @ X5 @ B2 ) )
     => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).

% subsetI
thf(fact_3291_subsetI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( member_nat @ X5 @ B2 ) )
     => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).

% subsetI
thf(fact_3292_succ__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% succ_corr
thf(fact_3293_pred__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Px: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Px ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Px ) ) ) ).

% pred_corr
thf(fact_3294_succ__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% succ_correct
thf(fact_3295_pred__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% pred_correct
thf(fact_3296_diff__self,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ A )
      = zero_zero_complex ) ).

% diff_self
thf(fact_3297_diff__self,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% diff_self
thf(fact_3298_diff__self,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% diff_self
thf(fact_3299_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_3300_diff__0__right,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ zero_zero_complex )
      = A ) ).

% diff_0_right
thf(fact_3301_diff__0__right,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_0_right
thf(fact_3302_diff__0__right,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_0_right
thf(fact_3303_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_3304_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_3305_diff__zero,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ zero_zero_complex )
      = A ) ).

% diff_zero
thf(fact_3306_diff__zero,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_zero
thf(fact_3307_diff__zero,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_zero
thf(fact_3308_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_3309_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_3310_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_3311_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_3312_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_3313_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_3314_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_3315_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_3316_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_3317_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_3318_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_3319_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_3320_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_3321_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_3322_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_3323_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_3324_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_3325_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_3326_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_3327_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_3328_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_3329_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_3330_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_3331_diff__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_3332_diff__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_3333_diff__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_3334_add__diff__cancel,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_3335_add__diff__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_3336_add__diff__cancel,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_3337_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_3338_minus__mod__self2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_mod_self2
thf(fact_3339_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_Suc_Suc
thf(fact_3340_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).

% Suc_diff_diff
thf(fact_3341_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_3342_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_3343_diff__diff__cancel,axiom,
    ! [I3: nat,N: nat] :
      ( ( ord_less_eq_nat @ I3 @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
        = I3 ) ) ).

% diff_diff_cancel
thf(fact_3344_diff__diff__left,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J ) @ K )
      = ( minus_minus_nat @ I3 @ ( plus_plus_nat @ J @ K ) ) ) ).

% diff_diff_left
thf(fact_3345_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_3346_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_3347_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_3348_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_3349_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_3350_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_3351_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_3352_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_3353_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_3354_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_3355_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_3356_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_3357_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_3358_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_3359_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_3360_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
    = zero_zero_complex ) ).

% diff_numeral_special(9)
thf(fact_3361_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_real @ one_one_real @ one_one_real )
    = zero_zero_real ) ).

% diff_numeral_special(9)
thf(fact_3362_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_3363_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_3364_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_3365_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_3366_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_3367_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_3368_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_3369_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_3370_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_3371_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_3372_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_3373_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_3374_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
      = ( ord_less_nat @ M @ N ) ) ).

% zero_less_diff
thf(fact_3375_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% diff_is_0_eq
thf(fact_3376_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_3377_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I3: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_3378_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I3: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I3 ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_3379_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I3: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_3380_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_3381_Suc__pred,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% Suc_pred
thf(fact_3382_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I3: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I3 )
        = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I3 ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_3383_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I3: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I3 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ ( suc @ J ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_3384_Suc__diff__1,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
        = N ) ) ).

% Suc_diff_1
thf(fact_3385_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_3386_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_3387_odd__Suc__minus__one,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% odd_Suc_minus_one
thf(fact_3388_even__diff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).

% even_diff_nat
thf(fact_3389_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_3390_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_3391_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_3392_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( minus_minus_nat @ N @ one_one_nat ) ) ) ).

% odd_two_times_div_two_nat
thf(fact_3393_complete__real,axiom,
    ! [S3: set_real] :
      ( ? [X3: real] : ( member_real @ X3 @ S3 )
     => ( ? [Z5: real] :
          ! [X5: real] :
            ( ( member_real @ X5 @ S3 )
           => ( ord_less_eq_real @ X5 @ Z5 ) )
       => ? [Y4: real] :
            ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ord_less_eq_real @ X3 @ Y4 ) )
            & ! [Z5: real] :
                ( ! [X5: real] :
                    ( ( member_real @ X5 @ S3 )
                   => ( ord_less_eq_real @ X5 @ Z5 ) )
               => ( ord_less_eq_real @ Y4 @ Z5 ) ) ) ) ) ).

% complete_real
thf(fact_3394_verit__la__generic,axiom,
    ! [A: int,X: int] :
      ( ( ord_less_eq_int @ A @ X )
      | ( A = X )
      | ( ord_less_eq_int @ X @ A ) ) ).

% verit_la_generic
thf(fact_3395_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N )
       => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% dvd_diff_nat
thf(fact_3396_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ M @ N )
     => ( ( dvd_dvd_nat @ N @ M )
       => ( M = N ) ) ) ).

% dvd_antisym
thf(fact_3397_Collect__mono__iff,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
      = ( ! [X6: real] :
            ( ( P @ X6 )
           => ( Q @ X6 ) ) ) ) ).

% Collect_mono_iff
thf(fact_3398_Collect__mono__iff,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
      = ( ! [X6: list_nat] :
            ( ( P @ X6 )
           => ( Q @ X6 ) ) ) ) ).

% Collect_mono_iff
thf(fact_3399_Collect__mono__iff,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
      = ( ! [X6: set_nat] :
            ( ( P @ X6 )
           => ( Q @ X6 ) ) ) ) ).

% Collect_mono_iff
thf(fact_3400_Collect__mono__iff,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
      = ( ! [X6: int] :
            ( ( P @ X6 )
           => ( Q @ X6 ) ) ) ) ).

% Collect_mono_iff
thf(fact_3401_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X6: nat] :
            ( ( P @ X6 )
           => ( Q @ X6 ) ) ) ) ).

% Collect_mono_iff
thf(fact_3402_set__eq__subset,axiom,
    ( ( ^ [Y3: set_nat,Z: set_nat] : Y3 = Z )
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ A6 @ B7 )
          & ( ord_less_eq_set_nat @ B7 @ A6 ) ) ) ) ).

% set_eq_subset
thf(fact_3403_subset__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C5 )
       => ( ord_less_eq_set_nat @ A2 @ C5 ) ) ) ).

% subset_trans
thf(fact_3404_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_3405_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_3406_Collect__mono,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X5: set_nat] :
          ( ( P @ X5 )
         => ( Q @ X5 ) )
     => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_3407_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_3408_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_3409_subset__refl,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).

% subset_refl
thf(fact_3410_subset__iff,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A6: set_complex,B7: set_complex] :
        ! [T2: complex] :
          ( ( member_complex @ T2 @ A6 )
         => ( member_complex @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_3411_subset__iff,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B7: set_real] :
        ! [T2: real] :
          ( ( member_real @ T2 @ A6 )
         => ( member_real @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_3412_subset__iff,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A6: set_set_nat,B7: set_set_nat] :
        ! [T2: set_nat] :
          ( ( member_set_nat @ T2 @ A6 )
         => ( member_set_nat @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_3413_subset__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B7: set_int] :
        ! [T2: int] :
          ( ( member_int @ T2 @ A6 )
         => ( member_int @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_3414_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
        ! [T2: nat] :
          ( ( member_nat @ T2 @ A6 )
         => ( member_nat @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_3415_equalityD2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
     => ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).

% equalityD2
thf(fact_3416_equalityD1,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
     => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).

% equalityD1
thf(fact_3417_subset__eq,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A6: set_complex,B7: set_complex] :
        ! [X6: complex] :
          ( ( member_complex @ X6 @ A6 )
         => ( member_complex @ X6 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_3418_subset__eq,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B7: set_real] :
        ! [X6: real] :
          ( ( member_real @ X6 @ A6 )
         => ( member_real @ X6 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_3419_subset__eq,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A6: set_set_nat,B7: set_set_nat] :
        ! [X6: set_nat] :
          ( ( member_set_nat @ X6 @ A6 )
         => ( member_set_nat @ X6 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_3420_subset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B7: set_int] :
        ! [X6: int] :
          ( ( member_int @ X6 @ A6 )
         => ( member_int @ X6 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_3421_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
        ! [X6: nat] :
          ( ( member_nat @ X6 @ A6 )
         => ( member_nat @ X6 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_3422_equalityE,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 = B2 )
     => ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
         => ~ ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).

% equalityE
thf(fact_3423_subsetD,axiom,
    ! [A2: set_complex,B2: set_complex,C: complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ B2 )
     => ( ( member_complex @ C @ A2 )
       => ( member_complex @ C @ B2 ) ) ) ).

% subsetD
thf(fact_3424_subsetD,axiom,
    ! [A2: set_real,B2: set_real,C: real] :
      ( ( ord_less_eq_set_real @ A2 @ B2 )
     => ( ( member_real @ C @ A2 )
       => ( member_real @ C @ B2 ) ) ) ).

% subsetD
thf(fact_3425_subsetD,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
     => ( ( member_set_nat @ C @ A2 )
       => ( member_set_nat @ C @ B2 ) ) ) ).

% subsetD
thf(fact_3426_subsetD,axiom,
    ! [A2: set_int,B2: set_int,C: int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ( member_int @ C @ A2 )
       => ( member_int @ C @ B2 ) ) ) ).

% subsetD
thf(fact_3427_subsetD,axiom,
    ! [A2: set_nat,B2: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( member_nat @ C @ A2 )
       => ( member_nat @ C @ B2 ) ) ) ).

% subsetD
thf(fact_3428_in__mono,axiom,
    ! [A2: set_complex,B2: set_complex,X: complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ B2 )
     => ( ( member_complex @ X @ A2 )
       => ( member_complex @ X @ B2 ) ) ) ).

% in_mono
thf(fact_3429_in__mono,axiom,
    ! [A2: set_real,B2: set_real,X: real] :
      ( ( ord_less_eq_set_real @ A2 @ B2 )
     => ( ( member_real @ X @ A2 )
       => ( member_real @ X @ B2 ) ) ) ).

% in_mono
thf(fact_3430_in__mono,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,X: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
     => ( ( member_set_nat @ X @ A2 )
       => ( member_set_nat @ X @ B2 ) ) ) ).

% in_mono
thf(fact_3431_in__mono,axiom,
    ! [A2: set_int,B2: set_int,X: int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ( member_int @ X @ A2 )
       => ( member_int @ X @ B2 ) ) ) ).

% in_mono
thf(fact_3432_in__mono,axiom,
    ! [A2: set_nat,B2: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( member_nat @ X @ A2 )
       => ( member_nat @ X @ B2 ) ) ) ).

% in_mono
thf(fact_3433_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_3434_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_3435_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_3436_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_3437_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_3438_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_3439_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_3440_diff__commute,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K ) @ J ) ) ).

% diff_commute
thf(fact_3441_inf__period_I1_J,axiom,
    ! [P: real > $o,D4: real,Q: real > $o] :
      ( ! [X5: real,K2: real] :
          ( ( P @ X5 )
          = ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K2 @ D4 ) ) ) )
     => ( ! [X5: real,K2: real] :
            ( ( Q @ X5 )
            = ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K2 @ D4 ) ) ) )
       => ! [X3: real,K4: real] :
            ( ( ( P @ X3 )
              & ( Q @ X3 ) )
            = ( ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3442_inf__period_I1_J,axiom,
    ! [P: rat > $o,D4: rat,Q: rat > $o] :
      ( ! [X5: rat,K2: rat] :
          ( ( P @ X5 )
          = ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K2 @ D4 ) ) ) )
     => ( ! [X5: rat,K2: rat] :
            ( ( Q @ X5 )
            = ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K2 @ D4 ) ) ) )
       => ! [X3: rat,K4: rat] :
            ( ( ( P @ X3 )
              & ( Q @ X3 ) )
            = ( ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3443_inf__period_I1_J,axiom,
    ! [P: int > $o,D4: int,Q: int > $o] :
      ( ! [X5: int,K2: int] :
          ( ( P @ X5 )
          = ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D4 ) ) ) )
     => ( ! [X5: int,K2: int] :
            ( ( Q @ X5 )
            = ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D4 ) ) ) )
       => ! [X3: int,K4: int] :
            ( ( ( P @ X3 )
              & ( Q @ X3 ) )
            = ( ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3444_inf__period_I2_J,axiom,
    ! [P: real > $o,D4: real,Q: real > $o] :
      ( ! [X5: real,K2: real] :
          ( ( P @ X5 )
          = ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K2 @ D4 ) ) ) )
     => ( ! [X5: real,K2: real] :
            ( ( Q @ X5 )
            = ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K2 @ D4 ) ) ) )
       => ! [X3: real,K4: real] :
            ( ( ( P @ X3 )
              | ( Q @ X3 ) )
            = ( ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3445_inf__period_I2_J,axiom,
    ! [P: rat > $o,D4: rat,Q: rat > $o] :
      ( ! [X5: rat,K2: rat] :
          ( ( P @ X5 )
          = ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K2 @ D4 ) ) ) )
     => ( ! [X5: rat,K2: rat] :
            ( ( Q @ X5 )
            = ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K2 @ D4 ) ) ) )
       => ! [X3: rat,K4: rat] :
            ( ( ( P @ X3 )
              | ( Q @ X3 ) )
            = ( ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3446_inf__period_I2_J,axiom,
    ! [P: int > $o,D4: int,Q: int > $o] :
      ( ! [X5: int,K2: int] :
          ( ( P @ X5 )
          = ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D4 ) ) ) )
     => ( ! [X5: int,K2: int] :
            ( ( Q @ X5 )
            = ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D4 ) ) ) )
       => ! [X3: int,K4: int] :
            ( ( ( P @ X3 )
              | ( Q @ X3 ) )
            = ( ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3447_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_3448_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_3449_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_3450_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_3451_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_3452_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_3453_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_3454_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_3455_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_3456_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_3457_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_3458_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_3459_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y3: complex,Z: complex] : Y3 = Z )
    = ( ^ [A3: complex,B3: complex] :
          ( ( minus_minus_complex @ A3 @ B3 )
          = zero_zero_complex ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3460_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y3: real,Z: real] : Y3 = Z )
    = ( ^ [A3: real,B3: real] :
          ( ( minus_minus_real @ A3 @ B3 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3461_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y3: rat,Z: rat] : Y3 = Z )
    = ( ^ [A3: rat,B3: rat] :
          ( ( minus_minus_rat @ A3 @ B3 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3462_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y3: int,Z: int] : Y3 = Z )
    = ( ^ [A3: int,B3: int] :
          ( ( minus_minus_int @ A3 @ B3 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3463_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_3464_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_3465_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_3466_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_3467_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_3468_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_3469_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_3470_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_3471_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_3472_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_3473_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_3474_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_3475_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_3476_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_3477_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_3478_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_3479_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_3480_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_3481_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_3482_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_3483_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_3484_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_3485_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_3486_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_3487_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_3488_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_3489_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_3490_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_3491_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_3492_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_3493_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_3494_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_3495_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_3496_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_3497_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_3498_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_3499_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_3500_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_3501_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_3502_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_3503_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_3504_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_3505_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_3506_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_3507_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_3508_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_3509_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_3510_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_3511_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_3512_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_3513_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_3514_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_3515_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_3516_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_3517_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_3518_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_3519_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_3520_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_3521_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_3522_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_3523_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_3524_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_3525_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_3526_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I3: nat] :
      ( ( P @ K )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ ( minus_minus_nat @ K @ I3 ) ) ) ) ).

% zero_induct_lemma
thf(fact_3527_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_3528_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M )
          = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
thf(fact_3529_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ord_less_nat @ M @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_3530_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).

% less_imp_diff_less
thf(fact_3531_dvd__minus__self,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
      = ( ( ord_less_nat @ N @ M )
        | ( dvd_dvd_nat @ M @ N ) ) ) ).

% dvd_minus_self
thf(fact_3532_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_3533_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_3534_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_3535_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_3536_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_3537_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_3538_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_3539_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_3540_dvd__diffD,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
     => ( ( dvd_dvd_nat @ K @ N )
       => ( ( ord_less_eq_nat @ N @ M )
         => ( dvd_dvd_nat @ K @ M ) ) ) ) ).

% dvd_diffD
thf(fact_3541_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
     => ( ( dvd_dvd_nat @ K @ M )
       => ( ( ord_less_eq_nat @ N @ M )
         => ( dvd_dvd_nat @ K @ N ) ) ) ) ).

% dvd_diffD1
thf(fact_3542_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M = N ) ) ) ) ).

% eq_diff_iff
thf(fact_3543_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% le_diff_iff
thf(fact_3544_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_3545_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_3546_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_3547_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_3548_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_3549_less__eq__dvd__minus,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( dvd_dvd_nat @ M @ N )
        = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).

% less_eq_dvd_minus
thf(fact_3550_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
      = M ) ).

% diff_add_inverse2
thf(fact_3551_diff__add__inverse,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
      = M ) ).

% diff_add_inverse
thf(fact_3552_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_cancel2
thf(fact_3553_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% Nat.diff_cancel
thf(fact_3554_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
      = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% diff_mult_distrib
thf(fact_3555_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
      = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% diff_mult_distrib2
thf(fact_3556_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X5: nat,Y4: nat] :
      ( ( dvd_dvd_nat @ D3 @ A )
      & ( dvd_dvd_nat @ D3 @ B )
      & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X5 ) @ ( times_times_nat @ B @ Y4 ) )
          = D3 )
        | ( ( minus_minus_nat @ ( times_times_nat @ B @ X5 ) @ ( times_times_nat @ A @ Y4 ) )
          = D3 ) ) ) ).

% bezout1_nat
thf(fact_3557_le__iff__diff__le__0,axiom,
    ( ord_less_eq_real
    = ( ^ [A3: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).

% le_iff_diff_le_0
thf(fact_3558_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A3: rat,B3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_3559_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_3560_less__iff__diff__less__0,axiom,
    ( ord_less_real
    = ( ^ [A3: real,B3: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).

% less_iff_diff_less_0
thf(fact_3561_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A3: rat,B3: rat] : ( ord_less_rat @ ( minus_minus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_3562_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_3563_inf__period_I3_J,axiom,
    ! [D: code_integer,D4: code_integer,T: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ D4 )
     => ! [X3: code_integer,K4: code_integer] :
          ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ T ) )
          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_3564_inf__period_I3_J,axiom,
    ! [D: real,D4: real,T: real] :
      ( ( dvd_dvd_real @ D @ D4 )
     => ! [X3: real,K4: real] :
          ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ T ) )
          = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_3565_inf__period_I3_J,axiom,
    ! [D: rat,D4: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D4 )
     => ! [X3: rat,K4: rat] :
          ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ T ) )
          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_3566_inf__period_I3_J,axiom,
    ! [D: int,D4: int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X3: int,K4: int] :
          ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ T ) )
          = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_3567_inf__period_I4_J,axiom,
    ! [D: code_integer,D4: code_integer,T: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ D4 )
     => ! [X3: code_integer,K4: code_integer] :
          ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ T ) ) )
          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X3 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_3568_inf__period_I4_J,axiom,
    ! [D: real,D4: real,T: real] :
      ( ( dvd_dvd_real @ D @ D4 )
     => ! [X3: real,K4: real] :
          ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ T ) ) )
          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X3 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_3569_inf__period_I4_J,axiom,
    ! [D: rat,D4: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D4 )
     => ! [X3: rat,K4: rat] :
          ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ T ) ) )
          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_3570_inf__period_I4_J,axiom,
    ! [D: int,D4: int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X3: int,K4: int] :
          ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ T ) ) )
          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X3 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_3571_add__le__add__imp__diff__le,axiom,
    ! [I3: real,K: real,N: real,J: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K ) @ N )
     => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
       => ( ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K ) @ N )
         => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
           => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_3572_add__le__add__imp__diff__le,axiom,
    ! [I3: rat,K: rat,N: rat,J: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I3 @ K ) @ N )
     => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
       => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I3 @ K ) @ N )
         => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
           => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_3573_add__le__add__imp__diff__le,axiom,
    ! [I3: nat,K: nat,N: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ N )
         => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_3574_add__le__add__imp__diff__le,axiom,
    ! [I3: int,K: int,N: int,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I3 @ K ) @ N )
     => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I3 @ K ) @ N )
         => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
           => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_3575_add__le__imp__le__diff,axiom,
    ! [I3: real,K: real,N: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K ) @ N )
     => ( ord_less_eq_real @ I3 @ ( minus_minus_real @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_3576_add__le__imp__le__diff,axiom,
    ! [I3: rat,K: rat,N: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I3 @ K ) @ N )
     => ( ord_less_eq_rat @ I3 @ ( minus_minus_rat @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_3577_add__le__imp__le__diff,axiom,
    ! [I3: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ N )
     => ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_3578_add__le__imp__le__diff,axiom,
    ! [I3: int,K: int,N: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I3 @ K ) @ N )
     => ( ord_less_eq_int @ I3 @ ( minus_minus_int @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_3579_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_3580_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_3581_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_3582_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_3583_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_3584_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_3585_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_3586_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_3587_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_3588_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_3589_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_3590_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_3591_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_3592_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_3593_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_3594_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_3595_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_3596_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_3597_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_3598_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_3599_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_3600_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_3601_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_3602_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_3603_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_3604_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_3605_square__diff__square__factored,axiom,
    ! [X: real,Y: real] :
      ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
      = ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_3606_square__diff__square__factored,axiom,
    ! [X: rat,Y: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
      = ( times_times_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_rat @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_3607_square__diff__square__factored,axiom,
    ! [X: int,Y: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
      = ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_3608_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_3609_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_3610_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_3611_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_3612_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_3613_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_3614_mult__diff__mult,axiom,
    ! [X: real,Y: real,A: real,B: real] :
      ( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A @ B ) )
      = ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_3615_mult__diff__mult,axiom,
    ! [X: rat,Y: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ Y ) @ ( times_times_rat @ A @ B ) )
      = ( plus_plus_rat @ ( times_times_rat @ X @ ( minus_minus_rat @ Y @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_3616_mult__diff__mult,axiom,
    ! [X: int,Y: int,A: int,B: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ Y ) @ ( times_times_int @ A @ B ) )
      = ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_3617_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_3618_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M @ N ) ) ) ).

% Suc_diff_Suc
thf(fact_3619_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_3620_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_3621_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_3622_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_3623_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_3624_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).

% diff_less
thf(fact_3625_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_3626_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_3627_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M @ N ) ) ) ) ).

% less_diff_iff
thf(fact_3628_diff__add__0,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_3629_less__diff__conv,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ).

% less_diff_conv
thf(fact_3630_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_3631_le__diff__conv,axiom,
    ! [J: nat,K: nat,I3: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I3 @ K ) ) ) ).

% le_diff_conv
thf(fact_3632_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I3: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_3633_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I3: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J ) @ K )
        = ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_3634_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I3: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I3 ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I3 ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_3635_Nat_Ole__imp__diff__is__add,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ( ( minus_minus_nat @ J @ I3 )
          = K )
        = ( J
          = ( plus_plus_nat @ K @ I3 ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_3636_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_3637_mod__geq,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( modulo_modulo_nat @ M @ N )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).

% mod_geq
thf(fact_3638_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M4: nat,N3: nat] : ( if_nat @ ( ord_less_nat @ M4 @ N3 ) @ M4 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M4 @ N3 ) @ N3 ) ) ) ) ).

% mod_if
thf(fact_3639_le__mod__geq,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( modulo_modulo_nat @ M @ N )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).

% le_mod_geq
thf(fact_3640_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q2: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( ( modulo_modulo_nat @ M @ Q2 )
          = ( modulo_modulo_nat @ N @ Q2 ) )
        = ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_3641_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va2 @ Vb @ Vc ) ) ).

% VEBT_internal.minNull.simps(5)
thf(fact_3642_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_3643_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_3644_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_3645_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_3646_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_3647_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_3648_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_3649_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_3650_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_3651_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_3652_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_3653_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_3654_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
          = A ) )
      & ( ( Z2 != zero_zero_complex )
       => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_3655_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
          = A ) )
      & ( ( Z2 != zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_3656_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
          = A ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_3657_diff__frac__eq,axiom,
    ! [Y: complex,Z2: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z2 != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z2 ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z2 ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z2 ) ) ) ) ) ).

% diff_frac_eq
thf(fact_3658_diff__frac__eq,axiom,
    ! [Y: real,Z2: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z2 ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) ) ) ) ).

% diff_frac_eq
thf(fact_3659_diff__frac__eq,axiom,
    ! [Y: rat,Z2: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z2 ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) ) ) ) ).

% diff_frac_eq
thf(fact_3660_diff__divide__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z2 ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).

% diff_divide_eq_iff
thf(fact_3661_diff__divide__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z2 ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).

% diff_divide_eq_iff
thf(fact_3662_diff__divide__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z2 ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ).

% diff_divide_eq_iff
thf(fact_3663_divide__diff__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z2 ) @ Y )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% divide_diff_eq_iff
thf(fact_3664_divide__diff__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ X @ Z2 ) @ Y )
        = ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% divide_diff_eq_iff
thf(fact_3665_divide__diff__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z2 ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% divide_diff_eq_iff
thf(fact_3666_square__diff__one__factored,axiom,
    ! [X: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ one_one_complex )
      = ( times_times_complex @ ( plus_plus_complex @ X @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ).

% square_diff_one_factored
thf(fact_3667_square__diff__one__factored,axiom,
    ! [X: real] :
      ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
      = ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).

% square_diff_one_factored
thf(fact_3668_square__diff__one__factored,axiom,
    ! [X: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
      = ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).

% square_diff_one_factored
thf(fact_3669_square__diff__one__factored,axiom,
    ! [X: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
      = ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).

% square_diff_one_factored
thf(fact_3670_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_3671_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_3672_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_3673_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_3674_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_3675_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_3676_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_3677_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_3678_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_3679_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_3680_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_3681_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_3682_diff__Suc__less,axiom,
    ! [N: nat,I3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) @ N ) ) ).

% diff_Suc_less
thf(fact_3683_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 ) )
        & ! [D2: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D2 ) )
           => ( P @ D2 ) ) ) ) ).

% nat_diff_split
thf(fact_3684_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 ) )
            | ? [D2: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D2 ) )
                & ~ ( P @ D2 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_3685_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I3: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I3 @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_3686_nat__eq__add__iff1,axiom,
    ! [J: nat,I3: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I3 )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J ) @ U ) @ M )
          = N ) ) ) ).

% nat_eq_add_iff1
thf(fact_3687_nat__eq__add__iff2,axiom,
    ! [I3: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( M
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I3 ) @ U ) @ N ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_3688_nat__le__add__iff1,axiom,
    ! [J: nat,I3: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I3 )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_le_add_iff1
thf(fact_3689_nat__le__add__iff2,axiom,
    ! [I3: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I3 ) @ U ) @ N ) ) ) ) ).

% nat_le_add_iff2
thf(fact_3690_nat__diff__add__eq1,axiom,
    ! [J: nat,I3: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I3 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_diff_add_eq1
thf(fact_3691_nat__diff__add__eq2,axiom,
    ! [I3: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I3 ) @ U ) @ N ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_3692_dvd__minus__add,axiom,
    ! [Q2: nat,N: nat,R2: nat,M: nat] :
      ( ( ord_less_eq_nat @ Q2 @ N )
     => ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R2 @ M ) )
       => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q2 ) )
          = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q2 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_3693_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X3: real] :
      ( ( ord_less_real @ X3 @ Z4 )
     => ~ ( ord_less_real @ T @ X3 ) ) ).

% minf(7)
thf(fact_3694_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ X3 @ Z4 )
     => ~ ( ord_less_rat @ T @ X3 ) ) ).

% minf(7)
thf(fact_3695_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X3: num] :
      ( ( ord_less_num @ X3 @ Z4 )
     => ~ ( ord_less_num @ T @ X3 ) ) ).

% minf(7)
thf(fact_3696_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z4 )
     => ~ ( ord_less_nat @ T @ X3 ) ) ).

% minf(7)
thf(fact_3697_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ Z4 )
     => ~ ( ord_less_int @ T @ X3 ) ) ).

% minf(7)
thf(fact_3698_minf_I7_J,axiom,
    ! [T: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X3 @ Z4 )
     => ~ ( ord_le72135733267957522d_enat @ T @ X3 ) ) ).

% minf(7)
thf(fact_3699_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X3: real] :
      ( ( ord_less_real @ X3 @ Z4 )
     => ( ord_less_real @ X3 @ T ) ) ).

% minf(5)
thf(fact_3700_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ X3 @ Z4 )
     => ( ord_less_rat @ X3 @ T ) ) ).

% minf(5)
thf(fact_3701_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X3: num] :
      ( ( ord_less_num @ X3 @ Z4 )
     => ( ord_less_num @ X3 @ T ) ) ).

% minf(5)
thf(fact_3702_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z4 )
     => ( ord_less_nat @ X3 @ T ) ) ).

% minf(5)
thf(fact_3703_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ Z4 )
     => ( ord_less_int @ X3 @ T ) ) ).

% minf(5)
thf(fact_3704_minf_I5_J,axiom,
    ! [T: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X3 @ Z4 )
     => ( ord_le72135733267957522d_enat @ X3 @ T ) ) ).

% minf(5)
thf(fact_3705_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X3: real] :
      ( ( ord_less_real @ X3 @ Z4 )
     => ( X3 != T ) ) ).

% minf(4)
thf(fact_3706_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ X3 @ Z4 )
     => ( X3 != T ) ) ).

% minf(4)
thf(fact_3707_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X3: num] :
      ( ( ord_less_num @ X3 @ Z4 )
     => ( X3 != T ) ) ).

% minf(4)
thf(fact_3708_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z4 )
     => ( X3 != T ) ) ).

% minf(4)
thf(fact_3709_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ Z4 )
     => ( X3 != T ) ) ).

% minf(4)
thf(fact_3710_minf_I4_J,axiom,
    ! [T: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X3 @ Z4 )
     => ( X3 != T ) ) ).

% minf(4)
thf(fact_3711_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X3: real] :
      ( ( ord_less_real @ X3 @ Z4 )
     => ( X3 != T ) ) ).

% minf(3)
thf(fact_3712_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ X3 @ Z4 )
     => ( X3 != T ) ) ).

% minf(3)
thf(fact_3713_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X3: num] :
      ( ( ord_less_num @ X3 @ Z4 )
     => ( X3 != T ) ) ).

% minf(3)
thf(fact_3714_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z4 )
     => ( X3 != T ) ) ).

% minf(3)
thf(fact_3715_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ Z4 )
     => ( X3 != T ) ) ).

% minf(3)
thf(fact_3716_minf_I3_J,axiom,
    ! [T: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X3 @ Z4 )
     => ( X3 != T ) ) ).

% minf(3)
thf(fact_3717_minf_I2_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z5: real] :
        ! [X5: real] :
          ( ( ord_less_real @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: real] :
          ! [X5: real] :
            ( ( ord_less_real @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: real] :
          ! [X3: real] :
            ( ( ord_less_real @ X3 @ Z4 )
           => ( ( ( P @ X3 )
                | ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                | ( Q6 @ X3 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3718_minf_I2_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z5: rat] :
        ! [X5: rat] :
          ( ( ord_less_rat @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z4 )
           => ( ( ( P @ X3 )
                | ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                | ( Q6 @ X3 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3719_minf_I2_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z5: num] :
        ! [X5: num] :
          ( ( ord_less_num @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: num] :
          ! [X5: num] :
            ( ( ord_less_num @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z4 )
           => ( ( ( P @ X3 )
                | ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                | ( Q6 @ X3 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3720_minf_I2_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z5: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z4 )
           => ( ( ( P @ X3 )
                | ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                | ( Q6 @ X3 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3721_minf_I2_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z5: int] :
        ! [X5: int] :
          ( ( ord_less_int @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z4 )
           => ( ( ( P @ X3 )
                | ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                | ( Q6 @ X3 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3722_minf_I2_J,axiom,
    ! [P: extended_enat > $o,P6: extended_enat > $o,Q: extended_enat > $o,Q6: extended_enat > $o] :
      ( ? [Z5: extended_enat] :
        ! [X5: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: extended_enat] :
          ! [X5: extended_enat] :
            ( ( ord_le72135733267957522d_enat @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: extended_enat] :
          ! [X3: extended_enat] :
            ( ( ord_le72135733267957522d_enat @ X3 @ Z4 )
           => ( ( ( P @ X3 )
                | ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                | ( Q6 @ X3 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3723_minf_I1_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z5: real] :
        ! [X5: real] :
          ( ( ord_less_real @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: real] :
          ! [X5: real] :
            ( ( ord_less_real @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: real] :
          ! [X3: real] :
            ( ( ord_less_real @ X3 @ Z4 )
           => ( ( ( P @ X3 )
                & ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                & ( Q6 @ X3 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3724_minf_I1_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z5: rat] :
        ! [X5: rat] :
          ( ( ord_less_rat @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z4 )
           => ( ( ( P @ X3 )
                & ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                & ( Q6 @ X3 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3725_minf_I1_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z5: num] :
        ! [X5: num] :
          ( ( ord_less_num @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: num] :
          ! [X5: num] :
            ( ( ord_less_num @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z4 )
           => ( ( ( P @ X3 )
                & ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                & ( Q6 @ X3 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3726_minf_I1_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z5: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z4 )
           => ( ( ( P @ X3 )
                & ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                & ( Q6 @ X3 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3727_minf_I1_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z5: int] :
        ! [X5: int] :
          ( ( ord_less_int @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z4 )
           => ( ( ( P @ X3 )
                & ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                & ( Q6 @ X3 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3728_minf_I1_J,axiom,
    ! [P: extended_enat > $o,P6: extended_enat > $o,Q: extended_enat > $o,Q6: extended_enat > $o] :
      ( ? [Z5: extended_enat] :
        ! [X5: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ X5 @ Z5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: extended_enat] :
          ! [X5: extended_enat] :
            ( ( ord_le72135733267957522d_enat @ X5 @ Z5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: extended_enat] :
          ! [X3: extended_enat] :
            ( ( ord_le72135733267957522d_enat @ X3 @ Z4 )
           => ( ( ( P @ X3 )
                & ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                & ( Q6 @ X3 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3729_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X3: real] :
      ( ( ord_less_real @ Z4 @ X3 )
     => ( ord_less_real @ T @ X3 ) ) ).

% pinf(7)
thf(fact_3730_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ Z4 @ X3 )
     => ( ord_less_rat @ T @ X3 ) ) ).

% pinf(7)
thf(fact_3731_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X3: num] :
      ( ( ord_less_num @ Z4 @ X3 )
     => ( ord_less_num @ T @ X3 ) ) ).

% pinf(7)
thf(fact_3732_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z4 @ X3 )
     => ( ord_less_nat @ T @ X3 ) ) ).

% pinf(7)
thf(fact_3733_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X3: int] :
      ( ( ord_less_int @ Z4 @ X3 )
     => ( ord_less_int @ T @ X3 ) ) ).

% pinf(7)
thf(fact_3734_pinf_I7_J,axiom,
    ! [T: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z4 @ X3 )
     => ( ord_le72135733267957522d_enat @ T @ X3 ) ) ).

% pinf(7)
thf(fact_3735_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X3: real] :
      ( ( ord_less_real @ Z4 @ X3 )
     => ~ ( ord_less_real @ X3 @ T ) ) ).

% pinf(5)
thf(fact_3736_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ Z4 @ X3 )
     => ~ ( ord_less_rat @ X3 @ T ) ) ).

% pinf(5)
thf(fact_3737_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X3: num] :
      ( ( ord_less_num @ Z4 @ X3 )
     => ~ ( ord_less_num @ X3 @ T ) ) ).

% pinf(5)
thf(fact_3738_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z4 @ X3 )
     => ~ ( ord_less_nat @ X3 @ T ) ) ).

% pinf(5)
thf(fact_3739_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X3: int] :
      ( ( ord_less_int @ Z4 @ X3 )
     => ~ ( ord_less_int @ X3 @ T ) ) ).

% pinf(5)
thf(fact_3740_pinf_I5_J,axiom,
    ! [T: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z4 @ X3 )
     => ~ ( ord_le72135733267957522d_enat @ X3 @ T ) ) ).

% pinf(5)
thf(fact_3741_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X3: real] :
      ( ( ord_less_real @ Z4 @ X3 )
     => ( X3 != T ) ) ).

% pinf(4)
thf(fact_3742_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ Z4 @ X3 )
     => ( X3 != T ) ) ).

% pinf(4)
thf(fact_3743_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X3: num] :
      ( ( ord_less_num @ Z4 @ X3 )
     => ( X3 != T ) ) ).

% pinf(4)
thf(fact_3744_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z4 @ X3 )
     => ( X3 != T ) ) ).

% pinf(4)
thf(fact_3745_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X3: int] :
      ( ( ord_less_int @ Z4 @ X3 )
     => ( X3 != T ) ) ).

% pinf(4)
thf(fact_3746_pinf_I4_J,axiom,
    ! [T: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z4 @ X3 )
     => ( X3 != T ) ) ).

% pinf(4)
thf(fact_3747_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X3: real] :
      ( ( ord_less_real @ Z4 @ X3 )
     => ( X3 != T ) ) ).

% pinf(3)
thf(fact_3748_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ Z4 @ X3 )
     => ( X3 != T ) ) ).

% pinf(3)
thf(fact_3749_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X3: num] :
      ( ( ord_less_num @ Z4 @ X3 )
     => ( X3 != T ) ) ).

% pinf(3)
thf(fact_3750_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z4 @ X3 )
     => ( X3 != T ) ) ).

% pinf(3)
thf(fact_3751_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X3: int] :
      ( ( ord_less_int @ Z4 @ X3 )
     => ( X3 != T ) ) ).

% pinf(3)
thf(fact_3752_pinf_I3_J,axiom,
    ! [T: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z4 @ X3 )
     => ( X3 != T ) ) ).

% pinf(3)
thf(fact_3753_pinf_I2_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z5: real] :
        ! [X5: real] :
          ( ( ord_less_real @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: real] :
          ! [X5: real] :
            ( ( ord_less_real @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: real] :
          ! [X3: real] :
            ( ( ord_less_real @ Z4 @ X3 )
           => ( ( ( P @ X3 )
                | ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                | ( Q6 @ X3 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3754_pinf_I2_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z5: rat] :
        ! [X5: rat] :
          ( ( ord_less_rat @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z4 @ X3 )
           => ( ( ( P @ X3 )
                | ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                | ( Q6 @ X3 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3755_pinf_I2_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z5: num] :
        ! [X5: num] :
          ( ( ord_less_num @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: num] :
          ! [X5: num] :
            ( ( ord_less_num @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z4 @ X3 )
           => ( ( ( P @ X3 )
                | ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                | ( Q6 @ X3 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3756_pinf_I2_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z5: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z4 @ X3 )
           => ( ( ( P @ X3 )
                | ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                | ( Q6 @ X3 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3757_pinf_I2_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z5: int] :
        ! [X5: int] :
          ( ( ord_less_int @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z4 @ X3 )
           => ( ( ( P @ X3 )
                | ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                | ( Q6 @ X3 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3758_pinf_I2_J,axiom,
    ! [P: extended_enat > $o,P6: extended_enat > $o,Q: extended_enat > $o,Q6: extended_enat > $o] :
      ( ? [Z5: extended_enat] :
        ! [X5: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: extended_enat] :
          ! [X5: extended_enat] :
            ( ( ord_le72135733267957522d_enat @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: extended_enat] :
          ! [X3: extended_enat] :
            ( ( ord_le72135733267957522d_enat @ Z4 @ X3 )
           => ( ( ( P @ X3 )
                | ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                | ( Q6 @ X3 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3759_pinf_I1_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z5: real] :
        ! [X5: real] :
          ( ( ord_less_real @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: real] :
          ! [X5: real] :
            ( ( ord_less_real @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: real] :
          ! [X3: real] :
            ( ( ord_less_real @ Z4 @ X3 )
           => ( ( ( P @ X3 )
                & ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                & ( Q6 @ X3 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3760_pinf_I1_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z5: rat] :
        ! [X5: rat] :
          ( ( ord_less_rat @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z4 @ X3 )
           => ( ( ( P @ X3 )
                & ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                & ( Q6 @ X3 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3761_pinf_I1_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z5: num] :
        ! [X5: num] :
          ( ( ord_less_num @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: num] :
          ! [X5: num] :
            ( ( ord_less_num @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z4 @ X3 )
           => ( ( ( P @ X3 )
                & ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                & ( Q6 @ X3 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3762_pinf_I1_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z5: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z4 @ X3 )
           => ( ( ( P @ X3 )
                & ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                & ( Q6 @ X3 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3763_pinf_I1_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z5: int] :
        ! [X5: int] :
          ( ( ord_less_int @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z4 @ X3 )
           => ( ( ( P @ X3 )
                & ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                & ( Q6 @ X3 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3764_pinf_I1_J,axiom,
    ! [P: extended_enat > $o,P6: extended_enat > $o,Q: extended_enat > $o,Q6: extended_enat > $o] :
      ( ? [Z5: extended_enat] :
        ! [X5: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ Z5 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z5: extended_enat] :
          ! [X5: extended_enat] :
            ( ( ord_le72135733267957522d_enat @ Z5 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z4: extended_enat] :
          ! [X3: extended_enat] :
            ( ( ord_le72135733267957522d_enat @ Z4 @ X3 )
           => ( ( ( P @ X3 )
                & ( Q @ X3 ) )
              = ( ( P6 @ X3 )
                & ( Q6 @ X3 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3765_mod__nat__eqI,axiom,
    ! [R2: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ R2 @ N )
     => ( ( ord_less_eq_nat @ R2 @ M )
       => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R2 ) )
         => ( ( modulo_modulo_nat @ M @ N )
            = R2 ) ) ) ) ).

% mod_nat_eqI
thf(fact_3766_VEBT__internal_OminNull_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ( X
       != ( vEBT_Leaf @ $false @ $false ) )
     => ( ! [Uv2: $o] :
            ( X
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.cases
thf(fact_3767_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X ) ).

% vebt_member.simps(3)
thf(fact_3768_modulo__nat__def,axiom,
    ( modulo_modulo_nat
    = ( ^ [M4: nat,N3: nat] : ( minus_minus_nat @ M4 @ ( times_times_nat @ ( divide_divide_nat @ M4 @ N3 ) @ N3 ) ) ) ) ).

% modulo_nat_def
thf(fact_3769_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X )
     => ( ! [Uv2: $o] :
            ( X
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).

% VEBT_internal.minNull.elims(3)
thf(fact_3770_conj__le__cong,axiom,
    ! [X: int,X8: int,P: $o,P6: $o] :
      ( ( X = X8 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
         => ( P = P6 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
            & P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X8 )
            & P6 ) ) ) ) ).

% conj_le_cong
thf(fact_3771_imp__le__cong,axiom,
    ! [X: int,X8: int,P: $o,P6: $o] :
      ( ( X = X8 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
         => ( P = P6 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
           => P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X8 )
           => P6 ) ) ) ) ).

% imp_le_cong
thf(fact_3772_frac__le__eq,axiom,
    ! [Y: real,Z2: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z2 ) )
          = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) @ zero_zero_real ) ) ) ) ).

% frac_le_eq
thf(fact_3773_frac__le__eq,axiom,
    ! [Y: rat,Z2: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z2 ) )
          = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).

% frac_le_eq
thf(fact_3774_frac__less__eq,axiom,
    ! [Y: real,Z2: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z2 ) )
          = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z2 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) @ zero_zero_real ) ) ) ) ).

% frac_less_eq
thf(fact_3775_frac__less__eq,axiom,
    ! [Y: rat,Z2: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z2 ) )
          = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z2 ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).

% frac_less_eq
thf(fact_3776_power2__commute,axiom,
    ! [X: complex,Y: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_complex @ ( minus_minus_complex @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_3777_power2__commute,axiom,
    ! [X: real,Y: real] :
      ( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ ( minus_minus_real @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_3778_power2__commute,axiom,
    ! [X: rat,Y: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ ( minus_minus_rat @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_3779_power2__commute,axiom,
    ! [X: int,Y: int] :
      ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ ( minus_minus_int @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_3780_power__diff,axiom,
    ! [A: complex,N: nat,M: nat] :
      ( ( A != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_3781_power__diff,axiom,
    ! [A: real,N: nat,M: nat] :
      ( ( A != zero_zero_real )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_3782_power__diff,axiom,
    ! [A: rat,N: nat,M: nat] :
      ( ( A != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_3783_power__diff,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_3784_power__diff,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_3785_div__geq,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( ( divide_divide_nat @ M @ N )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).

% div_geq
thf(fact_3786_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M4: nat,N3: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M4 @ N3 )
            | ( N3 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M4 @ N3 ) @ N3 ) ) ) ) ) ).

% div_if
thf(fact_3787_Suc__pred_H,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( N
        = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_3788_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_3789_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M4: nat,N3: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N3 ) ) ) ) ) ).

% add_eq_if
thf(fact_3790_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X ) ).

% vebt_member.simps(4)
thf(fact_3791_nat__less__add__iff1,axiom,
    ! [J: nat,I3: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I3 )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I3 @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_less_add_iff1
thf(fact_3792_nat__less__add__iff2,axiom,
    ! [I3: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I3 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I3 ) @ U ) @ N ) ) ) ) ).

% nat_less_add_iff2
thf(fact_3793_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M4: nat,N3: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N3 ) ) ) ) ) ).

% mult_eq_if
thf(fact_3794_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: $o] :
      ( ( ( vEBT_VEBT_minNull @ X )
        = Y )
     => ( ( ( X
            = ( vEBT_Leaf @ $false @ $false ) )
         => ~ Y )
       => ( ( ? [Uv2: $o] :
                ( X
                = ( vEBT_Leaf @ $true @ Uv2 ) )
           => Y )
         => ( ( ? [Uu2: $o] :
                  ( X
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
             => Y )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ~ Y )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                 => Y ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
thf(fact_3795_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S: real] :
      ( ( ord_less_eq_real @ U @ V )
     => ( ( ord_less_eq_real @ zero_zero_real @ R2 )
       => ( ( ord_less_eq_real @ R2 @ S )
         => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_3796_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S: rat] :
      ( ( ord_less_eq_rat @ U @ V )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
       => ( ( ord_less_eq_rat @ R2 @ S )
         => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_3797_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
       != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_3798_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
       != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_3799_power__diff__power__eq,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
            = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
            = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3800_power__diff__power__eq,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
            = ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
            = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3801_power__eq__if,axiom,
    ( power_power_complex
    = ( ^ [P5: complex,M4: nat] : ( if_complex @ ( M4 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P5 @ ( power_power_complex @ P5 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3802_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P5: real,M4: nat] : ( if_real @ ( M4 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P5 @ ( power_power_real @ P5 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3803_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P5: rat,M4: nat] : ( if_rat @ ( M4 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P5 @ ( power_power_rat @ P5 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3804_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P5: nat,M4: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3805_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P5: int,M4: nat] : ( if_int @ ( M4 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3806_power__minus__mult,axiom,
    ! [N: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_complex @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_3807_power__minus__mult,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_real @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_3808_power__minus__mult,axiom,
    ! [N: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_rat @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_3809_power__minus__mult,axiom,
    ! [N: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_nat @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_3810_power__minus__mult,axiom,
    ! [N: nat,A: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_int @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_3811_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).

% diff_le_diff_pow
thf(fact_3812_le__div__geq,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( divide_divide_nat @ M @ N )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).

% le_div_geq
thf(fact_3813_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( suc @ zero_zero_nat ) )
     => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_mod_4_div_2
thf(fact_3814_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X3: real] :
      ( ( ord_less_real @ X3 @ Z4 )
     => ~ ( ord_less_eq_real @ T @ X3 ) ) ).

% minf(8)
thf(fact_3815_minf_I8_J,axiom,
    ! [T: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X3 @ Z4 )
     => ~ ( ord_le2932123472753598470d_enat @ T @ X3 ) ) ).

% minf(8)
thf(fact_3816_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ X3 @ Z4 )
     => ~ ( ord_less_eq_rat @ T @ X3 ) ) ).

% minf(8)
thf(fact_3817_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X3: num] :
      ( ( ord_less_num @ X3 @ Z4 )
     => ~ ( ord_less_eq_num @ T @ X3 ) ) ).

% minf(8)
thf(fact_3818_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z4 )
     => ~ ( ord_less_eq_nat @ T @ X3 ) ) ).

% minf(8)
thf(fact_3819_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ Z4 )
     => ~ ( ord_less_eq_int @ T @ X3 ) ) ).

% minf(8)
thf(fact_3820_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X3: real] :
      ( ( ord_less_real @ X3 @ Z4 )
     => ( ord_less_eq_real @ X3 @ T ) ) ).

% minf(6)
thf(fact_3821_minf_I6_J,axiom,
    ! [T: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X3 @ Z4 )
     => ( ord_le2932123472753598470d_enat @ X3 @ T ) ) ).

% minf(6)
thf(fact_3822_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ X3 @ Z4 )
     => ( ord_less_eq_rat @ X3 @ T ) ) ).

% minf(6)
thf(fact_3823_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X3: num] :
      ( ( ord_less_num @ X3 @ Z4 )
     => ( ord_less_eq_num @ X3 @ T ) ) ).

% minf(6)
thf(fact_3824_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z4 )
     => ( ord_less_eq_nat @ X3 @ T ) ) ).

% minf(6)
thf(fact_3825_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ Z4 )
     => ( ord_less_eq_int @ X3 @ T ) ) ).

% minf(6)
thf(fact_3826_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X3: real] :
      ( ( ord_less_real @ Z4 @ X3 )
     => ( ord_less_eq_real @ T @ X3 ) ) ).

% pinf(8)
thf(fact_3827_pinf_I8_J,axiom,
    ! [T: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z4 @ X3 )
     => ( ord_le2932123472753598470d_enat @ T @ X3 ) ) ).

% pinf(8)
thf(fact_3828_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ Z4 @ X3 )
     => ( ord_less_eq_rat @ T @ X3 ) ) ).

% pinf(8)
thf(fact_3829_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X3: num] :
      ( ( ord_less_num @ Z4 @ X3 )
     => ( ord_less_eq_num @ T @ X3 ) ) ).

% pinf(8)
thf(fact_3830_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z4 @ X3 )
     => ( ord_less_eq_nat @ T @ X3 ) ) ).

% pinf(8)
thf(fact_3831_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X3: int] :
      ( ( ord_less_int @ Z4 @ X3 )
     => ( ord_less_eq_int @ T @ X3 ) ) ).

% pinf(8)
thf(fact_3832_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z4: real] :
    ! [X3: real] :
      ( ( ord_less_real @ Z4 @ X3 )
     => ~ ( ord_less_eq_real @ X3 @ T ) ) ).

% pinf(6)
thf(fact_3833_pinf_I6_J,axiom,
    ! [T: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z4 @ X3 )
     => ~ ( ord_le2932123472753598470d_enat @ X3 @ T ) ) ).

% pinf(6)
thf(fact_3834_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z4: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ Z4 @ X3 )
     => ~ ( ord_less_eq_rat @ X3 @ T ) ) ).

% pinf(6)
thf(fact_3835_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z4: num] :
    ! [X3: num] :
      ( ( ord_less_num @ Z4 @ X3 )
     => ~ ( ord_less_eq_num @ X3 @ T ) ) ).

% pinf(6)
thf(fact_3836_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z4: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z4 @ X3 )
     => ~ ( ord_less_eq_nat @ X3 @ T ) ) ).

% pinf(6)
thf(fact_3837_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z4: int] :
    ! [X3: int] :
      ( ( ord_less_int @ Z4 @ X3 )
     => ~ ( ord_less_eq_int @ X3 @ T ) ) ).

% pinf(6)
thf(fact_3838_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N2: nat] :
            ( X
           != ( suc @ N2 ) ) ) ).

% list_decode.cases
thf(fact_3839_power2__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_3840_power2__diff,axiom,
    ! [X: real,Y: real] :
      ( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_3841_power2__diff,axiom,
    ! [X: int,Y: int] :
      ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_3842_power2__diff,axiom,
    ! [X: rat,Y: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( 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 ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_3843_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
        = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_3844_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
        = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_3845_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
        = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_3846_zdvd__mono,axiom,
    ! [K: int,M: int,T: int] :
      ( ( K != zero_zero_int )
     => ( ( dvd_dvd_int @ M @ T )
        = ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).

% zdvd_mono
thf(fact_3847_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_3848_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_3849_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_3850_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_3851_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% even_mask_div_iff'
thf(fact_3852_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% even_mask_div_iff'
thf(fact_3853_even__mask__div__iff_H,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% even_mask_div_iff'
thf(fact_3854_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
          = zero_z3403309356797280102nteger )
        | ( ord_less_eq_nat @ M @ N ) ) ) ).

% even_mask_div_iff
thf(fact_3855_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          = zero_zero_nat )
        | ( ord_less_eq_nat @ M @ N ) ) ) ).

% even_mask_div_iff
thf(fact_3856_even__mask__div__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
          = zero_zero_int )
        | ( ord_less_eq_nat @ M @ N ) ) ) ).

% even_mask_div_iff
thf(fact_3857_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_nat
        @ ( zero_n2687167440665602831ol_nat
          @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
             != zero_zero_nat )
            & ( ord_less_eq_nat @ N @ M ) ) )
        @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_3858_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_int
        @ ( zero_n2684676970156552555ol_int
          @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
             != zero_zero_int )
            & ( ord_less_eq_nat @ N @ M ) ) )
        @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_3859_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( times_3573771949741848930nteger
        @ ( zero_n356916108424825756nteger
          @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
             != zero_z3403309356797280102nteger )
            & ( ord_less_eq_nat @ N @ M ) ) )
        @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_3860_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M )
        | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
          = zero_z3403309356797280102nteger )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_3861_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M )
        | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          = zero_zero_nat )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_3862_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M )
        | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
          = zero_zero_int )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_3863_minf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z4: code_integer] :
    ! [X3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X3 @ Z4 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3864_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z4: real] :
    ! [X3: real] :
      ( ( ord_less_real @ X3 @ Z4 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3865_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z4: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ X3 @ Z4 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3866_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z4: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z4 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3867_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z4: int] :
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ Z4 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3868_minf_I10_J,axiom,
    ! [D: extended_enat,S: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X3 @ Z4 )
     => ( ( ~ ( dvd_dv3785147216227455552d_enat @ D @ ( plus_p3455044024723400733d_enat @ X3 @ S ) ) )
        = ( ~ ( dvd_dv3785147216227455552d_enat @ D @ ( plus_p3455044024723400733d_enat @ X3 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3869_minf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z4: code_integer] :
    ! [X3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X3 @ Z4 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S ) ) ) ) ).

% minf(9)
thf(fact_3870_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z4: real] :
    ! [X3: real] :
      ( ( ord_less_real @ X3 @ Z4 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) ) ) ).

% minf(9)
thf(fact_3871_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z4: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ X3 @ Z4 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) ) ) ) ).

% minf(9)
thf(fact_3872_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z4: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ X3 @ Z4 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) ) ) ).

% minf(9)
thf(fact_3873_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z4: int] :
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ Z4 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) ) ) ).

% minf(9)
thf(fact_3874_minf_I9_J,axiom,
    ! [D: extended_enat,S: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X3 @ Z4 )
     => ( ( dvd_dv3785147216227455552d_enat @ D @ ( plus_p3455044024723400733d_enat @ X3 @ S ) )
        = ( dvd_dv3785147216227455552d_enat @ D @ ( plus_p3455044024723400733d_enat @ X3 @ S ) ) ) ) ).

% minf(9)
thf(fact_3875_pinf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z4: code_integer] :
    ! [X3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z4 @ X3 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3876_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z4: real] :
    ! [X3: real] :
      ( ( ord_less_real @ Z4 @ X3 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3877_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z4: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ Z4 @ X3 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3878_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z4: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z4 @ X3 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3879_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z4: int] :
    ! [X3: int] :
      ( ( ord_less_int @ Z4 @ X3 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3880_pinf_I10_J,axiom,
    ! [D: extended_enat,S: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z4 @ X3 )
     => ( ( ~ ( dvd_dv3785147216227455552d_enat @ D @ ( plus_p3455044024723400733d_enat @ X3 @ S ) ) )
        = ( ~ ( dvd_dv3785147216227455552d_enat @ D @ ( plus_p3455044024723400733d_enat @ X3 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3881_pinf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z4: code_integer] :
    ! [X3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z4 @ X3 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X3 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3882_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z4: real] :
    ! [X3: real] :
      ( ( ord_less_real @ Z4 @ X3 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X3 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3883_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z4: rat] :
    ! [X3: rat] :
      ( ( ord_less_rat @ Z4 @ X3 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X3 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3884_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z4: nat] :
    ! [X3: nat] :
      ( ( ord_less_nat @ Z4 @ X3 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X3 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3885_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z4: int] :
    ! [X3: int] :
      ( ( ord_less_int @ Z4 @ X3 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X3 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3886_pinf_I9_J,axiom,
    ! [D: extended_enat,S: extended_enat] :
    ? [Z4: extended_enat] :
    ! [X3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z4 @ X3 )
     => ( ( dvd_dv3785147216227455552d_enat @ D @ ( plus_p3455044024723400733d_enat @ X3 @ S ) )
        = ( dvd_dv3785147216227455552d_enat @ D @ ( plus_p3455044024723400733d_enat @ X3 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3887_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
thf(fact_3888_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
        & ( ( X = Mi )
          | ( X = Ma )
          | ( ( ord_less_nat @ X @ Ma )
            & ( ord_less_nat @ Mi @ X )
            & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% member_inv
thf(fact_3889_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ Deg )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          | ( X = Mi )
          | ( X = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
thf(fact_3890_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
     => ( ( Mi != Ma )
       => ( ( ord_less_nat @ Mi @ Ma )
          & ? [M3: nat] :
              ( ( ( some_nat @ M3 )
                = ( vEBT_vebt_mint @ Summary ) )
              & ( ord_less_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% mintlistlength
thf(fact_3891_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = none_nat ) ) ) ) ).

% succ_list_to_short
thf(fact_3892_pred__list__to__short,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X @ Ma )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = none_nat ) ) ) ) ).

% pred_list_to_short
thf(fact_3893_vebt__pred_Osimps_I3_J,axiom,
    ! [B: $o,A: $o,Va2: nat] :
      ( ( B
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( A
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) )
              = none_nat ) ) ) ) ) ).

% vebt_pred.simps(3)
thf(fact_3894_vebt__succ_Osimps_I1_J,axiom,
    ! [B: $o,Uu: $o] :
      ( ( B
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
          = none_nat ) ) ) ).

% vebt_succ.simps(1)
thf(fact_3895_Diff__empty,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
      = A2 ) ).

% Diff_empty
thf(fact_3896_Diff__empty,axiom,
    ! [A2: set_o] :
      ( ( minus_minus_set_o @ A2 @ bot_bot_set_o )
      = A2 ) ).

% Diff_empty
thf(fact_3897_Diff__empty,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
      = A2 ) ).

% Diff_empty
thf(fact_3898_Diff__empty,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
      = A2 ) ).

% Diff_empty
thf(fact_3899_empty__Diff,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
      = bot_bot_set_real ) ).

% empty_Diff
thf(fact_3900_empty__Diff,axiom,
    ! [A2: set_o] :
      ( ( minus_minus_set_o @ bot_bot_set_o @ A2 )
      = bot_bot_set_o ) ).

% empty_Diff
thf(fact_3901_empty__Diff,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
      = bot_bot_set_int ) ).

% empty_Diff
thf(fact_3902_empty__Diff,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_3903_Diff__cancel,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ A2 @ A2 )
      = bot_bot_set_real ) ).

% Diff_cancel
thf(fact_3904_Diff__cancel,axiom,
    ! [A2: set_o] :
      ( ( minus_minus_set_o @ A2 @ A2 )
      = bot_bot_set_o ) ).

% Diff_cancel
thf(fact_3905_Diff__cancel,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ A2 @ A2 )
      = bot_bot_set_int ) ).

% Diff_cancel
thf(fact_3906_Diff__cancel,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ A2 )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_3907_idiff__0__right,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
      = N ) ).

% idiff_0_right
thf(fact_3908_idiff__0,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
      = zero_z5237406670263579293d_enat ) ).

% idiff_0
thf(fact_3909_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
     => ( ( Mi = Ma )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
             => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
          & ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 ) ) ) ) ).

% mi_eq_ma_no_ch
thf(fact_3910_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
     => ( ( ord_less_eq_nat @ Ma @ X )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = none_nat ) ) ) ).

% geqmaxNone
thf(fact_3911_insert__simp__mima,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        | ( X = Ma ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% insert_simp_mima
thf(fact_3912_Diff__eq__empty__iff,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( ( minus_minus_set_real @ A2 @ B2 )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ A2 @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_3913_Diff__eq__empty__iff,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( ( minus_minus_set_o @ A2 @ B2 )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ A2 @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_3914_Diff__eq__empty__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ( minus_minus_set_int @ A2 @ B2 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A2 @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_3915_Diff__eq__empty__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ( minus_minus_set_nat @ A2 @ B2 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_3916_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
     => ( ( ord_less_eq_nat @ Mi @ Ma )
        & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).

% mi_ma_2_deg
thf(fact_3917_succ__min,axiom,
    ! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( some_nat @ Mi ) ) ) ) ).

% succ_min
thf(fact_3918_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( some_nat @ Ma ) ) ) ) ).

% pred_max
thf(fact_3919_Diff__mono,axiom,
    ! [A2: set_nat,C5: set_nat,D4: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C5 )
     => ( ( ord_less_eq_set_nat @ D4 @ B2 )
       => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ C5 @ D4 ) ) ) ) ).

% Diff_mono
thf(fact_3920_Diff__subset,axiom,
    ! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ A2 ) ).

% Diff_subset
thf(fact_3921_double__diff,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C5 )
       => ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C5 @ A2 ) )
          = A2 ) ) ) ).

% double_diff
thf(fact_3922_minus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( minus_minus_int @ K @ zero_zero_int )
      = K ) ).

% minus_int_code(1)
thf(fact_3923_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
      = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).

% int_distrib(4)
thf(fact_3924_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] :
      ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
      = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).

% int_distrib(3)
thf(fact_3925_psubset__imp__ex__mem,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ( ord_less_set_complex @ A2 @ B2 )
     => ? [B5: complex] : ( member_complex @ B5 @ ( minus_811609699411566653omplex @ B2 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_3926_psubset__imp__ex__mem,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( ord_less_set_real @ A2 @ B2 )
     => ? [B5: real] : ( member_real @ B5 @ ( minus_minus_set_real @ B2 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_3927_psubset__imp__ex__mem,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ( ord_less_set_set_nat @ A2 @ B2 )
     => ? [B5: set_nat] : ( member_set_nat @ B5 @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_3928_psubset__imp__ex__mem,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ord_less_set_int @ A2 @ B2 )
     => ? [B5: int] : ( member_int @ B5 @ ( minus_minus_set_int @ B2 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_3929_psubset__imp__ex__mem,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ? [B5: nat] : ( member_nat @ B5 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_3930_signed__take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).

% signed_take_bit_diff
thf(fact_3931_add__diff__assoc__enat,axiom,
    ! [Z2: extended_enat,Y: extended_enat,X: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Z2 @ Y )
     => ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y @ Z2 ) )
        = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ Z2 ) ) ) ).

% add_diff_assoc_enat
thf(fact_3932_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 ) ) ) )
       => ( ? [Z5: int] :
            ! [X5: int] :
              ( ( ord_less_int @ X5 @ Z5 )
             => ( ( P @ X5 )
                = ( P1 @ X5 ) ) )
         => ( ? [X_1: int] : ( P1 @ X_1 )
           => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).

% minusinfinity
thf(fact_3933_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 ) ) ) )
       => ( ? [Z5: int] :
            ! [X5: int] :
              ( ( ord_less_int @ Z5 @ X5 )
             => ( ( P @ X5 )
                = ( P6 @ X5 ) ) )
         => ( ? [X_1: int] : ( P6 @ X_1 )
           => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).

% plusinfinity
thf(fact_3934_vebt__mint_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A5: $o,B5: $o] :
          ( X
         != ( vEBT_Leaf @ A5 @ B5 ) )
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
            ( X
           != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).

% vebt_mint.cases
thf(fact_3935_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va2 @ Vb ) @ X )
      = ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
thf(fact_3936_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( some_nat @ Mi ) ) ).

% vebt_mint.simps(3)
thf(fact_3937_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( some_nat @ Ma ) ) ).

% vebt_maxt.simps(3)
thf(fact_3938_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 )
         => ! [X3: int] :
              ( ( P @ X3 )
             => ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_3939_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_3940_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) ) ).

% vebt_insert.simps(4)
thf(fact_3941_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_3942_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_3943_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
     => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).

% signed_take_bit_int_less_eq
thf(fact_3944_vebt__mint_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X )
        = Y )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ~ ( ( A5
                 => ( Y
                    = ( some_nat @ zero_zero_nat ) ) )
                & ( ~ A5
                 => ( ( B5
                     => ( Y
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B5
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y
                 != ( some_nat @ Mi2 ) ) ) ) ) ) ).

% vebt_mint.elims
thf(fact_3945_vebt__maxt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X )
        = Y )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ~ ( ( B5
                 => ( Y
                    = ( some_nat @ one_one_nat ) ) )
                & ( ~ B5
                 => ( ( A5
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A5
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y
                 != ( some_nat @ Ma2 ) ) ) ) ) ) ).

% vebt_maxt.elims
thf(fact_3946_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_3947_is__succ__in__set__def,axiom,
    ( vEBT_is_succ_in_set
    = ( ^ [Xs: set_nat,X6: nat,Y6: nat] :
          ( ( member_nat @ Y6 @ Xs )
          & ( ord_less_nat @ X6 @ Y6 )
          & ! [Z3: nat] :
              ( ( member_nat @ Z3 @ Xs )
             => ( ( ord_less_nat @ X6 @ Z3 )
               => ( ord_less_eq_nat @ Y6 @ Z3 ) ) ) ) ) ) ).

% is_succ_in_set_def
thf(fact_3948_is__pred__in__set__def,axiom,
    ( vEBT_is_pred_in_set
    = ( ^ [Xs: set_nat,X6: nat,Y6: nat] :
          ( ( member_nat @ Y6 @ Xs )
          & ( ord_less_nat @ Y6 @ X6 )
          & ! [Z3: nat] :
              ( ( member_nat @ Z3 @ Xs )
             => ( ( ord_less_nat @ Z3 @ X6 )
               => ( ord_less_eq_nat @ Z3 @ Y6 ) ) ) ) ) ) ).

% is_pred_in_set_def
thf(fact_3949_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
      = none_nat ) ).

% vebt_succ.simps(4)
thf(fact_3950_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf )
      = none_nat ) ).

% vebt_pred.simps(5)
thf(fact_3951_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,N: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) )
      = none_nat ) ).

% vebt_succ.simps(2)
thf(fact_3952_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] :
      ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat )
      = none_nat ) ).

% vebt_pred.simps(1)
thf(fact_3953_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va2 ) @ Vb )
      = none_nat ) ).

% vebt_pred.simps(4)
thf(fact_3954_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va2: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va2 )
      = none_nat ) ).

% vebt_succ.simps(3)
thf(fact_3955_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X4 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ~ ? [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 )
                         => ! [I2: nat] :
                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
                                    = I2 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
                                & ! [X5: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X5 @ N )
                                        = I2 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
                                   => ( ( 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 @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
thf(fact_3956_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N ) )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X4 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ~ ? [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 )
                         => ! [I2: nat] :
                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
                                    = I2 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
                                & ! [X5: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X5 @ N )
                                        = I2 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
                                   => ( ( 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 @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
thf(fact_3957_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
      = none_nat ) ).

% vebt_succ.simps(5)
thf(fact_3958_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
      = none_nat ) ).

% vebt_pred.simps(6)
thf(fact_3959_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( ( vEBT_invar_vebt @ A1 @ A22 )
     => ( ( ? [A5: $o,B5: $o] :
              ( A1
              = ( vEBT_Leaf @ A5 @ B5 ) )
         => ( A22
           != ( suc @ zero_zero_nat ) ) )
       => ( ! [TreeList4: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,M3: nat,Deg2: nat] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList4 @ Summary3 ) )
             => ( ( A22 = Deg2 )
               => ( ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                     => ( vEBT_invar_vebt @ X3 @ N2 ) )
                 => ( ( vEBT_invar_vebt @ Summary3 @ M3 )
                   => ( ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                     => ( ( M3 = N2 )
                       => ( ( Deg2
                            = ( plus_plus_nat @ N2 @ M3 ) )
                         => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X_1 )
                           => ~ ! [X3: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                                 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList4: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,M3: nat,Deg2: nat] :
                ( ( A1
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList4 @ Summary3 ) )
               => ( ( A22 = Deg2 )
                 => ( ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                       => ( vEBT_invar_vebt @ X3 @ N2 ) )
                   => ( ( vEBT_invar_vebt @ Summary3 @ M3 )
                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                       => ( ( M3
                            = ( suc @ N2 ) )
                         => ( ( Deg2
                              = ( plus_plus_nat @ N2 @ M3 ) )
                           => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X_1 )
                             => ~ ! [X3: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                                   => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList4: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList4 @ Summary3 ) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                         => ( vEBT_invar_vebt @ X3 @ N2 ) )
                     => ( ( vEBT_invar_vebt @ Summary3 @ M3 )
                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                         => ( ( M3 = N2 )
                           => ( ( Deg2
                                = ( plus_plus_nat @ N2 @ M3 ) )
                             => ( ! [I4: nat] :
                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                   => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ X4 ) )
                                      = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X3: vEBT_VEBT] :
                                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
                                 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                   => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I4: nat] :
                                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                             => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
                                                    = I4 )
                                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
                                                & ! [X3: nat] :
                                                    ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
                                                        = I4 )
                                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) )
                                                   => ( ( ord_less_nat @ Mi2 @ X3 )
                                                      & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList4: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList4 @ Summary3 ) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X3: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                           => ( vEBT_invar_vebt @ X3 @ N2 ) )
                       => ( ( vEBT_invar_vebt @ Summary3 @ M3 )
                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                           => ( ( M3
                                = ( suc @ N2 ) )
                             => ( ( Deg2
                                  = ( plus_plus_nat @ N2 @ M3 ) )
                               => ( ! [I4: nat] :
                                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                     => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ X4 ) )
                                        = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X3: vEBT_VEBT] :
                                          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                                         => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
                                   => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                     => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I4: nat] :
                                                ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
                                                      = I4 )
                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
                                                  & ! [X3: nat] :
                                                      ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
                                                          = I4 )
                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) )
                                                     => ( ( ord_less_nat @ Mi2 @ X3 )
                                                        & ( ord_less_eq_nat @ X3 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
thf(fact_3960_invar__vebt_Osimps,axiom,
    ( vEBT_invar_vebt
    = ( ^ [A12: vEBT_VEBT,A23: nat] :
          ( ( ? [A3: $o,B3: $o] :
                ( A12
                = ( vEBT_Leaf @ A3 @ B3 ) )
            & ( A23
              = ( suc @ zero_zero_nat ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary4: vEBT_VEBT] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList3 @ Summary4 ) )
              & ! [X6: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X6 @ N3 ) )
              & ( vEBT_invar_vebt @ Summary4 @ N3 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
              & ( A23
                = ( plus_plus_nat @ N3 @ N3 ) )
              & ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary4 @ X4 )
              & ! [X6: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X4 ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary4: vEBT_VEBT] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList3 @ Summary4 ) )
              & ! [X6: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X6 @ N3 ) )
              & ( vEBT_invar_vebt @ Summary4 @ ( suc @ N3 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
              & ( A23
                = ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) )
              & ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary4 @ X4 )
              & ! [X6: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X4 ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary4: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A12
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList3 @ Summary4 ) )
              & ! [X6: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X6 @ N3 ) )
              & ( vEBT_invar_vebt @ Summary4 @ N3 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
              & ( A23
                = ( plus_plus_nat @ N3 @ N3 ) )
              & ! [I: nat] :
                  ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
                 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ X4 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary4 @ I ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X6: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                   => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X4 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I: nat] :
                    ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
                          = I )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
                      & ! [X6: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X6 @ N3 )
                              = I )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ X6 @ N3 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X6 )
                            & ( ord_less_eq_nat @ X6 @ Ma3 ) ) ) ) ) ) )
          | ? [TreeList3: list_VEBT_VEBT,N3: nat,Summary4: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A12
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList3 @ Summary4 ) )
              & ! [X6: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                 => ( vEBT_invar_vebt @ X6 @ N3 ) )
              & ( vEBT_invar_vebt @ Summary4 @ ( suc @ N3 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
              & ( A23
                = ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) )
              & ! [I: nat] :
                  ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
                 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ X4 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary4 @ I ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X6: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                   => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X4 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I: nat] :
                    ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
                          = I )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
                      & ! [X6: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X6 @ N3 )
                              = I )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ X6 @ N3 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X6 )
                            & ( ord_less_eq_nat @ X6 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
thf(fact_3961_vebt__pred_Osimps_I2_J,axiom,
    ! [A: $o,Uw: $o] :
      ( ( A
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
          = none_nat ) ) ) ).

% vebt_pred.simps(2)
thf(fact_3962_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_3963_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_3964_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
     => ( ( N
          = ( suc @ ( suc @ Va2 ) ) )
       => ( ~ ( ord_less_nat @ Ma @ Mi )
         => ( ( Ma != Mi )
           => ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ).

% nested_mint
thf(fact_3965_divmod__step__eq,axiom,
    ! [L: num,R2: nat,Q2: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_3966_divmod__step__eq,axiom,
    ! [L: num,R2: int,Q2: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q2 @ R2 ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q2 @ R2 ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_3967_divmod__step__eq,axiom,
    ! [L: num,R2: code_integer,Q2: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q2 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q2 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_3968_del__single__cont,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( X = Ma ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) ) ) ) ).

% del_single_cont
thf(fact_3969_delt__out__of__range,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X @ Mi )
        | ( ord_less_nat @ Ma @ X ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% delt_out_of_range
thf(fact_3970_inrange,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).

% inrange
thf(fact_3971_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
     => ( ( Mi != Ma )
       => ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
          = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% summaxma
thf(fact_3972_delete__pres__valid,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ T @ X ) @ N ) ) ).

% delete_pres_valid
thf(fact_3973_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T @ X ) @ Y )
        = ( ( X != Y )
          & ( vEBT_V8194947554948674370ptions @ T @ Y ) ) ) ) ).

% dele_bmo_cont_corr
thf(fact_3974_dele__member__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_vebt_member @ ( vEBT_vebt_delete @ T @ X ) @ Y )
        = ( ( X != Y )
          & ( vEBT_vebt_member @ T @ Y ) ) ) ) ).

% dele_member_cont_corr
thf(fact_3975_DiffI,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( ( member_complex @ C @ A2 )
     => ( ~ ( member_complex @ C @ B2 )
       => ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ) ).

% DiffI
thf(fact_3976_DiffI,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( ( member_real @ C @ A2 )
     => ( ~ ( member_real @ C @ B2 )
       => ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ).

% DiffI
thf(fact_3977_DiffI,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ A2 )
     => ( ~ ( member_set_nat @ C @ B2 )
       => ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ).

% DiffI
thf(fact_3978_DiffI,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( ( member_int @ C @ A2 )
     => ( ~ ( member_int @ C @ B2 )
       => ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ).

% DiffI
thf(fact_3979_DiffI,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ A2 )
     => ( ~ ( member_nat @ C @ B2 )
       => ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).

% DiffI
thf(fact_3980_Diff__iff,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
      = ( ( member_complex @ C @ A2 )
        & ~ ( member_complex @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_3981_Diff__iff,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
      = ( ( member_real @ C @ A2 )
        & ~ ( member_real @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_3982_Diff__iff,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
      = ( ( member_set_nat @ C @ A2 )
        & ~ ( member_set_nat @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_3983_Diff__iff,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
      = ( ( member_int @ C @ A2 )
        & ~ ( member_int @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_3984_Diff__iff,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
      = ( ( member_nat @ C @ A2 )
        & ~ ( member_nat @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_3985_Diff__idemp,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ B2 )
      = ( minus_minus_set_nat @ A2 @ B2 ) ) ).

% Diff_idemp
thf(fact_3986_atLeastatMost__empty__iff2,axiom,
    ! [A: $o,B: $o] :
      ( ( bot_bot_set_o
        = ( set_or8904488021354931149Most_o @ A @ B ) )
      = ( ~ ( ord_less_eq_o @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3987_atLeastatMost__empty__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_set_nat
        = ( set_or4548717258645045905et_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3988_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat
        = ( set_or633870826150836451st_rat @ A @ B ) )
      = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3989_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num
        = ( set_or7049704709247886629st_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3990_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat
        = ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3991_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int
        = ( set_or1266510415728281911st_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3992_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real
        = ( set_or1222579329274155063t_real @ A @ B ) )
      = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3993_atLeastatMost__empty__iff,axiom,
    ! [A: $o,B: $o] :
      ( ( ( set_or8904488021354931149Most_o @ A @ B )
        = bot_bot_set_o )
      = ( ~ ( ord_less_eq_o @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3994_atLeastatMost__empty__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ A @ B )
        = bot_bot_set_set_nat )
      = ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3995_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( set_or633870826150836451st_rat @ A @ B )
        = bot_bot_set_rat )
      = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3996_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( ( set_or7049704709247886629st_num @ A @ B )
        = bot_bot_set_num )
      = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3997_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3998_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( set_or1266510415728281911st_int @ A @ B )
        = bot_bot_set_int )
      = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3999_atLeastatMost__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( set_or1222579329274155063t_real @ A @ B )
        = bot_bot_set_real )
      = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4000_atLeastatMost__empty,axiom,
    ! [B: $o,A: $o] :
      ( ( ord_less_o @ B @ A )
     => ( ( set_or8904488021354931149Most_o @ A @ B )
        = bot_bot_set_o ) ) ).

% atLeastatMost_empty
thf(fact_4001_atLeastatMost__empty,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( set_or633870826150836451st_rat @ A @ B )
        = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
thf(fact_4002_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( set_or7049704709247886629st_num @ A @ B )
        = bot_bot_set_num ) ) ).

% atLeastatMost_empty
thf(fact_4003_atLeastatMost__empty,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( ( set_or5403411693681687835d_enat @ A @ B )
        = bot_bo7653980558646680370d_enat ) ) ).

% atLeastatMost_empty
thf(fact_4004_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( set_or1269000886237332187st_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
thf(fact_4005_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( set_or1266510415728281911st_int @ A @ B )
        = bot_bot_set_int ) ) ).

% atLeastatMost_empty
thf(fact_4006_atLeastatMost__empty,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( set_or1222579329274155063t_real @ A @ B )
        = bot_bot_set_real ) ) ).

% atLeastatMost_empty
thf(fact_4007_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( ( some_nat @ ( the_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_4008_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_4009_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( ( some_num @ ( the_num @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_4010_DiffE,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
     => ~ ( ( member_complex @ C @ A2 )
         => ( member_complex @ C @ B2 ) ) ) ).

% DiffE
thf(fact_4011_DiffE,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
     => ~ ( ( member_real @ C @ A2 )
         => ( member_real @ C @ B2 ) ) ) ).

% DiffE
thf(fact_4012_DiffE,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
     => ~ ( ( member_set_nat @ C @ A2 )
         => ( member_set_nat @ C @ B2 ) ) ) ).

% DiffE
thf(fact_4013_DiffE,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
     => ~ ( ( member_int @ C @ A2 )
         => ( member_int @ C @ B2 ) ) ) ).

% DiffE
thf(fact_4014_DiffE,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
     => ~ ( ( member_nat @ C @ A2 )
         => ( member_nat @ C @ B2 ) ) ) ).

% DiffE
thf(fact_4015_DiffD1,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
     => ( member_complex @ C @ A2 ) ) ).

% DiffD1
thf(fact_4016_DiffD1,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
     => ( member_real @ C @ A2 ) ) ).

% DiffD1
thf(fact_4017_DiffD1,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
     => ( member_set_nat @ C @ A2 ) ) ).

% DiffD1
thf(fact_4018_DiffD1,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
     => ( member_int @ C @ A2 ) ) ).

% DiffD1
thf(fact_4019_DiffD1,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
     => ( member_nat @ C @ A2 ) ) ).

% DiffD1
thf(fact_4020_DiffD2,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
     => ~ ( member_complex @ C @ B2 ) ) ).

% DiffD2
thf(fact_4021_DiffD2,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
     => ~ ( member_real @ C @ B2 ) ) ).

% DiffD2
thf(fact_4022_DiffD2,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
     => ~ ( member_set_nat @ C @ B2 ) ) ).

% DiffD2
thf(fact_4023_DiffD2,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
     => ~ ( member_int @ C @ B2 ) ) ).

% DiffD2
thf(fact_4024_DiffD2,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
     => ~ ( member_nat @ C @ B2 ) ) ).

% DiffD2
thf(fact_4025_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc8306885398267862888on_nat] :
      ( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
          ( X
         != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
     => ( ! [Uw2: nat > nat > nat,V2: nat] :
            ( X
           != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F2: nat > nat > nat,A5: nat,B5: nat] :
              ( X
             != ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A5 ) @ ( some_nat @ B5 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_4026_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc5542196010084753463at_nat] :
      ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
          ( X
         != ( produc2899441246263362727at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
     => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
            ( X
           != ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
       => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A5: product_prod_nat_nat,B5: product_prod_nat_nat] :
              ( X
             != ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A5 ) @ ( some_P7363390416028606310at_nat @ B5 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_4027_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc1193250871479095198on_num] :
      ( ! [Uu2: num > num > num,Uv2: option_num] :
          ( X
         != ( produc5778274026573060048on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
     => ( ! [Uw2: num > num > num,V2: num] :
            ( X
           != ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
       => ~ ! [F2: num > num > num,A5: num,B5: num] :
              ( X
             != ( produc5778274026573060048on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ A5 ) @ ( some_num @ B5 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_4028_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc2233624965454879586on_nat] :
      ( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
          ( X
         != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
     => ( ! [Uw2: nat > nat > $o,V2: nat] :
            ( X
           != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F2: nat > nat > $o,X5: nat,Y4: nat] :
              ( X
             != ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X5 ) @ ( some_nat @ Y4 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_4029_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc5491161045314408544at_nat] :
      ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
          ( X
         != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
     => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
            ( X
           != ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
       => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,X5: product_prod_nat_nat,Y4: product_prod_nat_nat] :
              ( X
             != ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X5 ) @ ( some_P7363390416028606310at_nat @ Y4 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_4030_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc7036089656553540234on_num] :
      ( ! [Uu2: num > num > $o,Uv2: option_num] :
          ( X
         != ( produc3576312749637752826on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
     => ( ! [Uw2: num > num > $o,V2: num] :
            ( X
           != ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
       => ~ ! [F2: num > num > $o,X5: num,Y4: num] :
              ( X
             != ( produc3576312749637752826on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ X5 ) @ ( some_num @ Y4 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_4031_option_Oexpand,axiom,
    ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
      ( ( ( Option = none_P5556105721700978146at_nat )
        = ( Option2 = none_P5556105721700978146at_nat ) )
     => ( ( ( Option != none_P5556105721700978146at_nat )
         => ( ( Option2 != none_P5556105721700978146at_nat )
           => ( ( the_Pr8591224930841456533at_nat @ Option )
              = ( the_Pr8591224930841456533at_nat @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_4032_option_Oexpand,axiom,
    ! [Option: option_nat,Option2: option_nat] :
      ( ( ( Option = none_nat )
        = ( Option2 = none_nat ) )
     => ( ( ( Option != none_nat )
         => ( ( Option2 != none_nat )
           => ( ( the_nat @ Option )
              = ( the_nat @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_4033_option_Oexpand,axiom,
    ! [Option: option_num,Option2: option_num] :
      ( ( ( Option = none_num )
        = ( Option2 = none_num ) )
     => ( ( ( Option != none_num )
         => ( ( Option2 != none_num )
           => ( ( the_num @ Option )
              = ( the_num @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_4034_ex__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M4: nat] :
            ( ( ord_less_eq_nat @ M4 @ N )
            & ( P @ M4 ) ) )
      = ( ? [X6: nat] :
            ( ( member_nat @ X6 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
            & ( P @ X6 ) ) ) ) ).

% ex_nat_less
thf(fact_4035_all__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M4: nat] :
            ( ( ord_less_eq_nat @ M4 @ N )
           => ( P @ M4 ) ) )
      = ( ! [X6: nat] :
            ( ( member_nat @ X6 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
           => ( P @ X6 ) ) ) ) ).

% all_nat_less
thf(fact_4036_VEBT__internal_Oinsert_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A5: $o,B5: $o,X5: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X5 ) )
     => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT,X5: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList4 @ Summary3 ) @ X5 ) ) ) ).

% VEBT_internal.insert'.cases
thf(fact_4037_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option
        = ( some_nat @ ( the_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_4038_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option
        = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_4039_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option
        = ( some_num @ ( the_num @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_4040_atLeastatMost__psubset__iff,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
      ( ( ord_le2529575680413868914d_enat @ ( set_or5403411693681687835d_enat @ A @ B ) @ ( set_or5403411693681687835d_enat @ C @ D ) )
      = ( ( ~ ( ord_le2932123472753598470d_enat @ A @ B )
          | ( ( ord_le2932123472753598470d_enat @ C @ A )
            & ( ord_le2932123472753598470d_enat @ B @ D )
            & ( ( ord_le72135733267957522d_enat @ C @ A )
              | ( ord_le72135733267957522d_enat @ B @ D ) ) ) )
        & ( ord_le2932123472753598470d_enat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4041_atLeastatMost__psubset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_set_nat @ A @ B )
          | ( ( ord_less_eq_set_nat @ C @ A )
            & ( ord_less_eq_set_nat @ B @ D )
            & ( ( ord_less_set_nat @ C @ A )
              | ( ord_less_set_nat @ B @ D ) ) ) )
        & ( ord_less_eq_set_nat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4042_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_4043_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_4044_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_4045_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_4046_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_4047_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A5: $o,B5: $o,X5: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X5 ) )
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList4: list_VEBT_VEBT,S2: vEBT_VEBT,X5: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList4 @ S2 ) @ X5 ) ) ) ) ).

% VEBT_internal.naive_member.cases
thf(fact_4048_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X5: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X5 ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList4: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList4 @ Vc2 ) @ X5 ) )
           => ~ ! [V2: nat,TreeList4: list_VEBT_VEBT,Vd2: vEBT_VEBT,X5: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList4 @ Vd2 ) @ X5 ) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
thf(fact_4049_vebt__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A5: $o,B5: $o,X5: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X5 ) )
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X5: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X5 ) )
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X5: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X5 ) )
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X5 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT,X5: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) @ X5 ) ) ) ) ) ) ).

% vebt_member.cases
thf(fact_4050_vebt__insert_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A5: $o,B5: $o,X5: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X5 ) )
     => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X5: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) @ X5 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X5: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ X5 ) )
         => ( ! [V2: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT,X5: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList4 @ Summary3 ) @ X5 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT,X5: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) @ X5 ) ) ) ) ) ) ).

% vebt_insert.cases
thf(fact_4051_vebt__pred_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) )
     => ( ! [A5: $o,Uw2: $o] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A5: $o,B5: $o,Va: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ ( suc @ ( suc @ Va ) ) ) )
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Vb2 ) )
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Vf2 ) )
             => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT,X5: nat] :
                      ( X
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) @ X5 ) ) ) ) ) ) ) ) ).

% vebt_pred.cases
thf(fact_4052_vebt__succ_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,B5: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B5 ) @ zero_zero_nat ) )
     => ( ! [Uv2: $o,Uw2: $o,N2: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N2 ) ) )
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va3 ) )
         => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve2 ) )
           => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT,X5: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) @ X5 ) ) ) ) ) ) ) ).

% vebt_succ.cases
thf(fact_4053_del__x__mi__lets__in__not__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                     => ( ~ ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
thf(fact_4054_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
               => ( ~ ( vEBT_VEBT_minNull @ Newnode )
                 => ( ( Newlist
                      = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                   => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                     => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
thf(fact_4055_vebt__delete_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A5: $o,B5: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ zero_zero_nat ) )
     => ( ! [A5: $o,B5: $o] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A5: $o,B5: $o,N2: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ ( suc @ ( suc @ N2 ) ) ) )
         => ( ! [Deg2: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT,Uu2: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList4 @ Summary3 ) @ Uu2 ) )
           => ( ! [Mi2: nat,Ma2: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X5: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) @ X5 ) )
             => ( ! [Mi2: nat,Ma2: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X5: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ X5 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT,X5: nat] :
                      ( X
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) @ X5 ) ) ) ) ) ) ) ) ).

% vebt_delete.cases
thf(fact_4056_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) ) ).

% vebt_delete.simps(6)
thf(fact_4057_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) ) ).

% vebt_delete.simps(5)
thf(fact_4058_delete__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X ) )
        = ( minus_minus_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% delete_correct
thf(fact_4059_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
thf(fact_4060_set__vebt_H__def,axiom,
    ( vEBT_VEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).

% set_vebt'_def
thf(fact_4061_insert__absorb2,axiom,
    ! [X: nat,A2: set_nat] :
      ( ( insert_nat @ X @ ( insert_nat @ X @ A2 ) )
      = ( insert_nat @ X @ A2 ) ) ).

% insert_absorb2
thf(fact_4062_insert__absorb2,axiom,
    ! [X: int,A2: set_int] :
      ( ( insert_int @ X @ ( insert_int @ X @ A2 ) )
      = ( insert_int @ X @ A2 ) ) ).

% insert_absorb2
thf(fact_4063_insert__absorb2,axiom,
    ! [X: real,A2: set_real] :
      ( ( insert_real @ X @ ( insert_real @ X @ A2 ) )
      = ( insert_real @ X @ A2 ) ) ).

% insert_absorb2
thf(fact_4064_insert__absorb2,axiom,
    ! [X: $o,A2: set_o] :
      ( ( insert_o @ X @ ( insert_o @ X @ A2 ) )
      = ( insert_o @ X @ A2 ) ) ).

% insert_absorb2
thf(fact_4065_insert__iff,axiom,
    ! [A: $o,B: $o,A2: set_o] :
      ( ( member_o @ A @ ( insert_o @ B @ A2 ) )
      = ( ( A = B )
        | ( member_o @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_4066_insert__iff,axiom,
    ! [A: complex,B: complex,A2: set_complex] :
      ( ( member_complex @ A @ ( insert_complex @ B @ A2 ) )
      = ( ( A = B )
        | ( member_complex @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_4067_insert__iff,axiom,
    ! [A: real,B: real,A2: set_real] :
      ( ( member_real @ A @ ( insert_real @ B @ A2 ) )
      = ( ( A = B )
        | ( member_real @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_4068_insert__iff,axiom,
    ! [A: set_nat,B: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
      = ( ( A = B )
        | ( member_set_nat @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_4069_insert__iff,axiom,
    ! [A: nat,B: nat,A2: set_nat] :
      ( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
      = ( ( A = B )
        | ( member_nat @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_4070_insert__iff,axiom,
    ! [A: int,B: int,A2: set_int] :
      ( ( member_int @ A @ ( insert_int @ B @ A2 ) )
      = ( ( A = B )
        | ( member_int @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_4071_insertCI,axiom,
    ! [A: $o,B2: set_o,B: $o] :
      ( ( ~ ( member_o @ A @ B2 )
       => ( A = B ) )
     => ( member_o @ A @ ( insert_o @ B @ B2 ) ) ) ).

% insertCI
thf(fact_4072_insertCI,axiom,
    ! [A: complex,B2: set_complex,B: complex] :
      ( ( ~ ( member_complex @ A @ B2 )
       => ( A = B ) )
     => ( member_complex @ A @ ( insert_complex @ B @ B2 ) ) ) ).

% insertCI
thf(fact_4073_insertCI,axiom,
    ! [A: real,B2: set_real,B: real] :
      ( ( ~ ( member_real @ A @ B2 )
       => ( A = B ) )
     => ( member_real @ A @ ( insert_real @ B @ B2 ) ) ) ).

% insertCI
thf(fact_4074_insertCI,axiom,
    ! [A: set_nat,B2: set_set_nat,B: set_nat] :
      ( ( ~ ( member_set_nat @ A @ B2 )
       => ( A = B ) )
     => ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).

% insertCI
thf(fact_4075_insertCI,axiom,
    ! [A: nat,B2: set_nat,B: nat] :
      ( ( ~ ( member_nat @ A @ B2 )
       => ( A = B ) )
     => ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).

% insertCI
thf(fact_4076_insertCI,axiom,
    ! [A: int,B2: set_int,B: int] :
      ( ( ~ ( member_int @ A @ B2 )
       => ( A = B ) )
     => ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).

% insertCI
thf(fact_4077_list__update__overwrite,axiom,
    ! [Xs2: list_VEBT_VEBT,I3: nat,X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I3 @ X ) @ I3 @ Y )
      = ( list_u1324408373059187874T_VEBT @ Xs2 @ I3 @ Y ) ) ).

% list_update_overwrite
thf(fact_4078_pred__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y6: nat] :
                ( ( vEBT_vebt_member @ T @ Y6 )
                & ( ord_less_nat @ Y6 @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% pred_empty
thf(fact_4079_succ__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y6: nat] :
                ( ( vEBT_vebt_member @ T @ Y6 )
                & ( ord_less_nat @ X @ Y6 ) ) )
          = bot_bot_set_nat ) ) ) ).

% succ_empty
thf(fact_4080_delete__correct_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X ) )
        = ( minus_minus_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% delete_correct'
thf(fact_4081_singletonI,axiom,
    ! [A: complex] : ( member_complex @ A @ ( insert_complex @ A @ bot_bot_set_complex ) ) ).

% singletonI
thf(fact_4082_singletonI,axiom,
    ! [A: set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).

% singletonI
thf(fact_4083_singletonI,axiom,
    ! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).

% singletonI
thf(fact_4084_singletonI,axiom,
    ! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).

% singletonI
thf(fact_4085_singletonI,axiom,
    ! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_4086_singletonI,axiom,
    ! [A: int] : ( member_int @ A @ ( insert_int @ A @ bot_bot_set_int ) ) ).

% singletonI
thf(fact_4087_insert__subset,axiom,
    ! [X: $o,A2: set_o,B2: set_o] :
      ( ( ord_less_eq_set_o @ ( insert_o @ X @ A2 ) @ B2 )
      = ( ( member_o @ X @ B2 )
        & ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ).

% insert_subset
thf(fact_4088_insert__subset,axiom,
    ! [X: complex,A2: set_complex,B2: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( insert_complex @ X @ A2 ) @ B2 )
      = ( ( member_complex @ X @ B2 )
        & ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ) ).

% insert_subset
thf(fact_4089_insert__subset,axiom,
    ! [X: real,A2: set_real,B2: set_real] :
      ( ( ord_less_eq_set_real @ ( insert_real @ X @ A2 ) @ B2 )
      = ( ( member_real @ X @ B2 )
        & ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).

% insert_subset
thf(fact_4090_insert__subset,axiom,
    ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X @ A2 ) @ B2 )
      = ( ( member_set_nat @ X @ B2 )
        & ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).

% insert_subset
thf(fact_4091_insert__subset,axiom,
    ! [X: int,A2: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ ( insert_int @ X @ A2 ) @ B2 )
      = ( ( member_int @ X @ B2 )
        & ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).

% insert_subset
thf(fact_4092_insert__subset,axiom,
    ! [X: nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
      = ( ( member_nat @ X @ B2 )
        & ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).

% insert_subset
thf(fact_4093_insert__Diff1,axiom,
    ! [X: $o,B2: set_o,A2: set_o] :
      ( ( member_o @ X @ B2 )
     => ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ B2 )
        = ( minus_minus_set_o @ A2 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_4094_insert__Diff1,axiom,
    ! [X: complex,B2: set_complex,A2: set_complex] :
      ( ( member_complex @ X @ B2 )
     => ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A2 ) @ B2 )
        = ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_4095_insert__Diff1,axiom,
    ! [X: real,B2: set_real,A2: set_real] :
      ( ( member_real @ X @ B2 )
     => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
        = ( minus_minus_set_real @ A2 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_4096_insert__Diff1,axiom,
    ! [X: set_nat,B2: set_set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ X @ B2 )
     => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B2 )
        = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_4097_insert__Diff1,axiom,
    ! [X: int,B2: set_int,A2: set_int] :
      ( ( member_int @ X @ B2 )
     => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B2 )
        = ( minus_minus_set_int @ A2 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_4098_insert__Diff1,axiom,
    ! [X: nat,B2: set_nat,A2: set_nat] :
      ( ( member_nat @ X @ B2 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
        = ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_4099_Diff__insert0,axiom,
    ! [X: $o,A2: set_o,B2: set_o] :
      ( ~ ( member_o @ X @ A2 )
     => ( ( minus_minus_set_o @ A2 @ ( insert_o @ X @ B2 ) )
        = ( minus_minus_set_o @ A2 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_4100_Diff__insert0,axiom,
    ! [X: complex,A2: set_complex,B2: set_complex] :
      ( ~ ( member_complex @ X @ A2 )
     => ( ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ B2 ) )
        = ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_4101_Diff__insert0,axiom,
    ! [X: real,A2: set_real,B2: set_real] :
      ( ~ ( member_real @ X @ A2 )
     => ( ( minus_minus_set_real @ A2 @ ( insert_real @ X @ B2 ) )
        = ( minus_minus_set_real @ A2 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_4102_Diff__insert0,axiom,
    ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A2 )
     => ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
        = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_4103_Diff__insert0,axiom,
    ! [X: int,A2: set_int,B2: set_int] :
      ( ~ ( member_int @ X @ A2 )
     => ( ( minus_minus_set_int @ A2 @ ( insert_int @ X @ B2 ) )
        = ( minus_minus_set_int @ A2 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_4104_Diff__insert0,axiom,
    ! [X: nat,A2: set_nat,B2: set_nat] :
      ( ~ ( member_nat @ X @ A2 )
     => ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
        = ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_4105_length__list__update,axiom,
    ! [Xs2: list_VEBT_VEBT,I3: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I3 @ X ) )
      = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).

% length_list_update
thf(fact_4106_length__list__update,axiom,
    ! [Xs2: list_o,I3: nat,X: $o] :
      ( ( size_size_list_o @ ( list_update_o @ Xs2 @ I3 @ X ) )
      = ( size_size_list_o @ Xs2 ) ) ).

% length_list_update
thf(fact_4107_length__list__update,axiom,
    ! [Xs2: list_nat,I3: nat,X: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs2 @ I3 @ X ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_list_update
thf(fact_4108_length__list__update,axiom,
    ! [Xs2: list_int,I3: nat,X: int] :
      ( ( size_size_list_int @ ( list_update_int @ Xs2 @ I3 @ X ) )
      = ( size_size_list_int @ Xs2 ) ) ).

% length_list_update
thf(fact_4109_nth__list__update__neq,axiom,
    ! [I3: nat,J: nat,Xs2: list_int,X: int] :
      ( ( I3 != J )
     => ( ( nth_int @ ( list_update_int @ Xs2 @ I3 @ X ) @ J )
        = ( nth_int @ Xs2 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_4110_nth__list__update__neq,axiom,
    ! [I3: nat,J: nat,Xs2: list_nat,X: nat] :
      ( ( I3 != J )
     => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I3 @ X ) @ J )
        = ( nth_nat @ Xs2 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_4111_nth__list__update__neq,axiom,
    ! [I3: nat,J: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( I3 != J )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I3 @ X ) @ J )
        = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_4112_list__update__id,axiom,
    ! [Xs2: list_int,I3: nat] :
      ( ( list_update_int @ Xs2 @ I3 @ ( nth_int @ Xs2 @ I3 ) )
      = Xs2 ) ).

% list_update_id
thf(fact_4113_list__update__id,axiom,
    ! [Xs2: list_nat,I3: nat] :
      ( ( list_update_nat @ Xs2 @ I3 @ ( nth_nat @ Xs2 @ I3 ) )
      = Xs2 ) ).

% list_update_id
thf(fact_4114_list__update__id,axiom,
    ! [Xs2: list_VEBT_VEBT,I3: nat] :
      ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I3 @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) )
      = Xs2 ) ).

% list_update_id
thf(fact_4115_singleton__conv,axiom,
    ! [A: list_nat] :
      ( ( collect_list_nat
        @ ^ [X6: list_nat] : X6 = A )
      = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).

% singleton_conv
thf(fact_4116_singleton__conv,axiom,
    ! [A: set_nat] :
      ( ( collect_set_nat
        @ ^ [X6: set_nat] : X6 = A )
      = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).

% singleton_conv
thf(fact_4117_singleton__conv,axiom,
    ! [A: real] :
      ( ( collect_real
        @ ^ [X6: real] : X6 = A )
      = ( insert_real @ A @ bot_bot_set_real ) ) ).

% singleton_conv
thf(fact_4118_singleton__conv,axiom,
    ! [A: $o] :
      ( ( collect_o
        @ ^ [X6: $o] : X6 = A )
      = ( insert_o @ A @ bot_bot_set_o ) ) ).

% singleton_conv
thf(fact_4119_singleton__conv,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ^ [X6: nat] : X6 = A )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_4120_singleton__conv,axiom,
    ! [A: int] :
      ( ( collect_int
        @ ^ [X6: int] : X6 = A )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% singleton_conv
thf(fact_4121_singleton__conv2,axiom,
    ! [A: list_nat] :
      ( ( collect_list_nat
        @ ( ^ [Y3: list_nat,Z: list_nat] : Y3 = Z
          @ A ) )
      = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).

% singleton_conv2
thf(fact_4122_singleton__conv2,axiom,
    ! [A: set_nat] :
      ( ( collect_set_nat
        @ ( ^ [Y3: set_nat,Z: set_nat] : Y3 = Z
          @ A ) )
      = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).

% singleton_conv2
thf(fact_4123_singleton__conv2,axiom,
    ! [A: real] :
      ( ( collect_real
        @ ( ^ [Y3: real,Z: real] : Y3 = Z
          @ A ) )
      = ( insert_real @ A @ bot_bot_set_real ) ) ).

% singleton_conv2
thf(fact_4124_singleton__conv2,axiom,
    ! [A: $o] :
      ( ( collect_o
        @ ( ^ [Y3: $o,Z: $o] : Y3 = Z
          @ A ) )
      = ( insert_o @ A @ bot_bot_set_o ) ) ).

% singleton_conv2
thf(fact_4125_singleton__conv2,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ( ^ [Y3: nat,Z: nat] : Y3 = Z
          @ A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_4126_singleton__conv2,axiom,
    ! [A: int] :
      ( ( collect_int
        @ ( ^ [Y3: int,Z: int] : Y3 = Z
          @ A ) )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% singleton_conv2
thf(fact_4127_singleton__insert__inj__eq_H,axiom,
    ! [A: real,A2: set_real,B: real] :
      ( ( ( insert_real @ A @ A2 )
        = ( insert_real @ B @ bot_bot_set_real ) )
      = ( ( A = B )
        & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4128_singleton__insert__inj__eq_H,axiom,
    ! [A: $o,A2: set_o,B: $o] :
      ( ( ( insert_o @ A @ A2 )
        = ( insert_o @ B @ bot_bot_set_o ) )
      = ( ( A = B )
        & ( ord_less_eq_set_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4129_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A2: set_int,B: int] :
      ( ( ( insert_int @ A @ A2 )
        = ( insert_int @ B @ bot_bot_set_int ) )
      = ( ( A = B )
        & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4130_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A2: set_nat,B: nat] :
      ( ( ( insert_nat @ A @ A2 )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4131_singleton__insert__inj__eq,axiom,
    ! [B: real,A: real,A2: set_real] :
      ( ( ( insert_real @ B @ bot_bot_set_real )
        = ( insert_real @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4132_singleton__insert__inj__eq,axiom,
    ! [B: $o,A: $o,A2: set_o] :
      ( ( ( insert_o @ B @ bot_bot_set_o )
        = ( insert_o @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4133_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A2: set_int] :
      ( ( ( insert_int @ B @ bot_bot_set_int )
        = ( insert_int @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4134_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A2: set_nat] :
      ( ( ( insert_nat @ B @ bot_bot_set_nat )
        = ( insert_nat @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4135_atLeastAtMost__singleton__iff,axiom,
    ! [A: $o,B: $o,C: $o] :
      ( ( ( set_or8904488021354931149Most_o @ A @ B )
        = ( insert_o @ C @ bot_bot_set_o ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_4136_atLeastAtMost__singleton__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A @ B )
        = ( insert_nat @ C @ bot_bot_set_nat ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_4137_atLeastAtMost__singleton__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( set_or1266510415728281911st_int @ A @ B )
        = ( insert_int @ C @ bot_bot_set_int ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_4138_atLeastAtMost__singleton__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( set_or1222579329274155063t_real @ A @ B )
        = ( insert_real @ C @ bot_bot_set_real ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_4139_atLeastAtMost__singleton,axiom,
    ! [A: $o] :
      ( ( set_or8904488021354931149Most_o @ A @ A )
      = ( insert_o @ A @ bot_bot_set_o ) ) ).

% atLeastAtMost_singleton
thf(fact_4140_atLeastAtMost__singleton,axiom,
    ! [A: nat] :
      ( ( set_or1269000886237332187st_nat @ A @ A )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% atLeastAtMost_singleton
thf(fact_4141_atLeastAtMost__singleton,axiom,
    ! [A: int] :
      ( ( set_or1266510415728281911st_int @ A @ A )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% atLeastAtMost_singleton
thf(fact_4142_atLeastAtMost__singleton,axiom,
    ! [A: real] :
      ( ( set_or1222579329274155063t_real @ A @ A )
      = ( insert_real @ A @ bot_bot_set_real ) ) ).

% atLeastAtMost_singleton
thf(fact_4143_insert__Diff__single,axiom,
    ! [A: real,A2: set_real] :
      ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
      = ( insert_real @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_4144_insert__Diff__single,axiom,
    ! [A: $o,A2: set_o] :
      ( ( insert_o @ A @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
      = ( insert_o @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_4145_insert__Diff__single,axiom,
    ! [A: int,A2: set_int] :
      ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
      = ( insert_int @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_4146_insert__Diff__single,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
      = ( insert_nat @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_4147_list__update__beyond,axiom,
    ! [Xs2: list_VEBT_VEBT,I3: nat,X: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ I3 )
     => ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I3 @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_4148_list__update__beyond,axiom,
    ! [Xs2: list_o,I3: nat,X: $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ I3 )
     => ( ( list_update_o @ Xs2 @ I3 @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_4149_list__update__beyond,axiom,
    ! [Xs2: list_nat,I3: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I3 )
     => ( ( list_update_nat @ Xs2 @ I3 @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_4150_list__update__beyond,axiom,
    ! [Xs2: list_int,I3: nat,X: int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ I3 )
     => ( ( list_update_int @ Xs2 @ I3 @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_4151_nth__list__update__eq,axiom,
    ! [I3: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I3 @ X ) @ I3 )
        = X ) ) ).

% nth_list_update_eq
thf(fact_4152_nth__list__update__eq,axiom,
    ! [I3: nat,Xs2: list_o,X: $o] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_o @ ( list_update_o @ Xs2 @ I3 @ X ) @ I3 )
        = X ) ) ).

% nth_list_update_eq
thf(fact_4153_nth__list__update__eq,axiom,
    ! [I3: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I3 @ X ) @ I3 )
        = X ) ) ).

% nth_list_update_eq
thf(fact_4154_nth__list__update__eq,axiom,
    ! [I3: nat,Xs2: list_int,X: int] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
     => ( ( nth_int @ ( list_update_int @ Xs2 @ I3 @ X ) @ I3 )
        = X ) ) ).

% nth_list_update_eq
thf(fact_4155_set__swap,axiom,
    ! [I3: nat,Xs2: list_VEBT_VEBT,J: nat] :
      ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I3 @ ( nth_VEBT_VEBT @ Xs2 @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) ) )
          = ( set_VEBT_VEBT2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_4156_set__swap,axiom,
    ! [I3: nat,Xs2: list_o,J: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs2 ) )
       => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs2 @ I3 @ ( nth_o @ Xs2 @ J ) ) @ J @ ( nth_o @ Xs2 @ I3 ) ) )
          = ( set_o2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_4157_set__swap,axiom,
    ! [I3: nat,Xs2: list_nat,J: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
       => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I3 @ ( nth_nat @ Xs2 @ J ) ) @ J @ ( nth_nat @ Xs2 @ I3 ) ) )
          = ( set_nat2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_4158_set__swap,axiom,
    ! [I3: nat,Xs2: list_int,J: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs2 ) )
       => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs2 @ I3 @ ( nth_int @ Xs2 @ J ) ) @ J @ ( nth_int @ Xs2 @ I3 ) ) )
          = ( set_int2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_4159_del__x__not__mia,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
               => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                  = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
                    @ ( vEBT_Node
                      @ ( some_P7363390416028606310at_nat
                        @ ( product_Pair_nat_nat @ Mi
                          @ ( if_nat @ ( X = Ma )
                            @ ( if_nat
                              @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                = none_nat )
                              @ Mi
                              @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                            @ Ma ) ) )
                      @ Deg
                      @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
                      @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                    @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
thf(fact_4160_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
               => ( ( vEBT_VEBT_minNull @ Newnode )
                 => ( ( Sn
                      = ( vEBT_vebt_delete @ Summary @ H2 ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                     => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X = Ma )
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ Sn )
                                      = none_nat )
                                    @ Mi
                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
                                  @ Ma ) ) )
                            @ Deg
                            @ Newlist
                            @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
thf(fact_4161_del__x__not__mi,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
               => ( ( Newlist
                    = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                   => ( ( ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X = Ma )
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                      = none_nat )
                                    @ Mi
                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                                  @ Ma ) ) )
                            @ Deg
                            @ Newlist
                            @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
                      & ( ~ ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
thf(fact_4162_del__x__mia,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
              @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                      @ ( if_nat
                        @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          = Ma )
                        @ ( if_nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            = none_nat )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                      @ ( if_nat
                        @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          = Ma )
                        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ Summary ) )
              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).

% del_x_mia
thf(fact_4163_del__x__mi__lets__in__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                     => ( ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( Sn
                            = ( vEBT_vebt_delete @ Summary @ H2 ) )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                            = ( vEBT_Node
                              @ ( some_P7363390416028606310at_nat
                                @ ( product_Pair_nat_nat @ Xn
                                  @ ( if_nat @ ( Xn = Ma )
                                    @ ( if_nat
                                      @ ( ( vEBT_vebt_maxt @ Sn )
                                        = none_nat )
                                      @ Xn
                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
                                    @ Ma ) ) )
                              @ Deg
                              @ Newlist
                              @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
thf(fact_4164_del__x__mi__lets__in,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Newnode ) )
                     => ( ( ( vEBT_VEBT_minNull @ Newnode )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                            = ( vEBT_Node
                              @ ( some_P7363390416028606310at_nat
                                @ ( product_Pair_nat_nat @ Xn
                                  @ ( if_nat @ ( Xn = Ma )
                                    @ ( if_nat
                                      @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                        = none_nat )
                                      @ Xn
                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                                    @ Ma ) ) )
                              @ Deg
                              @ Newlist
                              @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
                        & ( ~ ( vEBT_VEBT_minNull @ Newnode )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
thf(fact_4165_del__x__mi,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
                    = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
                      @ ( vEBT_Node
                        @ ( some_P7363390416028606310at_nat
                          @ ( product_Pair_nat_nat @ Xn
                            @ ( if_nat @ ( Xn = Ma )
                              @ ( if_nat
                                @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                                  = none_nat )
                                @ Xn
                                @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
                              @ Ma ) ) )
                        @ Deg
                        @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) )
                        @ ( vEBT_vebt_delete @ Summary @ H2 ) )
                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
thf(fact_4166_del__in__range,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_eq_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
              @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X = Mi )
                           => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                              = Ma ) )
                          & ( ( X != Mi )
                           => ( X = Ma ) ) )
                        @ ( if_nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            = none_nat )
                          @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X = Mi )
                           => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                              = Ma ) )
                          & ( ( X != Mi )
                           => ( X = Ma ) ) )
                        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ Summary ) )
              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).

% del_in_range
thf(fact_4167_pred__less__length__list,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X @ Ma )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
thf(fact_4168_pred__lesseq__max,axiom,
    ! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X @ Ma )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% pred_lesseq_max
thf(fact_4169_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% succ_greatereq_min
thf(fact_4170_set__diff__eq,axiom,
    ( minus_811609699411566653omplex
    = ( ^ [A6: set_complex,B7: set_complex] :
          ( collect_complex
          @ ^ [X6: complex] :
              ( ( member_complex @ X6 @ A6 )
              & ~ ( member_complex @ X6 @ B7 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4171_set__diff__eq,axiom,
    ( minus_minus_set_real
    = ( ^ [A6: set_real,B7: set_real] :
          ( collect_real
          @ ^ [X6: real] :
              ( ( member_real @ X6 @ A6 )
              & ~ ( member_real @ X6 @ B7 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4172_set__diff__eq,axiom,
    ( minus_7954133019191499631st_nat
    = ( ^ [A6: set_list_nat,B7: set_list_nat] :
          ( collect_list_nat
          @ ^ [X6: list_nat] :
              ( ( member_list_nat @ X6 @ A6 )
              & ~ ( member_list_nat @ X6 @ B7 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4173_set__diff__eq,axiom,
    ( minus_2163939370556025621et_nat
    = ( ^ [A6: set_set_nat,B7: set_set_nat] :
          ( collect_set_nat
          @ ^ [X6: set_nat] :
              ( ( member_set_nat @ X6 @ A6 )
              & ~ ( member_set_nat @ X6 @ B7 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4174_set__diff__eq,axiom,
    ( minus_minus_set_int
    = ( ^ [A6: set_int,B7: set_int] :
          ( collect_int
          @ ^ [X6: int] :
              ( ( member_int @ X6 @ A6 )
              & ~ ( member_int @ X6 @ B7 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4175_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( collect_nat
          @ ^ [X6: nat] :
              ( ( member_nat @ X6 @ A6 )
              & ~ ( member_nat @ X6 @ B7 ) ) ) ) ) ).

% set_diff_eq
thf(fact_4176_minus__set__def,axiom,
    ( minus_811609699411566653omplex
    = ( ^ [A6: set_complex,B7: set_complex] :
          ( collect_complex
          @ ( minus_8727706125548526216plex_o
            @ ^ [X6: complex] : ( member_complex @ X6 @ A6 )
            @ ^ [X6: complex] : ( member_complex @ X6 @ B7 ) ) ) ) ) ).

% minus_set_def
thf(fact_4177_minus__set__def,axiom,
    ( minus_minus_set_real
    = ( ^ [A6: set_real,B7: set_real] :
          ( collect_real
          @ ( minus_minus_real_o
            @ ^ [X6: real] : ( member_real @ X6 @ A6 )
            @ ^ [X6: real] : ( member_real @ X6 @ B7 ) ) ) ) ) ).

% minus_set_def
thf(fact_4178_minus__set__def,axiom,
    ( minus_7954133019191499631st_nat
    = ( ^ [A6: set_list_nat,B7: set_list_nat] :
          ( collect_list_nat
          @ ( minus_1139252259498527702_nat_o
            @ ^ [X6: list_nat] : ( member_list_nat @ X6 @ A6 )
            @ ^ [X6: list_nat] : ( member_list_nat @ X6 @ B7 ) ) ) ) ) ).

% minus_set_def
thf(fact_4179_minus__set__def,axiom,
    ( minus_2163939370556025621et_nat
    = ( ^ [A6: set_set_nat,B7: set_set_nat] :
          ( collect_set_nat
          @ ( minus_6910147592129066416_nat_o
            @ ^ [X6: set_nat] : ( member_set_nat @ X6 @ A6 )
            @ ^ [X6: set_nat] : ( member_set_nat @ X6 @ B7 ) ) ) ) ) ).

% minus_set_def
thf(fact_4180_minus__set__def,axiom,
    ( minus_minus_set_int
    = ( ^ [A6: set_int,B7: set_int] :
          ( collect_int
          @ ( minus_minus_int_o
            @ ^ [X6: int] : ( member_int @ X6 @ A6 )
            @ ^ [X6: int] : ( member_int @ X6 @ B7 ) ) ) ) ) ).

% minus_set_def
thf(fact_4181_minus__set__def,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( collect_nat
          @ ( minus_minus_nat_o
            @ ^ [X6: nat] : ( member_nat @ X6 @ A6 )
            @ ^ [X6: nat] : ( member_nat @ X6 @ B7 ) ) ) ) ) ).

% minus_set_def
thf(fact_4182_mult__commute__abs,axiom,
    ! [C: real] :
      ( ( ^ [X6: real] : ( times_times_real @ X6 @ C ) )
      = ( times_times_real @ C ) ) ).

% mult_commute_abs
thf(fact_4183_mult__commute__abs,axiom,
    ! [C: rat] :
      ( ( ^ [X6: rat] : ( times_times_rat @ X6 @ C ) )
      = ( times_times_rat @ C ) ) ).

% mult_commute_abs
thf(fact_4184_mult__commute__abs,axiom,
    ! [C: nat] :
      ( ( ^ [X6: nat] : ( times_times_nat @ X6 @ C ) )
      = ( times_times_nat @ C ) ) ).

% mult_commute_abs
thf(fact_4185_mult__commute__abs,axiom,
    ! [C: int] :
      ( ( ^ [X6: int] : ( times_times_int @ X6 @ C ) )
      = ( times_times_int @ C ) ) ).

% mult_commute_abs
thf(fact_4186_Collect__subset,axiom,
    ! [A2: set_complex,P: complex > $o] :
      ( ord_le211207098394363844omplex
      @ ( collect_complex
        @ ^ [X6: complex] :
            ( ( member_complex @ X6 @ A2 )
            & ( P @ X6 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_4187_Collect__subset,axiom,
    ! [A2: set_real,P: real > $o] :
      ( ord_less_eq_set_real
      @ ( collect_real
        @ ^ [X6: real] :
            ( ( member_real @ X6 @ A2 )
            & ( P @ X6 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_4188_Collect__subset,axiom,
    ! [A2: set_list_nat,P: list_nat > $o] :
      ( ord_le6045566169113846134st_nat
      @ ( collect_list_nat
        @ ^ [X6: list_nat] :
            ( ( member_list_nat @ X6 @ A2 )
            & ( P @ X6 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_4189_Collect__subset,axiom,
    ! [A2: set_set_nat,P: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X6: set_nat] :
            ( ( member_set_nat @ X6 @ A2 )
            & ( P @ X6 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_4190_Collect__subset,axiom,
    ! [A2: set_int,P: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X6: int] :
            ( ( member_int @ X6 @ A2 )
            & ( P @ X6 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_4191_Collect__subset,axiom,
    ! [A2: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X6: nat] :
            ( ( member_nat @ X6 @ A2 )
            & ( P @ X6 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_4192_less__eq__set__def,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A6: set_complex,B7: set_complex] :
          ( ord_le4573692005234683329plex_o
          @ ^ [X6: complex] : ( member_complex @ X6 @ A6 )
          @ ^ [X6: complex] : ( member_complex @ X6 @ B7 ) ) ) ) ).

% less_eq_set_def
thf(fact_4193_less__eq__set__def,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B7: set_real] :
          ( ord_less_eq_real_o
          @ ^ [X6: real] : ( member_real @ X6 @ A6 )
          @ ^ [X6: real] : ( member_real @ X6 @ B7 ) ) ) ) ).

% less_eq_set_def
thf(fact_4194_less__eq__set__def,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A6: set_set_nat,B7: set_set_nat] :
          ( ord_le3964352015994296041_nat_o
          @ ^ [X6: set_nat] : ( member_set_nat @ X6 @ A6 )
          @ ^ [X6: set_nat] : ( member_set_nat @ X6 @ B7 ) ) ) ) ).

% less_eq_set_def
thf(fact_4195_less__eq__set__def,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B7: set_int] :
          ( ord_less_eq_int_o
          @ ^ [X6: int] : ( member_int @ X6 @ A6 )
          @ ^ [X6: int] : ( member_int @ X6 @ B7 ) ) ) ) ).

% less_eq_set_def
thf(fact_4196_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X6: nat] : ( member_nat @ X6 @ A6 )
          @ ^ [X6: nat] : ( member_nat @ X6 @ B7 ) ) ) ) ).

% less_eq_set_def
thf(fact_4197_less__set__def,axiom,
    ( ord_less_set_complex
    = ( ^ [A6: set_complex,B7: set_complex] :
          ( ord_less_complex_o
          @ ^ [X6: complex] : ( member_complex @ X6 @ A6 )
          @ ^ [X6: complex] : ( member_complex @ X6 @ B7 ) ) ) ) ).

% less_set_def
thf(fact_4198_less__set__def,axiom,
    ( ord_less_set_real
    = ( ^ [A6: set_real,B7: set_real] :
          ( ord_less_real_o
          @ ^ [X6: real] : ( member_real @ X6 @ A6 )
          @ ^ [X6: real] : ( member_real @ X6 @ B7 ) ) ) ) ).

% less_set_def
thf(fact_4199_less__set__def,axiom,
    ( ord_less_set_set_nat
    = ( ^ [A6: set_set_nat,B7: set_set_nat] :
          ( ord_less_set_nat_o
          @ ^ [X6: set_nat] : ( member_set_nat @ X6 @ A6 )
          @ ^ [X6: set_nat] : ( member_set_nat @ X6 @ B7 ) ) ) ) ).

% less_set_def
thf(fact_4200_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ord_less_nat_o
          @ ^ [X6: nat] : ( member_nat @ X6 @ A6 )
          @ ^ [X6: nat] : ( member_nat @ X6 @ B7 ) ) ) ) ).

% less_set_def
thf(fact_4201_less__set__def,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B7: set_int] :
          ( ord_less_int_o
          @ ^ [X6: int] : ( member_int @ X6 @ A6 )
          @ ^ [X6: int] : ( member_int @ X6 @ B7 ) ) ) ) ).

% less_set_def
thf(fact_4202_empty__def,axiom,
    ( bot_bot_set_list_nat
    = ( collect_list_nat
      @ ^ [X6: list_nat] : $false ) ) ).

% empty_def
thf(fact_4203_empty__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat
      @ ^ [X6: set_nat] : $false ) ) ).

% empty_def
thf(fact_4204_empty__def,axiom,
    ( bot_bot_set_real
    = ( collect_real
      @ ^ [X6: real] : $false ) ) ).

% empty_def
thf(fact_4205_empty__def,axiom,
    ( bot_bot_set_o
    = ( collect_o
      @ ^ [X6: $o] : $false ) ) ).

% empty_def
thf(fact_4206_empty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X6: nat] : $false ) ) ).

% empty_def
thf(fact_4207_empty__def,axiom,
    ( bot_bot_set_int
    = ( collect_int
      @ ^ [X6: int] : $false ) ) ).

% empty_def
thf(fact_4208_list__update__swap,axiom,
    ! [I3: nat,I5: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT,X8: vEBT_VEBT] :
      ( ( I3 != I5 )
     => ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I3 @ X ) @ I5 @ X8 )
        = ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I5 @ X8 ) @ I3 @ X ) ) ) ).

% list_update_swap
thf(fact_4209_Collect__conv__if,axiom,
    ! [P: list_nat > $o,A: list_nat] :
      ( ( ( P @ A )
       => ( ( collect_list_nat
            @ ^ [X6: list_nat] :
                ( ( X6 = A )
                & ( P @ X6 ) ) )
          = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_list_nat
            @ ^ [X6: list_nat] :
                ( ( X6 = A )
                & ( P @ X6 ) ) )
          = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if
thf(fact_4210_Collect__conv__if,axiom,
    ! [P: set_nat > $o,A: set_nat] :
      ( ( ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X6: set_nat] :
                ( ( X6 = A )
                & ( P @ X6 ) ) )
          = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X6: set_nat] :
                ( ( X6 = A )
                & ( P @ X6 ) ) )
          = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_4211_Collect__conv__if,axiom,
    ! [P: real > $o,A: real] :
      ( ( ( P @ A )
       => ( ( collect_real
            @ ^ [X6: real] :
                ( ( X6 = A )
                & ( P @ X6 ) ) )
          = ( insert_real @ A @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_real
            @ ^ [X6: real] :
                ( ( X6 = A )
                & ( P @ X6 ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if
thf(fact_4212_Collect__conv__if,axiom,
    ! [P: $o > $o,A: $o] :
      ( ( ( P @ A )
       => ( ( collect_o
            @ ^ [X6: $o] :
                ( ( X6 = A )
                & ( P @ X6 ) ) )
          = ( insert_o @ A @ bot_bot_set_o ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_o
            @ ^ [X6: $o] :
                ( ( X6 = A )
                & ( P @ X6 ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if
thf(fact_4213_Collect__conv__if,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X6: nat] :
                ( ( X6 = A )
                & ( P @ X6 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X6: nat] :
                ( ( X6 = A )
                & ( P @ X6 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_4214_Collect__conv__if,axiom,
    ! [P: int > $o,A: int] :
      ( ( ( P @ A )
       => ( ( collect_int
            @ ^ [X6: int] :
                ( ( X6 = A )
                & ( P @ X6 ) ) )
          = ( insert_int @ A @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_int
            @ ^ [X6: int] :
                ( ( X6 = A )
                & ( P @ X6 ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if
thf(fact_4215_Collect__conv__if2,axiom,
    ! [P: list_nat > $o,A: list_nat] :
      ( ( ( P @ A )
       => ( ( collect_list_nat
            @ ^ [X6: list_nat] :
                ( ( A = X6 )
                & ( P @ X6 ) ) )
          = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_list_nat
            @ ^ [X6: list_nat] :
                ( ( A = X6 )
                & ( P @ X6 ) ) )
          = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if2
thf(fact_4216_Collect__conv__if2,axiom,
    ! [P: set_nat > $o,A: set_nat] :
      ( ( ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X6: set_nat] :
                ( ( A = X6 )
                & ( P @ X6 ) ) )
          = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X6: set_nat] :
                ( ( A = X6 )
                & ( P @ X6 ) ) )
          = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_4217_Collect__conv__if2,axiom,
    ! [P: real > $o,A: real] :
      ( ( ( P @ A )
       => ( ( collect_real
            @ ^ [X6: real] :
                ( ( A = X6 )
                & ( P @ X6 ) ) )
          = ( insert_real @ A @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_real
            @ ^ [X6: real] :
                ( ( A = X6 )
                & ( P @ X6 ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if2
thf(fact_4218_Collect__conv__if2,axiom,
    ! [P: $o > $o,A: $o] :
      ( ( ( P @ A )
       => ( ( collect_o
            @ ^ [X6: $o] :
                ( ( A = X6 )
                & ( P @ X6 ) ) )
          = ( insert_o @ A @ bot_bot_set_o ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_o
            @ ^ [X6: $o] :
                ( ( A = X6 )
                & ( P @ X6 ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if2
thf(fact_4219_Collect__conv__if2,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X6: nat] :
                ( ( A = X6 )
                & ( P @ X6 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X6: nat] :
                ( ( A = X6 )
                & ( P @ X6 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_4220_Collect__conv__if2,axiom,
    ! [P: int > $o,A: int] :
      ( ( ( P @ A )
       => ( ( collect_int
            @ ^ [X6: int] :
                ( ( A = X6 )
                & ( P @ X6 ) ) )
          = ( insert_int @ A @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_int
            @ ^ [X6: int] :
                ( ( A = X6 )
                & ( P @ X6 ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if2
thf(fact_4221_mk__disjoint__insert,axiom,
    ! [A: $o,A2: set_o] :
      ( ( member_o @ A @ A2 )
     => ? [B8: set_o] :
          ( ( A2
            = ( insert_o @ A @ B8 ) )
          & ~ ( member_o @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_4222_mk__disjoint__insert,axiom,
    ! [A: complex,A2: set_complex] :
      ( ( member_complex @ A @ A2 )
     => ? [B8: set_complex] :
          ( ( A2
            = ( insert_complex @ A @ B8 ) )
          & ~ ( member_complex @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_4223_mk__disjoint__insert,axiom,
    ! [A: real,A2: set_real] :
      ( ( member_real @ A @ A2 )
     => ? [B8: set_real] :
          ( ( A2
            = ( insert_real @ A @ B8 ) )
          & ~ ( member_real @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_4224_mk__disjoint__insert,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ A @ A2 )
     => ? [B8: set_set_nat] :
          ( ( A2
            = ( insert_set_nat @ A @ B8 ) )
          & ~ ( member_set_nat @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_4225_mk__disjoint__insert,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ? [B8: set_nat] :
          ( ( A2
            = ( insert_nat @ A @ B8 ) )
          & ~ ( member_nat @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_4226_mk__disjoint__insert,axiom,
    ! [A: int,A2: set_int] :
      ( ( member_int @ A @ A2 )
     => ? [B8: set_int] :
          ( ( A2
            = ( insert_int @ A @ B8 ) )
          & ~ ( member_int @ A @ B8 ) ) ) ).

% mk_disjoint_insert
thf(fact_4227_insert__commute,axiom,
    ! [X: nat,Y: nat,A2: set_nat] :
      ( ( insert_nat @ X @ ( insert_nat @ Y @ A2 ) )
      = ( insert_nat @ Y @ ( insert_nat @ X @ A2 ) ) ) ).

% insert_commute
thf(fact_4228_insert__commute,axiom,
    ! [X: int,Y: int,A2: set_int] :
      ( ( insert_int @ X @ ( insert_int @ Y @ A2 ) )
      = ( insert_int @ Y @ ( insert_int @ X @ A2 ) ) ) ).

% insert_commute
thf(fact_4229_insert__commute,axiom,
    ! [X: real,Y: real,A2: set_real] :
      ( ( insert_real @ X @ ( insert_real @ Y @ A2 ) )
      = ( insert_real @ Y @ ( insert_real @ X @ A2 ) ) ) ).

% insert_commute
thf(fact_4230_insert__commute,axiom,
    ! [X: $o,Y: $o,A2: set_o] :
      ( ( insert_o @ X @ ( insert_o @ Y @ A2 ) )
      = ( insert_o @ Y @ ( insert_o @ X @ A2 ) ) ) ).

% insert_commute
thf(fact_4231_insert__Collect,axiom,
    ! [A: $o,P: $o > $o] :
      ( ( insert_o @ A @ ( collect_o @ P ) )
      = ( collect_o
        @ ^ [U2: $o] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_4232_insert__Collect,axiom,
    ! [A: real,P: real > $o] :
      ( ( insert_real @ A @ ( collect_real @ P ) )
      = ( collect_real
        @ ^ [U2: real] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_4233_insert__Collect,axiom,
    ! [A: list_nat,P: list_nat > $o] :
      ( ( insert_list_nat @ A @ ( collect_list_nat @ P ) )
      = ( collect_list_nat
        @ ^ [U2: list_nat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_4234_insert__Collect,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( insert_set_nat @ A @ ( collect_set_nat @ P ) )
      = ( collect_set_nat
        @ ^ [U2: set_nat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_4235_insert__Collect,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( insert_nat @ A @ ( collect_nat @ P ) )
      = ( collect_nat
        @ ^ [U2: nat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_4236_insert__Collect,axiom,
    ! [A: int,P: int > $o] :
      ( ( insert_int @ A @ ( collect_int @ P ) )
      = ( collect_int
        @ ^ [U2: int] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_4237_insert__eq__iff,axiom,
    ! [A: $o,A2: set_o,B: $o,B2: set_o] :
      ( ~ ( member_o @ A @ A2 )
     => ( ~ ( member_o @ B @ B2 )
       => ( ( ( insert_o @ A @ A2 )
            = ( insert_o @ B @ B2 ) )
          = ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A = ~ B )
             => ? [C6: set_o] :
                  ( ( A2
                    = ( insert_o @ B @ C6 ) )
                  & ~ ( member_o @ B @ C6 )
                  & ( B2
                    = ( insert_o @ A @ C6 ) )
                  & ~ ( member_o @ A @ C6 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_4238_insert__eq__iff,axiom,
    ! [A: complex,A2: set_complex,B: complex,B2: set_complex] :
      ( ~ ( member_complex @ A @ A2 )
     => ( ~ ( member_complex @ B @ B2 )
       => ( ( ( insert_complex @ A @ A2 )
            = ( insert_complex @ B @ B2 ) )
          = ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C6: set_complex] :
                  ( ( A2
                    = ( insert_complex @ B @ C6 ) )
                  & ~ ( member_complex @ B @ C6 )
                  & ( B2
                    = ( insert_complex @ A @ C6 ) )
                  & ~ ( member_complex @ A @ C6 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_4239_insert__eq__iff,axiom,
    ! [A: real,A2: set_real,B: real,B2: set_real] :
      ( ~ ( member_real @ A @ A2 )
     => ( ~ ( member_real @ B @ B2 )
       => ( ( ( insert_real @ A @ A2 )
            = ( insert_real @ B @ B2 ) )
          = ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C6: set_real] :
                  ( ( A2
                    = ( insert_real @ B @ C6 ) )
                  & ~ ( member_real @ B @ C6 )
                  & ( B2
                    = ( insert_real @ A @ C6 ) )
                  & ~ ( member_real @ A @ C6 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_4240_insert__eq__iff,axiom,
    ! [A: set_nat,A2: set_set_nat,B: set_nat,B2: set_set_nat] :
      ( ~ ( member_set_nat @ A @ A2 )
     => ( ~ ( member_set_nat @ B @ B2 )
       => ( ( ( insert_set_nat @ A @ A2 )
            = ( insert_set_nat @ B @ B2 ) )
          = ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C6: set_set_nat] :
                  ( ( A2
                    = ( insert_set_nat @ B @ C6 ) )
                  & ~ ( member_set_nat @ B @ C6 )
                  & ( B2
                    = ( insert_set_nat @ A @ C6 ) )
                  & ~ ( member_set_nat @ A @ C6 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_4241_insert__eq__iff,axiom,
    ! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
      ( ~ ( member_nat @ A @ A2 )
     => ( ~ ( member_nat @ B @ B2 )
       => ( ( ( insert_nat @ A @ A2 )
            = ( insert_nat @ B @ B2 ) )
          = ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C6: set_nat] :
                  ( ( A2
                    = ( insert_nat @ B @ C6 ) )
                  & ~ ( member_nat @ B @ C6 )
                  & ( B2
                    = ( insert_nat @ A @ C6 ) )
                  & ~ ( member_nat @ A @ C6 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_4242_insert__eq__iff,axiom,
    ! [A: int,A2: set_int,B: int,B2: set_int] :
      ( ~ ( member_int @ A @ A2 )
     => ( ~ ( member_int @ B @ B2 )
       => ( ( ( insert_int @ A @ A2 )
            = ( insert_int @ B @ B2 ) )
          = ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C6: set_int] :
                  ( ( A2
                    = ( insert_int @ B @ C6 ) )
                  & ~ ( member_int @ B @ C6 )
                  & ( B2
                    = ( insert_int @ A @ C6 ) )
                  & ~ ( member_int @ A @ C6 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_4243_insert__absorb,axiom,
    ! [A: $o,A2: set_o] :
      ( ( member_o @ A @ A2 )
     => ( ( insert_o @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_4244_insert__absorb,axiom,
    ! [A: complex,A2: set_complex] :
      ( ( member_complex @ A @ A2 )
     => ( ( insert_complex @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_4245_insert__absorb,axiom,
    ! [A: real,A2: set_real] :
      ( ( member_real @ A @ A2 )
     => ( ( insert_real @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_4246_insert__absorb,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ A @ A2 )
     => ( ( insert_set_nat @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_4247_insert__absorb,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ( ( insert_nat @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_4248_insert__absorb,axiom,
    ! [A: int,A2: set_int] :
      ( ( member_int @ A @ A2 )
     => ( ( insert_int @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_4249_insert__ident,axiom,
    ! [X: $o,A2: set_o,B2: set_o] :
      ( ~ ( member_o @ X @ A2 )
     => ( ~ ( member_o @ X @ B2 )
       => ( ( ( insert_o @ X @ A2 )
            = ( insert_o @ X @ B2 ) )
          = ( A2 = B2 ) ) ) ) ).

% insert_ident
thf(fact_4250_insert__ident,axiom,
    ! [X: complex,A2: set_complex,B2: set_complex] :
      ( ~ ( member_complex @ X @ A2 )
     => ( ~ ( member_complex @ X @ B2 )
       => ( ( ( insert_complex @ X @ A2 )
            = ( insert_complex @ X @ B2 ) )
          = ( A2 = B2 ) ) ) ) ).

% insert_ident
thf(fact_4251_insert__ident,axiom,
    ! [X: real,A2: set_real,B2: set_real] :
      ( ~ ( member_real @ X @ A2 )
     => ( ~ ( member_real @ X @ B2 )
       => ( ( ( insert_real @ X @ A2 )
            = ( insert_real @ X @ B2 ) )
          = ( A2 = B2 ) ) ) ) ).

% insert_ident
thf(fact_4252_insert__ident,axiom,
    ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A2 )
     => ( ~ ( member_set_nat @ X @ B2 )
       => ( ( ( insert_set_nat @ X @ A2 )
            = ( insert_set_nat @ X @ B2 ) )
          = ( A2 = B2 ) ) ) ) ).

% insert_ident
thf(fact_4253_insert__ident,axiom,
    ! [X: nat,A2: set_nat,B2: set_nat] :
      ( ~ ( member_nat @ X @ A2 )
     => ( ~ ( member_nat @ X @ B2 )
       => ( ( ( insert_nat @ X @ A2 )
            = ( insert_nat @ X @ B2 ) )
          = ( A2 = B2 ) ) ) ) ).

% insert_ident
thf(fact_4254_insert__ident,axiom,
    ! [X: int,A2: set_int,B2: set_int] :
      ( ~ ( member_int @ X @ A2 )
     => ( ~ ( member_int @ X @ B2 )
       => ( ( ( insert_int @ X @ A2 )
            = ( insert_int @ X @ B2 ) )
          = ( A2 = B2 ) ) ) ) ).

% insert_ident
thf(fact_4255_insert__compr,axiom,
    ( insert_o
    = ( ^ [A3: $o,B7: set_o] :
          ( collect_o
          @ ^ [X6: $o] :
              ( ( X6 = A3 )
              | ( member_o @ X6 @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_4256_insert__compr,axiom,
    ( insert_complex
    = ( ^ [A3: complex,B7: set_complex] :
          ( collect_complex
          @ ^ [X6: complex] :
              ( ( X6 = A3 )
              | ( member_complex @ X6 @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_4257_insert__compr,axiom,
    ( insert_real
    = ( ^ [A3: real,B7: set_real] :
          ( collect_real
          @ ^ [X6: real] :
              ( ( X6 = A3 )
              | ( member_real @ X6 @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_4258_insert__compr,axiom,
    ( insert_list_nat
    = ( ^ [A3: list_nat,B7: set_list_nat] :
          ( collect_list_nat
          @ ^ [X6: list_nat] :
              ( ( X6 = A3 )
              | ( member_list_nat @ X6 @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_4259_insert__compr,axiom,
    ( insert_set_nat
    = ( ^ [A3: set_nat,B7: set_set_nat] :
          ( collect_set_nat
          @ ^ [X6: set_nat] :
              ( ( X6 = A3 )
              | ( member_set_nat @ X6 @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_4260_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A3: nat,B7: set_nat] :
          ( collect_nat
          @ ^ [X6: nat] :
              ( ( X6 = A3 )
              | ( member_nat @ X6 @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_4261_insert__compr,axiom,
    ( insert_int
    = ( ^ [A3: int,B7: set_int] :
          ( collect_int
          @ ^ [X6: int] :
              ( ( X6 = A3 )
              | ( member_int @ X6 @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_4262_Set_Oset__insert,axiom,
    ! [X: $o,A2: set_o] :
      ( ( member_o @ X @ A2 )
     => ~ ! [B8: set_o] :
            ( ( A2
              = ( insert_o @ X @ B8 ) )
           => ( member_o @ X @ B8 ) ) ) ).

% Set.set_insert
thf(fact_4263_Set_Oset__insert,axiom,
    ! [X: complex,A2: set_complex] :
      ( ( member_complex @ X @ A2 )
     => ~ ! [B8: set_complex] :
            ( ( A2
              = ( insert_complex @ X @ B8 ) )
           => ( member_complex @ X @ B8 ) ) ) ).

% Set.set_insert
thf(fact_4264_Set_Oset__insert,axiom,
    ! [X: real,A2: set_real] :
      ( ( member_real @ X @ A2 )
     => ~ ! [B8: set_real] :
            ( ( A2
              = ( insert_real @ X @ B8 ) )
           => ( member_real @ X @ B8 ) ) ) ).

% Set.set_insert
thf(fact_4265_Set_Oset__insert,axiom,
    ! [X: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ X @ A2 )
     => ~ ! [B8: set_set_nat] :
            ( ( A2
              = ( insert_set_nat @ X @ B8 ) )
           => ( member_set_nat @ X @ B8 ) ) ) ).

% Set.set_insert
thf(fact_4266_Set_Oset__insert,axiom,
    ! [X: nat,A2: set_nat] :
      ( ( member_nat @ X @ A2 )
     => ~ ! [B8: set_nat] :
            ( ( A2
              = ( insert_nat @ X @ B8 ) )
           => ( member_nat @ X @ B8 ) ) ) ).

% Set.set_insert
thf(fact_4267_Set_Oset__insert,axiom,
    ! [X: int,A2: set_int] :
      ( ( member_int @ X @ A2 )
     => ~ ! [B8: set_int] :
            ( ( A2
              = ( insert_int @ X @ B8 ) )
           => ( member_int @ X @ B8 ) ) ) ).

% Set.set_insert
thf(fact_4268_insertI2,axiom,
    ! [A: $o,B2: set_o,B: $o] :
      ( ( member_o @ A @ B2 )
     => ( member_o @ A @ ( insert_o @ B @ B2 ) ) ) ).

% insertI2
thf(fact_4269_insertI2,axiom,
    ! [A: complex,B2: set_complex,B: complex] :
      ( ( member_complex @ A @ B2 )
     => ( member_complex @ A @ ( insert_complex @ B @ B2 ) ) ) ).

% insertI2
thf(fact_4270_insertI2,axiom,
    ! [A: real,B2: set_real,B: real] :
      ( ( member_real @ A @ B2 )
     => ( member_real @ A @ ( insert_real @ B @ B2 ) ) ) ).

% insertI2
thf(fact_4271_insertI2,axiom,
    ! [A: set_nat,B2: set_set_nat,B: set_nat] :
      ( ( member_set_nat @ A @ B2 )
     => ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).

% insertI2
thf(fact_4272_insertI2,axiom,
    ! [A: nat,B2: set_nat,B: nat] :
      ( ( member_nat @ A @ B2 )
     => ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).

% insertI2
thf(fact_4273_insertI2,axiom,
    ! [A: int,B2: set_int,B: int] :
      ( ( member_int @ A @ B2 )
     => ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).

% insertI2
thf(fact_4274_insertI1,axiom,
    ! [A: $o,B2: set_o] : ( member_o @ A @ ( insert_o @ A @ B2 ) ) ).

% insertI1
thf(fact_4275_insertI1,axiom,
    ! [A: complex,B2: set_complex] : ( member_complex @ A @ ( insert_complex @ A @ B2 ) ) ).

% insertI1
thf(fact_4276_insertI1,axiom,
    ! [A: real,B2: set_real] : ( member_real @ A @ ( insert_real @ A @ B2 ) ) ).

% insertI1
thf(fact_4277_insertI1,axiom,
    ! [A: set_nat,B2: set_set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ B2 ) ) ).

% insertI1
thf(fact_4278_insertI1,axiom,
    ! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B2 ) ) ).

% insertI1
thf(fact_4279_insertI1,axiom,
    ! [A: int,B2: set_int] : ( member_int @ A @ ( insert_int @ A @ B2 ) ) ).

% insertI1
thf(fact_4280_insertE,axiom,
    ! [A: $o,B: $o,A2: set_o] :
      ( ( member_o @ A @ ( insert_o @ B @ A2 ) )
     => ( ( A = ~ B )
       => ( member_o @ A @ A2 ) ) ) ).

% insertE
thf(fact_4281_insertE,axiom,
    ! [A: complex,B: complex,A2: set_complex] :
      ( ( member_complex @ A @ ( insert_complex @ B @ A2 ) )
     => ( ( A != B )
       => ( member_complex @ A @ A2 ) ) ) ).

% insertE
thf(fact_4282_insertE,axiom,
    ! [A: real,B: real,A2: set_real] :
      ( ( member_real @ A @ ( insert_real @ B @ A2 ) )
     => ( ( A != B )
       => ( member_real @ A @ A2 ) ) ) ).

% insertE
thf(fact_4283_insertE,axiom,
    ! [A: set_nat,B: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
     => ( ( A != B )
       => ( member_set_nat @ A @ A2 ) ) ) ).

% insertE
thf(fact_4284_insertE,axiom,
    ! [A: nat,B: nat,A2: set_nat] :
      ( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
     => ( ( A != B )
       => ( member_nat @ A @ A2 ) ) ) ).

% insertE
thf(fact_4285_insertE,axiom,
    ! [A: int,B: int,A2: set_int] :
      ( ( member_int @ A @ ( insert_int @ B @ A2 ) )
     => ( ( A != B )
       => ( member_int @ A @ A2 ) ) ) ).

% insertE
thf(fact_4286_lambda__zero,axiom,
    ( ( ^ [H: complex] : zero_zero_complex )
    = ( times_times_complex @ zero_zero_complex ) ) ).

% lambda_zero
thf(fact_4287_lambda__zero,axiom,
    ( ( ^ [H: real] : zero_zero_real )
    = ( times_times_real @ zero_zero_real ) ) ).

% lambda_zero
thf(fact_4288_lambda__zero,axiom,
    ( ( ^ [H: rat] : zero_zero_rat )
    = ( times_times_rat @ zero_zero_rat ) ) ).

% lambda_zero
thf(fact_4289_lambda__zero,axiom,
    ( ( ^ [H: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_4290_lambda__zero,axiom,
    ( ( ^ [H: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_4291_lambda__one,axiom,
    ( ( ^ [X6: complex] : X6 )
    = ( times_times_complex @ one_one_complex ) ) ).

% lambda_one
thf(fact_4292_lambda__one,axiom,
    ( ( ^ [X6: real] : X6 )
    = ( times_times_real @ one_one_real ) ) ).

% lambda_one
thf(fact_4293_lambda__one,axiom,
    ( ( ^ [X6: rat] : X6 )
    = ( times_times_rat @ one_one_rat ) ) ).

% lambda_one
thf(fact_4294_lambda__one,axiom,
    ( ( ^ [X6: nat] : X6 )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_4295_lambda__one,axiom,
    ( ( ^ [X6: int] : X6 )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_4296_set__update__subset__insert,axiom,
    ! [Xs2: list_real,I3: nat,X: real] : ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs2 @ I3 @ X ) ) @ ( insert_real @ X @ ( set_real2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_4297_set__update__subset__insert,axiom,
    ! [Xs2: list_o,I3: nat,X: $o] : ( ord_less_eq_set_o @ ( set_o2 @ ( list_update_o @ Xs2 @ I3 @ X ) ) @ ( insert_o @ X @ ( set_o2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_4298_set__update__subset__insert,axiom,
    ! [Xs2: list_int,I3: nat,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I3 @ X ) ) @ ( insert_int @ X @ ( set_int2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_4299_set__update__subset__insert,axiom,
    ! [Xs2: list_VEBT_VEBT,I3: nat,X: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I3 @ X ) ) @ ( insert_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_4300_set__update__subset__insert,axiom,
    ! [Xs2: list_nat,I3: nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I3 @ X ) ) @ ( insert_nat @ X @ ( set_nat2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_4301_set__vebt__def,axiom,
    ( vEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).

% set_vebt_def
thf(fact_4302_singletonD,axiom,
    ! [B: complex,A: complex] :
      ( ( member_complex @ B @ ( insert_complex @ A @ bot_bot_set_complex ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_4303_singletonD,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_4304_singletonD,axiom,
    ! [B: real,A: real] :
      ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_4305_singletonD,axiom,
    ! [B: $o,A: $o] :
      ( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_4306_singletonD,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_4307_singletonD,axiom,
    ! [B: int,A: int] :
      ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_4308_singleton__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( member_complex @ B @ ( insert_complex @ A @ bot_bot_set_complex ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_4309_singleton__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_4310_singleton__iff,axiom,
    ! [B: real,A: real] :
      ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_4311_singleton__iff,axiom,
    ! [B: $o,A: $o] :
      ( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_4312_singleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_4313_singleton__iff,axiom,
    ! [B: int,A: int] :
      ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_4314_doubleton__eq__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( insert_real @ A @ ( insert_real @ B @ bot_bot_set_real ) )
        = ( insert_real @ C @ ( insert_real @ D @ bot_bot_set_real ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_4315_doubleton__eq__iff,axiom,
    ! [A: $o,B: $o,C: $o,D: $o] :
      ( ( ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) )
        = ( insert_o @ C @ ( insert_o @ D @ bot_bot_set_o ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_4316_doubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
        = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_4317_doubleton__eq__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) )
        = ( insert_int @ C @ ( insert_int @ D @ bot_bot_set_int ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_4318_insert__not__empty,axiom,
    ! [A: real,A2: set_real] :
      ( ( insert_real @ A @ A2 )
     != bot_bot_set_real ) ).

% insert_not_empty
thf(fact_4319_insert__not__empty,axiom,
    ! [A: $o,A2: set_o] :
      ( ( insert_o @ A @ A2 )
     != bot_bot_set_o ) ).

% insert_not_empty
thf(fact_4320_insert__not__empty,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( insert_nat @ A @ A2 )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_4321_insert__not__empty,axiom,
    ! [A: int,A2: set_int] :
      ( ( insert_int @ A @ A2 )
     != bot_bot_set_int ) ).

% insert_not_empty
thf(fact_4322_singleton__inject,axiom,
    ! [A: real,B: real] :
      ( ( ( insert_real @ A @ bot_bot_set_real )
        = ( insert_real @ B @ bot_bot_set_real ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_4323_singleton__inject,axiom,
    ! [A: $o,B: $o] :
      ( ( ( insert_o @ A @ bot_bot_set_o )
        = ( insert_o @ B @ bot_bot_set_o ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_4324_singleton__inject,axiom,
    ! [A: nat,B: nat] :
      ( ( ( insert_nat @ A @ bot_bot_set_nat )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_4325_singleton__inject,axiom,
    ! [A: int,B: int] :
      ( ( ( insert_int @ A @ bot_bot_set_int )
        = ( insert_int @ B @ bot_bot_set_int ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_4326_insert__mono,axiom,
    ! [C5: set_int,D4: set_int,A: int] :
      ( ( ord_less_eq_set_int @ C5 @ D4 )
     => ( ord_less_eq_set_int @ ( insert_int @ A @ C5 ) @ ( insert_int @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_4327_insert__mono,axiom,
    ! [C5: set_real,D4: set_real,A: real] :
      ( ( ord_less_eq_set_real @ C5 @ D4 )
     => ( ord_less_eq_set_real @ ( insert_real @ A @ C5 ) @ ( insert_real @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_4328_insert__mono,axiom,
    ! [C5: set_o,D4: set_o,A: $o] :
      ( ( ord_less_eq_set_o @ C5 @ D4 )
     => ( ord_less_eq_set_o @ ( insert_o @ A @ C5 ) @ ( insert_o @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_4329_insert__mono,axiom,
    ! [C5: set_nat,D4: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ C5 @ D4 )
     => ( ord_less_eq_set_nat @ ( insert_nat @ A @ C5 ) @ ( insert_nat @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_4330_subset__insert,axiom,
    ! [X: $o,A2: set_o,B2: set_o] :
      ( ~ ( member_o @ X @ A2 )
     => ( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ B2 ) )
        = ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ).

% subset_insert
thf(fact_4331_subset__insert,axiom,
    ! [X: complex,A2: set_complex,B2: set_complex] :
      ( ~ ( member_complex @ X @ A2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ ( insert_complex @ X @ B2 ) )
        = ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ) ).

% subset_insert
thf(fact_4332_subset__insert,axiom,
    ! [X: real,A2: set_real,B2: set_real] :
      ( ~ ( member_real @ X @ A2 )
     => ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B2 ) )
        = ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).

% subset_insert
thf(fact_4333_subset__insert,axiom,
    ! [X: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A2 )
     => ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
        = ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).

% subset_insert
thf(fact_4334_subset__insert,axiom,
    ! [X: int,A2: set_int,B2: set_int] :
      ( ~ ( member_int @ X @ A2 )
     => ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B2 ) )
        = ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).

% subset_insert
thf(fact_4335_subset__insert,axiom,
    ! [X: nat,A2: set_nat,B2: set_nat] :
      ( ~ ( member_nat @ X @ A2 )
     => ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
        = ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).

% subset_insert
thf(fact_4336_subset__insertI,axiom,
    ! [B2: set_int,A: int] : ( ord_less_eq_set_int @ B2 @ ( insert_int @ A @ B2 ) ) ).

% subset_insertI
thf(fact_4337_subset__insertI,axiom,
    ! [B2: set_real,A: real] : ( ord_less_eq_set_real @ B2 @ ( insert_real @ A @ B2 ) ) ).

% subset_insertI
thf(fact_4338_subset__insertI,axiom,
    ! [B2: set_o,A: $o] : ( ord_less_eq_set_o @ B2 @ ( insert_o @ A @ B2 ) ) ).

% subset_insertI
thf(fact_4339_subset__insertI,axiom,
    ! [B2: set_nat,A: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat @ A @ B2 ) ) ).

% subset_insertI
thf(fact_4340_subset__insertI2,axiom,
    ! [A2: set_int,B2: set_int,B: int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ B2 ) ) ) ).

% subset_insertI2
thf(fact_4341_subset__insertI2,axiom,
    ! [A2: set_real,B2: set_real,B: real] :
      ( ( ord_less_eq_set_real @ A2 @ B2 )
     => ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ B2 ) ) ) ).

% subset_insertI2
thf(fact_4342_subset__insertI2,axiom,
    ! [A2: set_o,B2: set_o,B: $o] :
      ( ( ord_less_eq_set_o @ A2 @ B2 )
     => ( ord_less_eq_set_o @ A2 @ ( insert_o @ B @ B2 ) ) ) ).

% subset_insertI2
thf(fact_4343_subset__insertI2,axiom,
    ! [A2: set_nat,B2: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).

% subset_insertI2
thf(fact_4344_insert__Diff__if,axiom,
    ! [X: $o,B2: set_o,A2: set_o] :
      ( ( ( member_o @ X @ B2 )
       => ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ B2 )
          = ( minus_minus_set_o @ A2 @ B2 ) ) )
      & ( ~ ( member_o @ X @ B2 )
       => ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ B2 )
          = ( insert_o @ X @ ( minus_minus_set_o @ A2 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_4345_insert__Diff__if,axiom,
    ! [X: complex,B2: set_complex,A2: set_complex] :
      ( ( ( member_complex @ X @ B2 )
       => ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A2 ) @ B2 )
          = ( minus_811609699411566653omplex @ A2 @ B2 ) ) )
      & ( ~ ( member_complex @ X @ B2 )
       => ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A2 ) @ B2 )
          = ( insert_complex @ X @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_4346_insert__Diff__if,axiom,
    ! [X: real,B2: set_real,A2: set_real] :
      ( ( ( member_real @ X @ B2 )
       => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
          = ( minus_minus_set_real @ A2 @ B2 ) ) )
      & ( ~ ( member_real @ X @ B2 )
       => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B2 )
          = ( insert_real @ X @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_4347_insert__Diff__if,axiom,
    ! [X: set_nat,B2: set_set_nat,A2: set_set_nat] :
      ( ( ( member_set_nat @ X @ B2 )
       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B2 )
          = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) )
      & ( ~ ( member_set_nat @ X @ B2 )
       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B2 )
          = ( insert_set_nat @ X @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_4348_insert__Diff__if,axiom,
    ! [X: int,B2: set_int,A2: set_int] :
      ( ( ( member_int @ X @ B2 )
       => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B2 )
          = ( minus_minus_set_int @ A2 @ B2 ) ) )
      & ( ~ ( member_int @ X @ B2 )
       => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B2 )
          = ( insert_int @ X @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_4349_insert__Diff__if,axiom,
    ! [X: nat,B2: set_nat,A2: set_nat] :
      ( ( ( member_nat @ X @ B2 )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
          = ( minus_minus_set_nat @ A2 @ B2 ) ) )
      & ( ~ ( member_nat @ X @ B2 )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
          = ( insert_nat @ X @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_4350_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numera1916890842035813515d_enat @ ( bit0 @ N ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) ) ).

% numeral_code(2)
thf(fact_4351_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_code(2)
thf(fact_4352_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).

% numeral_code(2)
thf(fact_4353_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).

% numeral_code(2)
thf(fact_4354_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).

% numeral_code(2)
thf(fact_4355_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).

% numeral_code(2)
thf(fact_4356_power__numeral__even,axiom,
    ! [Z2: complex,W: num] :
      ( ( power_power_complex @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_complex @ ( power_power_complex @ Z2 @ ( numeral_numeral_nat @ W ) ) @ ( power_power_complex @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_4357_power__numeral__even,axiom,
    ! [Z2: real,W: num] :
      ( ( power_power_real @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_real @ ( power_power_real @ Z2 @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_4358_power__numeral__even,axiom,
    ! [Z2: rat,W: num] :
      ( ( power_power_rat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_rat @ ( power_power_rat @ Z2 @ ( numeral_numeral_nat @ W ) ) @ ( power_power_rat @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_4359_power__numeral__even,axiom,
    ! [Z2: nat,W: num] :
      ( ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_nat @ ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_4360_power__numeral__even,axiom,
    ! [Z2: int,W: num] :
      ( ( power_power_int @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_int @ ( power_power_int @ Z2 @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_4361_subset__singletonD,axiom,
    ! [A2: set_real,X: real] :
      ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) )
     => ( ( A2 = bot_bot_set_real )
        | ( A2
          = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).

% subset_singletonD
thf(fact_4362_subset__singletonD,axiom,
    ! [A2: set_o,X: $o] :
      ( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) )
     => ( ( A2 = bot_bot_set_o )
        | ( A2
          = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).

% subset_singletonD
thf(fact_4363_subset__singletonD,axiom,
    ! [A2: set_int,X: int] :
      ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) )
     => ( ( A2 = bot_bot_set_int )
        | ( A2
          = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).

% subset_singletonD
thf(fact_4364_subset__singletonD,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
     => ( ( A2 = bot_bot_set_nat )
        | ( A2
          = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% subset_singletonD
thf(fact_4365_subset__singleton__iff,axiom,
    ! [X9: set_real,A: real] :
      ( ( ord_less_eq_set_real @ X9 @ ( insert_real @ A @ bot_bot_set_real ) )
      = ( ( X9 = bot_bot_set_real )
        | ( X9
          = ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).

% subset_singleton_iff
thf(fact_4366_subset__singleton__iff,axiom,
    ! [X9: set_o,A: $o] :
      ( ( ord_less_eq_set_o @ X9 @ ( insert_o @ A @ bot_bot_set_o ) )
      = ( ( X9 = bot_bot_set_o )
        | ( X9
          = ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).

% subset_singleton_iff
thf(fact_4367_subset__singleton__iff,axiom,
    ! [X9: set_int,A: int] :
      ( ( ord_less_eq_set_int @ X9 @ ( insert_int @ A @ bot_bot_set_int ) )
      = ( ( X9 = bot_bot_set_int )
        | ( X9
          = ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).

% subset_singleton_iff
thf(fact_4368_subset__singleton__iff,axiom,
    ! [X9: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ X9 @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( ( X9 = bot_bot_set_nat )
        | ( X9
          = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_4369_atLeastAtMost__singleton_H,axiom,
    ! [A: $o,B: $o] :
      ( ( A = B )
     => ( ( set_or8904488021354931149Most_o @ A @ B )
        = ( insert_o @ A @ bot_bot_set_o ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_4370_atLeastAtMost__singleton_H,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( ( set_or1269000886237332187st_nat @ A @ B )
        = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_4371_atLeastAtMost__singleton_H,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( ( set_or1266510415728281911st_int @ A @ B )
        = ( insert_int @ A @ bot_bot_set_int ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_4372_atLeastAtMost__singleton_H,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
     => ( ( set_or1222579329274155063t_real @ A @ B )
        = ( insert_real @ A @ bot_bot_set_real ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_4373_Diff__insert__absorb,axiom,
    ! [X: complex,A2: set_complex] :
      ( ~ ( member_complex @ X @ A2 )
     => ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A2 ) @ ( insert_complex @ X @ bot_bot_set_complex ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_4374_Diff__insert__absorb,axiom,
    ! [X: set_nat,A2: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A2 )
     => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_4375_Diff__insert__absorb,axiom,
    ! [X: real,A2: set_real] :
      ( ~ ( member_real @ X @ A2 )
     => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ ( insert_real @ X @ bot_bot_set_real ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_4376_Diff__insert__absorb,axiom,
    ! [X: $o,A2: set_o] :
      ( ~ ( member_o @ X @ A2 )
     => ( ( minus_minus_set_o @ ( insert_o @ X @ A2 ) @ ( insert_o @ X @ bot_bot_set_o ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_4377_Diff__insert__absorb,axiom,
    ! [X: int,A2: set_int] :
      ( ~ ( member_int @ X @ A2 )
     => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ ( insert_int @ X @ bot_bot_set_int ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_4378_Diff__insert__absorb,axiom,
    ! [X: nat,A2: set_nat] :
      ( ~ ( member_nat @ X @ A2 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_4379_Diff__insert2,axiom,
    ! [A2: set_real,A: real,B2: set_real] :
      ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) )
      = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_4380_Diff__insert2,axiom,
    ! [A2: set_o,A: $o,B2: set_o] :
      ( ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B2 ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_4381_Diff__insert2,axiom,
    ! [A2: set_int,A: int,B2: set_int] :
      ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_4382_Diff__insert2,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_4383_insert__Diff,axiom,
    ! [A: complex,A2: set_complex] :
      ( ( member_complex @ A @ A2 )
     => ( ( insert_complex @ A @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_4384_insert__Diff,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ A @ A2 )
     => ( ( insert_set_nat @ A @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_4385_insert__Diff,axiom,
    ! [A: real,A2: set_real] :
      ( ( member_real @ A @ A2 )
     => ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_4386_insert__Diff,axiom,
    ! [A: $o,A2: set_o] :
      ( ( member_o @ A @ A2 )
     => ( ( insert_o @ A @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_4387_insert__Diff,axiom,
    ! [A: int,A2: set_int] :
      ( ( member_int @ A @ A2 )
     => ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_4388_insert__Diff,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_4389_Diff__insert,axiom,
    ! [A2: set_real,A: real,B2: set_real] :
      ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) )
      = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ B2 ) @ ( insert_real @ A @ bot_bot_set_real ) ) ) ).

% Diff_insert
thf(fact_4390_Diff__insert,axiom,
    ! [A2: set_o,A: $o,B2: set_o] :
      ( ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B2 ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ B2 ) @ ( insert_o @ A @ bot_bot_set_o ) ) ) ).

% Diff_insert
thf(fact_4391_Diff__insert,axiom,
    ! [A2: set_int,A: int,B2: set_int] :
      ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ ( insert_int @ A @ bot_bot_set_int ) ) ) ).

% Diff_insert
thf(fact_4392_Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).

% Diff_insert
thf(fact_4393_subset__Diff__insert,axiom,
    ! [A2: set_o,B2: set_o,X: $o,C5: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ ( minus_minus_set_o @ B2 @ ( insert_o @ X @ C5 ) ) )
      = ( ( ord_less_eq_set_o @ A2 @ ( minus_minus_set_o @ B2 @ C5 ) )
        & ~ ( member_o @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_4394_subset__Diff__insert,axiom,
    ! [A2: set_complex,B2: set_complex,X: complex,C5: set_complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ ( minus_811609699411566653omplex @ B2 @ ( insert_complex @ X @ C5 ) ) )
      = ( ( ord_le211207098394363844omplex @ A2 @ ( minus_811609699411566653omplex @ B2 @ C5 ) )
        & ~ ( member_complex @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_4395_subset__Diff__insert,axiom,
    ! [A2: set_real,B2: set_real,X: real,C5: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ ( insert_real @ X @ C5 ) ) )
      = ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ C5 ) )
        & ~ ( member_real @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_4396_subset__Diff__insert,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,X: set_nat,C5: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ ( insert_set_nat @ X @ C5 ) ) )
      = ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ C5 ) )
        & ~ ( member_set_nat @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_4397_subset__Diff__insert,axiom,
    ! [A2: set_int,B2: set_int,X: int,C5: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B2 @ ( insert_int @ X @ C5 ) ) )
      = ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B2 @ C5 ) )
        & ~ ( member_int @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_4398_subset__Diff__insert,axiom,
    ! [A2: set_nat,B2: set_nat,X: nat,C5: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ ( insert_nat @ X @ C5 ) ) )
      = ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C5 ) )
        & ~ ( member_nat @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_4399_set__update__subsetI,axiom,
    ! [Xs2: list_complex,A2: set_complex,X: complex,I3: nat] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ord_le211207098394363844omplex @ ( set_complex2 @ ( list_update_complex @ Xs2 @ I3 @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_4400_set__update__subsetI,axiom,
    ! [Xs2: list_real,A2: set_real,X: real,I3: nat] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs2 @ I3 @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_4401_set__update__subsetI,axiom,
    ! [Xs2: list_set_nat,A2: set_set_nat,X: set_nat,I3: nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs2 ) @ A2 )
     => ( ( member_set_nat @ X @ A2 )
       => ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ ( list_update_set_nat @ Xs2 @ I3 @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_4402_set__update__subsetI,axiom,
    ! [Xs2: list_int,A2: set_int,X: int,I3: nat] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
     => ( ( member_int @ X @ A2 )
       => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I3 @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_4403_set__update__subsetI,axiom,
    ! [Xs2: list_VEBT_VEBT,A2: set_VEBT_VEBT,X: vEBT_VEBT,I3: nat] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
     => ( ( member_VEBT_VEBT @ X @ A2 )
       => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I3 @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_4404_set__update__subsetI,axiom,
    ! [Xs2: list_nat,A2: set_nat,X: nat,I3: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
     => ( ( member_nat @ X @ A2 )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I3 @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_4405_Diff__single__insert,axiom,
    ! [A2: set_real,X: real,B2: set_real] :
      ( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 )
     => ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_4406_Diff__single__insert,axiom,
    ! [A2: set_o,X: $o,B2: set_o] :
      ( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B2 )
     => ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_4407_Diff__single__insert,axiom,
    ! [A2: set_int,X: int,B2: set_int] :
      ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B2 )
     => ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_4408_Diff__single__insert,axiom,
    ! [A2: set_nat,X: nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 )
     => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_4409_subset__insert__iff,axiom,
    ! [A2: set_complex,X: complex,B2: set_complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ ( insert_complex @ X @ B2 ) )
      = ( ( ( member_complex @ X @ A2 )
         => ( ord_le211207098394363844omplex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) @ B2 ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_4410_subset__insert__iff,axiom,
    ! [A2: set_set_nat,X: set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
      = ( ( ( member_set_nat @ X @ A2 )
         => ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B2 ) )
        & ( ~ ( member_set_nat @ X @ A2 )
         => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_4411_subset__insert__iff,axiom,
    ! [A2: set_real,X: real,B2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B2 ) )
      = ( ( ( member_real @ X @ A2 )
         => ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_4412_subset__insert__iff,axiom,
    ! [A2: set_o,X: $o,B2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X @ B2 ) )
      = ( ( ( member_o @ X @ A2 )
         => ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B2 ) )
        & ( ~ ( member_o @ X @ A2 )
         => ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_4413_subset__insert__iff,axiom,
    ! [A2: set_int,X: int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B2 ) )
      = ( ( ( member_int @ X @ A2 )
         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B2 ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_4414_subset__insert__iff,axiom,
    ! [A2: set_nat,X: nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
      = ( ( ( member_nat @ X @ A2 )
         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) )
        & ( ~ ( member_nat @ X @ A2 )
         => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_4415_atLeast0__atMost__Suc,axiom,
    ! [N: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% atLeast0_atMost_Suc
thf(fact_4416_atLeastAtMost__insertL,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
        = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% atLeastAtMost_insertL
thf(fact_4417_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
        = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).

% atLeastAtMostSuc_conv
thf(fact_4418_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( set_or1269000886237332187st_nat @ M @ N )
        = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).

% Icc_eq_insert_lb_nat
thf(fact_4419_set__update__memI,axiom,
    ! [N: nat,Xs2: list_complex,X: complex] :
      ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
     => ( member_complex @ X @ ( set_complex2 @ ( list_update_complex @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_4420_set__update__memI,axiom,
    ! [N: nat,Xs2: list_real,X: real] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
     => ( member_real @ X @ ( set_real2 @ ( list_update_real @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_4421_set__update__memI,axiom,
    ! [N: nat,Xs2: list_set_nat,X: set_nat] :
      ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs2 ) )
     => ( member_set_nat @ X @ ( set_set_nat2 @ ( list_update_set_nat @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_4422_set__update__memI,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_4423_set__update__memI,axiom,
    ! [N: nat,Xs2: list_o,X: $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
     => ( member_o @ X @ ( set_o2 @ ( list_update_o @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_4424_set__update__memI,axiom,
    ! [N: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_4425_set__update__memI,axiom,
    ! [N: nat,Xs2: list_int,X: int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
     => ( member_int @ X @ ( set_int2 @ ( list_update_int @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_4426_list__update__same__conv,axiom,
    ! [I3: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I3 @ X )
          = Xs2 )
        = ( ( nth_VEBT_VEBT @ Xs2 @ I3 )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_4427_list__update__same__conv,axiom,
    ! [I3: nat,Xs2: list_o,X: $o] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
     => ( ( ( list_update_o @ Xs2 @ I3 @ X )
          = Xs2 )
        = ( ( nth_o @ Xs2 @ I3 )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_4428_list__update__same__conv,axiom,
    ! [I3: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ( list_update_nat @ Xs2 @ I3 @ X )
          = Xs2 )
        = ( ( nth_nat @ Xs2 @ I3 )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_4429_list__update__same__conv,axiom,
    ! [I3: nat,Xs2: list_int,X: int] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
     => ( ( ( list_update_int @ Xs2 @ I3 @ X )
          = Xs2 )
        = ( ( nth_int @ Xs2 @ I3 )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_4430_nth__list__update,axiom,
    ! [I3: nat,Xs2: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ( I3 = J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I3 @ X ) @ J )
            = X ) )
        & ( ( I3 != J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I3 @ X ) @ J )
            = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_4431_nth__list__update,axiom,
    ! [I3: nat,Xs2: list_o,X: $o,J: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_o @ ( list_update_o @ Xs2 @ I3 @ X ) @ J )
        = ( ( ( I3 = J )
           => X )
          & ( ( I3 != J )
           => ( nth_o @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_4432_nth__list__update,axiom,
    ! [I3: nat,Xs2: list_nat,J: nat,X: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ( I3 = J )
         => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I3 @ X ) @ J )
            = X ) )
        & ( ( I3 != J )
         => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I3 @ X ) @ J )
            = ( nth_nat @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_4433_nth__list__update,axiom,
    ! [I3: nat,Xs2: list_int,J: nat,X: int] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
     => ( ( ( I3 = J )
         => ( ( nth_int @ ( list_update_int @ Xs2 @ I3 @ X ) @ J )
            = X ) )
        & ( ( I3 != J )
         => ( ( nth_int @ ( list_update_int @ Xs2 @ I3 @ X ) @ J )
            = ( nth_int @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_4434_psubset__insert__iff,axiom,
    ! [A2: set_complex,X: complex,B2: set_complex] :
      ( ( ord_less_set_complex @ A2 @ ( insert_complex @ X @ B2 ) )
      = ( ( ( member_complex @ X @ B2 )
         => ( ord_less_set_complex @ A2 @ B2 ) )
        & ( ~ ( member_complex @ X @ B2 )
         => ( ( ( member_complex @ X @ A2 )
             => ( ord_less_set_complex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) @ B2 ) )
            & ( ~ ( member_complex @ X @ A2 )
             => ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_4435_psubset__insert__iff,axiom,
    ! [A2: set_set_nat,X: set_nat,B2: set_set_nat] :
      ( ( ord_less_set_set_nat @ A2 @ ( insert_set_nat @ X @ B2 ) )
      = ( ( ( member_set_nat @ X @ B2 )
         => ( ord_less_set_set_nat @ A2 @ B2 ) )
        & ( ~ ( member_set_nat @ X @ B2 )
         => ( ( ( member_set_nat @ X @ A2 )
             => ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B2 ) )
            & ( ~ ( member_set_nat @ X @ A2 )
             => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_4436_psubset__insert__iff,axiom,
    ! [A2: set_real,X: real,B2: set_real] :
      ( ( ord_less_set_real @ A2 @ ( insert_real @ X @ B2 ) )
      = ( ( ( member_real @ X @ B2 )
         => ( ord_less_set_real @ A2 @ B2 ) )
        & ( ~ ( member_real @ X @ B2 )
         => ( ( ( member_real @ X @ A2 )
             => ( ord_less_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B2 ) )
            & ( ~ ( member_real @ X @ A2 )
             => ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_4437_psubset__insert__iff,axiom,
    ! [A2: set_o,X: $o,B2: set_o] :
      ( ( ord_less_set_o @ A2 @ ( insert_o @ X @ B2 ) )
      = ( ( ( member_o @ X @ B2 )
         => ( ord_less_set_o @ A2 @ B2 ) )
        & ( ~ ( member_o @ X @ B2 )
         => ( ( ( member_o @ X @ A2 )
             => ( ord_less_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B2 ) )
            & ( ~ ( member_o @ X @ A2 )
             => ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_4438_psubset__insert__iff,axiom,
    ! [A2: set_int,X: int,B2: set_int] :
      ( ( ord_less_set_int @ A2 @ ( insert_int @ X @ B2 ) )
      = ( ( ( member_int @ X @ B2 )
         => ( ord_less_set_int @ A2 @ B2 ) )
        & ( ~ ( member_int @ X @ B2 )
         => ( ( ( member_int @ X @ A2 )
             => ( ord_less_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B2 ) )
            & ( ~ ( member_int @ X @ A2 )
             => ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_4439_psubset__insert__iff,axiom,
    ! [A2: set_nat,X: nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
      = ( ( ( member_nat @ X @ B2 )
         => ( ord_less_set_nat @ A2 @ B2 ) )
        & ( ~ ( member_nat @ X @ B2 )
         => ( ( ( member_nat @ X @ A2 )
             => ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) )
            & ( ~ ( member_nat @ X @ A2 )
             => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_4440_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 ) ) ) )
       => ( ( ? [X4: int] : ( P @ X4 ) )
          = ( ? [X6: int] :
                ( ( member_int @ X6 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                & ( P @ X6 ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_4441_aset_I7_J,axiom,
    ! [D4: int,A2: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X3: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A2 )
                 => ( X3
                   != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ T @ X3 )
           => ( ord_less_int @ T @ ( plus_plus_int @ X3 @ D4 ) ) ) ) ) ).

% aset(7)
thf(fact_4442_aset_I5_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ A2 )
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X3
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ X3 @ T )
             => ( ord_less_int @ ( plus_plus_int @ X3 @ D4 ) @ T ) ) ) ) ) ).

% aset(5)
thf(fact_4443_aset_I4_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ A2 )
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X3
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X3 != T )
             => ( ( plus_plus_int @ X3 @ D4 )
               != T ) ) ) ) ) ).

% aset(4)
thf(fact_4444_aset_I3_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X3
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X3 = T )
             => ( ( plus_plus_int @ X3 @ D4 )
                = T ) ) ) ) ) ).

% aset(3)
thf(fact_4445_bset_I7_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ B2 )
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B2 )
                   => ( X3
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ T @ X3 )
             => ( ord_less_int @ T @ ( minus_minus_int @ X3 @ D4 ) ) ) ) ) ) ).

% bset(7)
thf(fact_4446_bset_I5_J,axiom,
    ! [D4: int,B2: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X3: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B2 )
                 => ( X3
                   != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ X3 @ T )
           => ( ord_less_int @ ( minus_minus_int @ X3 @ D4 ) @ T ) ) ) ) ).

% bset(5)
thf(fact_4447_bset_I4_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ B2 )
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B2 )
                   => ( X3
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X3 != T )
             => ( ( minus_minus_int @ X3 @ D4 )
               != T ) ) ) ) ) ).

% bset(4)
thf(fact_4448_bset_I3_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B2 )
                   => ( X3
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X3 = T )
             => ( ( minus_minus_int @ X3 @ D4 )
                = T ) ) ) ) ) ).

% bset(3)
thf(fact_4449_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S ) @ X )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).

% VEBT_internal.naive_member.simps(3)
thf(fact_4450_aset_I8_J,axiom,
    ! [D4: int,A2: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X3: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A2 )
                 => ( X3
                   != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ T @ X3 )
           => ( ord_less_eq_int @ T @ ( plus_plus_int @ X3 @ D4 ) ) ) ) ) ).

% aset(8)
thf(fact_4451_aset_I6_J,axiom,
    ! [D4: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X3
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ X3 @ T )
             => ( ord_less_eq_int @ ( plus_plus_int @ X3 @ D4 ) @ T ) ) ) ) ) ).

% aset(6)
thf(fact_4452_bset_I8_J,axiom,
    ! [D4: int,T: int,B2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
       => ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B2 )
                   => ( X3
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ T @ X3 )
             => ( ord_less_eq_int @ T @ ( minus_minus_int @ X3 @ D4 ) ) ) ) ) ) ).

% bset(8)
thf(fact_4453_bset_I6_J,axiom,
    ! [D4: int,B2: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X3: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B2 )
                 => ( X3
                   != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ X3 @ T )
           => ( ord_less_eq_int @ ( minus_minus_int @ X3 @ D4 ) @ T ) ) ) ) ).

% bset(6)
thf(fact_4454_cpmi,axiom,
    ! [D4: int,P: int > $o,P6: int > $o,B2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z5: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z5 )
           => ( ( P @ X5 )
              = ( P6 @ X5 ) ) )
       => ( ! [X5: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb3: int] :
                      ( ( member_int @ Xb3 @ B2 )
                     => ( X5
                       != ( plus_plus_int @ Xb3 @ Xa ) ) ) )
             => ( ( P @ X5 )
               => ( P @ ( minus_minus_int @ X5 @ D4 ) ) ) )
         => ( ! [X5: int,K2: int] :
                ( ( P6 @ X5 )
                = ( P6 @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D4 ) ) ) )
           => ( ( ? [X4: int] : ( P @ X4 ) )
              = ( ? [X6: int] :
                    ( ( member_int @ X6 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P6 @ X6 ) )
                | ? [X6: int] :
                    ( ( member_int @ X6 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y6: int] :
                        ( ( member_int @ Y6 @ B2 )
                        & ( P @ ( plus_plus_int @ Y6 @ X6 ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_4455_cppi,axiom,
    ! [D4: int,P: int > $o,P6: int > $o,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z5: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z5 @ X5 )
           => ( ( P @ X5 )
              = ( P6 @ X5 ) ) )
       => ( ! [X5: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb3: int] :
                      ( ( member_int @ Xb3 @ A2 )
                     => ( X5
                       != ( minus_minus_int @ Xb3 @ Xa ) ) ) )
             => ( ( P @ X5 )
               => ( P @ ( plus_plus_int @ X5 @ D4 ) ) ) )
         => ( ! [X5: int,K2: int] :
                ( ( P6 @ X5 )
                = ( P6 @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D4 ) ) ) )
           => ( ( ? [X4: int] : ( P @ X4 ) )
              = ( ? [X6: int] :
                    ( ( member_int @ X6 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P6 @ X6 ) )
                | ? [X6: int] :
                    ( ( member_int @ X6 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y6: int] :
                        ( ( member_int @ Y6 @ A2 )
                        & ( P @ ( minus_minus_int @ Y6 @ X6 ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_4456_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd ) @ X )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).

% VEBT_internal.membermima.simps(5)
thf(fact_4457_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
      = ( ( X != Mi )
       => ( ( X != Ma )
         => ( ~ ( ord_less_nat @ X @ Mi )
            & ( ~ ( ord_less_nat @ X @ Mi )
             => ( ~ ( ord_less_nat @ Ma @ X )
                & ( ~ ( ord_less_nat @ Ma @ X )
                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                     => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
thf(fact_4458_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList @ Vc ) @ X )
      = ( ( X = Mi )
        | ( X = Ma )
        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
          & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ).

% VEBT_internal.membermima.simps(4)
thf(fact_4459_vebt__delete_Osimps_I7_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ( ord_less_nat @ X @ Mi )
          | ( ord_less_nat @ Ma @ X ) )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) )
      & ( ~ ( ( ord_less_nat @ X @ Mi )
            | ( ord_less_nat @ Ma @ X ) )
       => ( ( ( ( X = Mi )
              & ( X = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
              = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) )
          & ( ~ ( ( X = Mi )
                & ( X = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
              = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X = Mi )
                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X != Mi )
                             => ( X = Ma ) ) )
                          @ ( if_nat
                            @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              = none_nat )
                            @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                            @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va2 ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X = Mi )
                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X != Mi )
                             => ( X = Ma ) ) )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va2 ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ Summary ) )
                @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
thf(fact_4460_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A5 )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B5 )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList4: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList4 @ S2 ) )
               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
thf(fact_4461_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A5 )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B5 )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList4: list_VEBT_VEBT] :
              ( ? [S2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList4 @ S2 ) )
             => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
thf(fact_4462_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
        = Y )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ( Y
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B5 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
           => Y )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList4: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList4 @ S2 ) )
               => ( Y
                  = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
thf(fact_4463_vebt__delete_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X @ Xa2 )
        = Y )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ( ( Xa2 = zero_zero_nat )
             => ( Y
               != ( vEBT_Leaf @ $false @ B5 ) ) ) )
       => ( ! [A5: $o] :
              ( ? [B5: $o] :
                  ( X
                  = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( Xa2
                  = ( suc @ zero_zero_nat ) )
               => ( Y
                 != ( vEBT_Leaf @ A5 @ $false ) ) ) )
         => ( ! [A5: $o,B5: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A5 @ B5 ) )
               => ( ? [N2: nat] :
                      ( Xa2
                      = ( suc @ ( suc @ N2 ) ) )
                 => ( Y
                   != ( vEBT_Leaf @ A5 @ B5 ) ) ) )
           => ( ! [Deg2: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList4 @ Summary3 ) )
                 => ( Y
                   != ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList4 @ Summary3 ) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) )
                   => ( Y
                     != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) )
                     => ( Y
                       != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) )
                       => ~ ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa2 ) )
                             => ( Y
                                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) ) )
                            & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa2 ) )
                             => ( ( ( ( Xa2 = Mi2 )
                                    & ( Xa2 = Ma2 ) )
                                 => ( Y
                                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) ) )
                                & ( ~ ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                 => ( Y
                                    = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                                      @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some_P7363390416028606310at_nat
                                            @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa2 = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa2 != Mi2 )
                                                   => ( Xa2 = Ma2 ) ) )
                                                @ ( if_nat
                                                  @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                    = none_nat )
                                                  @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some_P7363390416028606310at_nat
                                            @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa2 = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa2 != Mi2 )
                                                   => ( Xa2 = Ma2 ) ) )
                                                @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ Summary3 ) )
                                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
thf(fact_4464_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
           => ~ ( ( Xa2 = Mi2 )
                | ( Xa2 = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList4: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList4 @ Vc2 ) )
             => ~ ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 )
                  | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) ) ) ) )
         => ~ ! [V2: nat,TreeList4: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList4 @ Vd2 ) )
               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
thf(fact_4465_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X @ Xa2 )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A5 )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B5 )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList4: list_VEBT_VEBT] :
              ( ? [Summary3: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) )
             => ~ ( ( Xa2 != Mi2 )
                 => ( ( Xa2 != Ma2 )
                   => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                      & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                       => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                          & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                           => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                               => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
thf(fact_4466_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ! [Uu2: $o,Uv2: $o] :
            ( X
           != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
               => ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList4: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList4 @ Vc2 ) )
                 => ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 )
                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) ) ) ) )
             => ~ ! [V2: nat,TreeList4: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList4 @ Vd2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
thf(fact_4467_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => Y )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
           => Y )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
               => ( Y
                  = ( ~ ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList4: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList4 @ Vc2 ) )
                 => ( Y
                    = ( ~ ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) ) ) ) ) ) )
             => ~ ! [V2: nat,TreeList4: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList4 @ Vd2 ) )
                   => ( Y
                      = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
thf(fact_4468_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X @ Xa2 )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A5 )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B5 )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList4: list_VEBT_VEBT] :
                    ( ? [Summary3: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) )
                   => ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
thf(fact_4469_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X @ Xa2 )
        = Y )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ( Y
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B5 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => Y )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
             => Y )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => Y )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList4: list_VEBT_VEBT] :
                    ( ? [Summary3: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) )
                   => ( Y
                      = ( ~ ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
thf(fact_4470_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X: nat,Mi: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = ( some_nat @ Ma ) ) )
      & ( ~ ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_pred.simps(7)
thf(fact_4471_vebt__succ_Osimps_I6_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = ( some_nat @ Mi ) ) )
      & ( ~ ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_succ.simps(6)
thf(fact_4472_vebt__delete_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,N: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N ) ) )
      = ( vEBT_Leaf @ A @ B ) ) ).

% vebt_delete.simps(3)
thf(fact_4473_vebt__delete_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat )
      = ( vEBT_Leaf @ $false @ B ) ) ).

% vebt_delete.simps(1)
thf(fact_4474_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Uu )
      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) ) ).

% vebt_delete.simps(4)
thf(fact_4475_vebt__pred_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_pred @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( ( Xa2 = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A5: $o] :
              ( ? [Uw2: $o] :
                  ( X
                  = ( vEBT_Leaf @ A5 @ Uw2 ) )
             => ( ( Xa2
                  = ( suc @ zero_zero_nat ) )
               => ~ ( ( A5
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A5
                     => ( Y = none_nat ) ) ) ) )
         => ( ! [A5: $o,B5: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A5 @ B5 ) )
               => ( ? [Va: nat] :
                      ( Xa2
                      = ( suc @ ( suc @ Va ) ) )
                 => ~ ( ( B5
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B5
                       => ( ( A5
                           => ( Y
                              = ( some_nat @ zero_zero_nat ) ) )
                          & ( ~ A5
                           => ( Y = none_nat ) ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                 => ( Y != none_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                   => ( Y != none_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) )
                       => ~ ( ( ( ord_less_nat @ Ma2 @ Xa2 )
                             => ( Y
                                = ( some_nat @ Ma2 ) ) )
                            & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        = none_nat )
                                      @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
thf(fact_4476_vebt__succ_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X @ Xa2 )
        = Y )
     => ( ! [Uu2: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ Uu2 @ B5 ) )
           => ( ( Xa2 = zero_zero_nat )
             => ~ ( ( B5
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B5
                   => ( Y = none_nat ) ) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] :
                ( X
                = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
           => ( ? [N2: nat] :
                  ( Xa2
                  = ( suc @ N2 ) )
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                 => ( Y != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) )
                     => ~ ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( Y
                              = ( some_nat @ Mi2 ) ) )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( Y
                              = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                                @ ( if_option_nat
                                  @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                     != none_nat )
                                    & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  @ ( if_option_nat
                                    @ ( ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                      = none_nat )
                                    @ none_nat
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
thf(fact_4477_vebt__delete_Osimps_I2_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) )
      = ( vEBT_Leaf @ A @ $false ) ) ).

% vebt_delete.simps(2)
thf(fact_4478_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list_VEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ord_less_nat @ X @ Mi )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_excp
thf(fact_4479_insert__simp__norm,axiom,
    ! [X: nat,Deg: nat,TreeList: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ord_less_nat @ Mi @ X )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_norm
thf(fact_4480_insert__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( sup_sup_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
          = ( vEBT_set_vebt @ ( vEBT_vebt_insert @ T @ X ) ) ) ) ) ).

% insert_correct
thf(fact_4481_insert__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( sup_sup_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
          = ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_insert @ T @ X ) ) ) ) ) ).

% insert_corr
thf(fact_4482_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa2 )
        = Y )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => ( Y
                    = ( vEBT_Leaf @ $true @ B5 ) ) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A5 @ $true ) ) )
                    & ( ( Xa2 != one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) )
             => ( Y
               != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) )
               => ( Y
                 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) ) )
           => ( ! [V2: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList4 @ Summary3 ) )
                 => ( Y
                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList4 @ Summary3 ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) )
                   => ( Y
                     != ( if_VEBT_VEBT
                        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                          & ~ ( ( Xa2 = Mi2 )
                              | ( Xa2 = Ma2 ) ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary3 ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
thf(fact_4483_vebt__succ_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ B5 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( ( B5
                     => ( Y
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B5
                     => ( Y = none_nat ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B5 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
               => ! [N2: nat] :
                    ( ( Xa2
                      = ( suc @ N2 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N2 ) ) ) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y = none_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa2 ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                     => ( ( Y = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa2 ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) )
                       => ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                             => ( Y
                                = ( some_nat @ Mi2 ) ) )
                            & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        = none_nat )
                                      @ none_nat
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
thf(fact_4484_vebt__pred_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_pred @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A5: $o,Uw2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A5 @ Uw2 ) )
               => ( ( Xa2
                    = ( suc @ zero_zero_nat ) )
                 => ( ( ( A5
                       => ( Y
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A5
                       => ( Y = none_nat ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A5: $o,B5: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A5 @ B5 ) )
                 => ! [Va: nat] :
                      ( ( Xa2
                        = ( suc @ ( suc @ Va ) ) )
                     => ( ( ( B5
                           => ( Y
                              = ( some_nat @ one_one_nat ) ) )
                          & ( ~ B5
                           => ( ( A5
                               => ( Y
                                  = ( some_nat @ zero_zero_nat ) ) )
                              & ( ~ A5
                               => ( Y = none_nat ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ ( suc @ ( suc @ Va ) ) ) ) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Xa2 ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                     => ( ( Y = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa2 ) ) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                       => ( ( Y = none_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa2 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) )
                         => ( ( ( ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( Y
                                  = ( some_nat @ Ma2 ) ) )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( Y
                                  = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                                    @ ( if_option_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( if_option_nat
                                        @ ( ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                          = none_nat )
                                        @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
                                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                    @ none_nat ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
thf(fact_4485_UnCI,axiom,
    ! [C: complex,B2: set_complex,A2: set_complex] :
      ( ( ~ ( member_complex @ C @ B2 )
       => ( member_complex @ C @ A2 ) )
     => ( member_complex @ C @ ( sup_sup_set_complex @ A2 @ B2 ) ) ) ).

% UnCI
thf(fact_4486_UnCI,axiom,
    ! [C: real,B2: set_real,A2: set_real] :
      ( ( ~ ( member_real @ C @ B2 )
       => ( member_real @ C @ A2 ) )
     => ( member_real @ C @ ( sup_sup_set_real @ A2 @ B2 ) ) ) ).

% UnCI
thf(fact_4487_UnCI,axiom,
    ! [C: set_nat,B2: set_set_nat,A2: set_set_nat] :
      ( ( ~ ( member_set_nat @ C @ B2 )
       => ( member_set_nat @ C @ A2 ) )
     => ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).

% UnCI
thf(fact_4488_UnCI,axiom,
    ! [C: int,B2: set_int,A2: set_int] :
      ( ( ~ ( member_int @ C @ B2 )
       => ( member_int @ C @ A2 ) )
     => ( member_int @ C @ ( sup_sup_set_int @ A2 @ B2 ) ) ) ).

% UnCI
thf(fact_4489_UnCI,axiom,
    ! [C: nat,B2: set_nat,A2: set_nat] :
      ( ( ~ ( member_nat @ C @ B2 )
       => ( member_nat @ C @ A2 ) )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% UnCI
thf(fact_4490_Un__iff,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( ( member_complex @ C @ ( sup_sup_set_complex @ A2 @ B2 ) )
      = ( ( member_complex @ C @ A2 )
        | ( member_complex @ C @ B2 ) ) ) ).

% Un_iff
thf(fact_4491_Un__iff,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( ( member_real @ C @ ( sup_sup_set_real @ A2 @ B2 ) )
      = ( ( member_real @ C @ A2 )
        | ( member_real @ C @ B2 ) ) ) ).

% Un_iff
thf(fact_4492_Un__iff,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
      = ( ( member_set_nat @ C @ A2 )
        | ( member_set_nat @ C @ B2 ) ) ) ).

% Un_iff
thf(fact_4493_Un__iff,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( ( member_int @ C @ ( sup_sup_set_int @ A2 @ B2 ) )
      = ( ( member_int @ C @ A2 )
        | ( member_int @ C @ B2 ) ) ) ).

% Un_iff
thf(fact_4494_Un__iff,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) )
      = ( ( member_nat @ C @ A2 )
        | ( member_nat @ C @ B2 ) ) ) ).

% Un_iff
thf(fact_4495_Un__empty,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( ( sup_sup_set_real @ A2 @ B2 )
        = bot_bot_set_real )
      = ( ( A2 = bot_bot_set_real )
        & ( B2 = bot_bot_set_real ) ) ) ).

% Un_empty
thf(fact_4496_Un__empty,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( ( sup_sup_set_o @ A2 @ B2 )
        = bot_bot_set_o )
      = ( ( A2 = bot_bot_set_o )
        & ( B2 = bot_bot_set_o ) ) ) ).

% Un_empty
thf(fact_4497_Un__empty,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ( sup_sup_set_nat @ A2 @ B2 )
        = bot_bot_set_nat )
      = ( ( A2 = bot_bot_set_nat )
        & ( B2 = bot_bot_set_nat ) ) ) ).

% Un_empty
thf(fact_4498_Un__empty,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ( sup_sup_set_int @ A2 @ B2 )
        = bot_bot_set_int )
      = ( ( A2 = bot_bot_set_int )
        & ( B2 = bot_bot_set_int ) ) ) ).

% Un_empty
thf(fact_4499_Un__subset__iff,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C5 )
      = ( ( ord_less_eq_set_nat @ A2 @ C5 )
        & ( ord_less_eq_set_nat @ B2 @ C5 ) ) ) ).

% Un_subset_iff
thf(fact_4500_Un__insert__left,axiom,
    ! [A: int,B2: set_int,C5: set_int] :
      ( ( sup_sup_set_int @ ( insert_int @ A @ B2 ) @ C5 )
      = ( insert_int @ A @ ( sup_sup_set_int @ B2 @ C5 ) ) ) ).

% Un_insert_left
thf(fact_4501_Un__insert__left,axiom,
    ! [A: real,B2: set_real,C5: set_real] :
      ( ( sup_sup_set_real @ ( insert_real @ A @ B2 ) @ C5 )
      = ( insert_real @ A @ ( sup_sup_set_real @ B2 @ C5 ) ) ) ).

% Un_insert_left
thf(fact_4502_Un__insert__left,axiom,
    ! [A: $o,B2: set_o,C5: set_o] :
      ( ( sup_sup_set_o @ ( insert_o @ A @ B2 ) @ C5 )
      = ( insert_o @ A @ ( sup_sup_set_o @ B2 @ C5 ) ) ) ).

% Un_insert_left
thf(fact_4503_Un__insert__left,axiom,
    ! [A: nat,B2: set_nat,C5: set_nat] :
      ( ( sup_sup_set_nat @ ( insert_nat @ A @ B2 ) @ C5 )
      = ( insert_nat @ A @ ( sup_sup_set_nat @ B2 @ C5 ) ) ) ).

% Un_insert_left
thf(fact_4504_Un__insert__right,axiom,
    ! [A2: set_int,A: int,B2: set_int] :
      ( ( sup_sup_set_int @ A2 @ ( insert_int @ A @ B2 ) )
      = ( insert_int @ A @ ( sup_sup_set_int @ A2 @ B2 ) ) ) ).

% Un_insert_right
thf(fact_4505_Un__insert__right,axiom,
    ! [A2: set_real,A: real,B2: set_real] :
      ( ( sup_sup_set_real @ A2 @ ( insert_real @ A @ B2 ) )
      = ( insert_real @ A @ ( sup_sup_set_real @ A2 @ B2 ) ) ) ).

% Un_insert_right
thf(fact_4506_Un__insert__right,axiom,
    ! [A2: set_o,A: $o,B2: set_o] :
      ( ( sup_sup_set_o @ A2 @ ( insert_o @ A @ B2 ) )
      = ( insert_o @ A @ ( sup_sup_set_o @ A2 @ B2 ) ) ) ).

% Un_insert_right
thf(fact_4507_Un__insert__right,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
      = ( insert_nat @ A @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% Un_insert_right
thf(fact_4508_Un__Diff__cancel,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
      = ( sup_sup_set_nat @ A2 @ B2 ) ) ).

% Un_Diff_cancel
thf(fact_4509_Un__Diff__cancel2,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B2 @ A2 ) @ A2 )
      = ( sup_sup_set_nat @ B2 @ A2 ) ) ).

% Un_Diff_cancel2
thf(fact_4510_max__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).

% max_Suc_Suc
thf(fact_4511_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_4512_max__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A )
      = A ) ).

% max_nat.left_neutral
thf(fact_4513_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_4514_max__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ zero_zero_nat )
      = A ) ).

% max_nat.right_neutral
thf(fact_4515_max__0L,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N )
      = N ) ).

% max_0L
thf(fact_4516_max__0R,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ N @ zero_zero_nat )
      = N ) ).

% max_0R
thf(fact_4517_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
          = ( numera6620942414471956472nteger @ V ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
          = ( numera6620942414471956472nteger @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_4518_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
       => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
          = ( numera1916890842035813515d_enat @ V ) ) )
      & ( ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
       => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
          = ( numera1916890842035813515d_enat @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_4519_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
          = ( numeral_numeral_real @ V ) ) )
      & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
          = ( numeral_numeral_real @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_4520_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ V ) ) )
      & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_4521_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
       => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
          = ( numeral_numeral_nat @ V ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
       => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
          = ( numeral_numeral_nat @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_4522_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
          = ( numeral_numeral_int @ V ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
          = ( numeral_numeral_int @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_4523_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ X ) )
      = ( numera6620942414471956472nteger @ X ) ) ).

% max_0_1(3)
thf(fact_4524_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X ) )
      = ( numera1916890842035813515d_enat @ X ) ) ).

% max_0_1(3)
thf(fact_4525_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X ) )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(3)
thf(fact_4526_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(3)
thf(fact_4527_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X ) )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(3)
thf(fact_4528_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X ) )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(3)
thf(fact_4529_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X ) @ zero_z3403309356797280102nteger )
      = ( numera6620942414471956472nteger @ X ) ) ).

% max_0_1(4)
thf(fact_4530_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ zero_z5237406670263579293d_enat )
      = ( numera1916890842035813515d_enat @ X ) ) ).

% max_0_1(4)
thf(fact_4531_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ zero_zero_real )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(4)
thf(fact_4532_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(4)
thf(fact_4533_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ zero_zero_int )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(4)
thf(fact_4534_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ zero_zero_rat )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(4)
thf(fact_4535_max__0__1_I2_J,axiom,
    ( ( ord_max_real @ one_one_real @ zero_zero_real )
    = one_one_real ) ).

% max_0_1(2)
thf(fact_4536_max__0__1_I2_J,axiom,
    ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
    = one_one_rat ) ).

% max_0_1(2)
thf(fact_4537_max__0__1_I2_J,axiom,
    ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
    = one_one_nat ) ).

% max_0_1(2)
thf(fact_4538_max__0__1_I2_J,axiom,
    ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat )
    = one_on7984719198319812577d_enat ) ).

% max_0_1(2)
thf(fact_4539_max__0__1_I2_J,axiom,
    ( ( ord_max_int @ one_one_int @ zero_zero_int )
    = one_one_int ) ).

% max_0_1(2)
thf(fact_4540_max__0__1_I2_J,axiom,
    ( ( ord_max_Code_integer @ one_one_Code_integer @ zero_z3403309356797280102nteger )
    = one_one_Code_integer ) ).

% max_0_1(2)
thf(fact_4541_max__0__1_I1_J,axiom,
    ( ( ord_max_real @ zero_zero_real @ one_one_real )
    = one_one_real ) ).

% max_0_1(1)
thf(fact_4542_max__0__1_I1_J,axiom,
    ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
    = one_one_rat ) ).

% max_0_1(1)
thf(fact_4543_max__0__1_I1_J,axiom,
    ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
    = one_one_nat ) ).

% max_0_1(1)
thf(fact_4544_max__0__1_I1_J,axiom,
    ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat )
    = one_on7984719198319812577d_enat ) ).

% max_0_1(1)
thf(fact_4545_max__0__1_I1_J,axiom,
    ( ( ord_max_int @ zero_zero_int @ one_one_int )
    = one_one_int ) ).

% max_0_1(1)
thf(fact_4546_max__0__1_I1_J,axiom,
    ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% max_0_1(1)
thf(fact_4547_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_integer @ one_one_Code_integer @ ( numera6620942414471956472nteger @ X ) )
      = ( numera6620942414471956472nteger @ X ) ) ).

% max_0_1(5)
thf(fact_4548_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X ) )
      = ( numera1916890842035813515d_enat @ X ) ) ).

% max_0_1(5)
thf(fact_4549_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X ) )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(5)
thf(fact_4550_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(5)
thf(fact_4551_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X ) )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(5)
thf(fact_4552_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(5)
thf(fact_4553_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ X ) ) ).

% max_0_1(6)
thf(fact_4554_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat )
      = ( numera1916890842035813515d_enat @ X ) ) ).

% max_0_1(6)
thf(fact_4555_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ one_one_real )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(6)
thf(fact_4556_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(6)
thf(fact_4557_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ one_one_int )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(6)
thf(fact_4558_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(6)
thf(fact_4559_Collect__disj__eq,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ( collect_real
        @ ^ [X6: real] :
            ( ( P @ X6 )
            | ( Q @ X6 ) ) )
      = ( sup_sup_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_4560_Collect__disj__eq,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X6: list_nat] :
            ( ( P @ X6 )
            | ( Q @ X6 ) ) )
      = ( sup_sup_set_list_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_4561_Collect__disj__eq,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X6: set_nat] :
            ( ( P @ X6 )
            | ( Q @ X6 ) ) )
      = ( sup_sup_set_set_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_4562_Collect__disj__eq,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( collect_int
        @ ^ [X6: int] :
            ( ( P @ X6 )
            | ( Q @ X6 ) ) )
      = ( sup_sup_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_4563_Collect__disj__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X6: nat] :
            ( ( P @ X6 )
            | ( Q @ X6 ) ) )
      = ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_4564_Un__def,axiom,
    ( sup_sup_set_complex
    = ( ^ [A6: set_complex,B7: set_complex] :
          ( collect_complex
          @ ^ [X6: complex] :
              ( ( member_complex @ X6 @ A6 )
              | ( member_complex @ X6 @ B7 ) ) ) ) ) ).

% Un_def
thf(fact_4565_Un__def,axiom,
    ( sup_sup_set_real
    = ( ^ [A6: set_real,B7: set_real] :
          ( collect_real
          @ ^ [X6: real] :
              ( ( member_real @ X6 @ A6 )
              | ( member_real @ X6 @ B7 ) ) ) ) ) ).

% Un_def
thf(fact_4566_Un__def,axiom,
    ( sup_sup_set_list_nat
    = ( ^ [A6: set_list_nat,B7: set_list_nat] :
          ( collect_list_nat
          @ ^ [X6: list_nat] :
              ( ( member_list_nat @ X6 @ A6 )
              | ( member_list_nat @ X6 @ B7 ) ) ) ) ) ).

% Un_def
thf(fact_4567_Un__def,axiom,
    ( sup_sup_set_set_nat
    = ( ^ [A6: set_set_nat,B7: set_set_nat] :
          ( collect_set_nat
          @ ^ [X6: set_nat] :
              ( ( member_set_nat @ X6 @ A6 )
              | ( member_set_nat @ X6 @ B7 ) ) ) ) ) ).

% Un_def
thf(fact_4568_Un__def,axiom,
    ( sup_sup_set_int
    = ( ^ [A6: set_int,B7: set_int] :
          ( collect_int
          @ ^ [X6: int] :
              ( ( member_int @ X6 @ A6 )
              | ( member_int @ X6 @ B7 ) ) ) ) ) ).

% Un_def
thf(fact_4569_Un__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( collect_nat
          @ ^ [X6: nat] :
              ( ( member_nat @ X6 @ A6 )
              | ( member_nat @ X6 @ B7 ) ) ) ) ) ).

% Un_def
thf(fact_4570_UnE,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( ( member_complex @ C @ ( sup_sup_set_complex @ A2 @ B2 ) )
     => ( ~ ( member_complex @ C @ A2 )
       => ( member_complex @ C @ B2 ) ) ) ).

% UnE
thf(fact_4571_UnE,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( ( member_real @ C @ ( sup_sup_set_real @ A2 @ B2 ) )
     => ( ~ ( member_real @ C @ A2 )
       => ( member_real @ C @ B2 ) ) ) ).

% UnE
thf(fact_4572_UnE,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
     => ( ~ ( member_set_nat @ C @ A2 )
       => ( member_set_nat @ C @ B2 ) ) ) ).

% UnE
thf(fact_4573_UnE,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( ( member_int @ C @ ( sup_sup_set_int @ A2 @ B2 ) )
     => ( ~ ( member_int @ C @ A2 )
       => ( member_int @ C @ B2 ) ) ) ).

% UnE
thf(fact_4574_UnE,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) )
     => ( ~ ( member_nat @ C @ A2 )
       => ( member_nat @ C @ B2 ) ) ) ).

% UnE
thf(fact_4575_UnI1,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( ( member_complex @ C @ A2 )
     => ( member_complex @ C @ ( sup_sup_set_complex @ A2 @ B2 ) ) ) ).

% UnI1
thf(fact_4576_UnI1,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( ( member_real @ C @ A2 )
     => ( member_real @ C @ ( sup_sup_set_real @ A2 @ B2 ) ) ) ).

% UnI1
thf(fact_4577_UnI1,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ A2 )
     => ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).

% UnI1
thf(fact_4578_UnI1,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( ( member_int @ C @ A2 )
     => ( member_int @ C @ ( sup_sup_set_int @ A2 @ B2 ) ) ) ).

% UnI1
thf(fact_4579_UnI1,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ A2 )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% UnI1
thf(fact_4580_UnI2,axiom,
    ! [C: complex,B2: set_complex,A2: set_complex] :
      ( ( member_complex @ C @ B2 )
     => ( member_complex @ C @ ( sup_sup_set_complex @ A2 @ B2 ) ) ) ).

% UnI2
thf(fact_4581_UnI2,axiom,
    ! [C: real,B2: set_real,A2: set_real] :
      ( ( member_real @ C @ B2 )
     => ( member_real @ C @ ( sup_sup_set_real @ A2 @ B2 ) ) ) ).

% UnI2
thf(fact_4582_UnI2,axiom,
    ! [C: set_nat,B2: set_set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ C @ B2 )
     => ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).

% UnI2
thf(fact_4583_UnI2,axiom,
    ! [C: int,B2: set_int,A2: set_int] :
      ( ( member_int @ C @ B2 )
     => ( member_int @ C @ ( sup_sup_set_int @ A2 @ B2 ) ) ) ).

% UnI2
thf(fact_4584_UnI2,axiom,
    ! [C: nat,B2: set_nat,A2: set_nat] :
      ( ( member_nat @ C @ B2 )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% UnI2
thf(fact_4585_bex__Un,axiom,
    ! [A2: set_nat,B2: set_nat,P: nat > $o] :
      ( ( ? [X6: nat] :
            ( ( member_nat @ X6 @ ( sup_sup_set_nat @ A2 @ B2 ) )
            & ( P @ X6 ) ) )
      = ( ? [X6: nat] :
            ( ( member_nat @ X6 @ A2 )
            & ( P @ X6 ) )
        | ? [X6: nat] :
            ( ( member_nat @ X6 @ B2 )
            & ( P @ X6 ) ) ) ) ).

% bex_Un
thf(fact_4586_ball__Un,axiom,
    ! [A2: set_nat,B2: set_nat,P: nat > $o] :
      ( ( ! [X6: nat] :
            ( ( member_nat @ X6 @ ( sup_sup_set_nat @ A2 @ B2 ) )
           => ( P @ X6 ) ) )
      = ( ! [X6: nat] :
            ( ( member_nat @ X6 @ A2 )
           => ( P @ X6 ) )
        & ! [X6: nat] :
            ( ( member_nat @ X6 @ B2 )
           => ( P @ X6 ) ) ) ) ).

% ball_Un
thf(fact_4587_Un__assoc,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C5 )
      = ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C5 ) ) ) ).

% Un_assoc
thf(fact_4588_Un__absorb,axiom,
    ! [A2: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ A2 )
      = A2 ) ).

% Un_absorb
thf(fact_4589_Un__commute,axiom,
    ( sup_sup_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] : ( sup_sup_set_nat @ B7 @ A6 ) ) ) ).

% Un_commute
thf(fact_4590_Un__left__absorb,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
      = ( sup_sup_set_nat @ A2 @ B2 ) ) ).

% Un_left_absorb
thf(fact_4591_Un__left__commute,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C5 ) )
      = ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ C5 ) ) ) ).

% Un_left_commute
thf(fact_4592_max__add__distrib__right,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( ( plus_plus_real @ X @ ( ord_max_real @ Y @ Z2 ) )
      = ( ord_max_real @ ( plus_plus_real @ X @ Y ) @ ( plus_plus_real @ X @ Z2 ) ) ) ).

% max_add_distrib_right
thf(fact_4593_max__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y @ Z2 ) )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z2 ) ) ) ).

% max_add_distrib_right
thf(fact_4594_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z2 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z2 ) ) ) ).

% max_add_distrib_right
thf(fact_4595_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z2 ) )
      = ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z2 ) ) ) ).

% max_add_distrib_right
thf(fact_4596_max__add__distrib__right,axiom,
    ! [X: code_integer,Y: code_integer,Z2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ X @ ( ord_max_Code_integer @ Y @ Z2 ) )
      = ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Y ) @ ( plus_p5714425477246183910nteger @ X @ Z2 ) ) ) ).

% max_add_distrib_right
thf(fact_4597_max__add__distrib__left,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( ( plus_plus_real @ ( ord_max_real @ X @ Y ) @ Z2 )
      = ( ord_max_real @ ( plus_plus_real @ X @ Z2 ) @ ( plus_plus_real @ Y @ Z2 ) ) ) ).

% max_add_distrib_left
thf(fact_4598_max__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y ) @ Z2 )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Z2 ) @ ( plus_plus_rat @ Y @ Z2 ) ) ) ).

% max_add_distrib_left
thf(fact_4599_max__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z2 )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Z2 ) @ ( plus_plus_nat @ Y @ Z2 ) ) ) ).

% max_add_distrib_left
thf(fact_4600_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z2 )
      = ( ord_max_int @ ( plus_plus_int @ X @ Z2 ) @ ( plus_plus_int @ Y @ Z2 ) ) ) ).

% max_add_distrib_left
thf(fact_4601_max__add__distrib__left,axiom,
    ! [X: code_integer,Y: code_integer,Z2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( ord_max_Code_integer @ X @ Y ) @ Z2 )
      = ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Z2 ) @ ( plus_p5714425477246183910nteger @ Y @ Z2 ) ) ) ).

% max_add_distrib_left
thf(fact_4602_max__diff__distrib__left,axiom,
    ! [X: code_integer,Y: code_integer,Z2: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( ord_max_Code_integer @ X @ Y ) @ Z2 )
      = ( ord_max_Code_integer @ ( minus_8373710615458151222nteger @ X @ Z2 ) @ ( minus_8373710615458151222nteger @ Y @ Z2 ) ) ) ).

% max_diff_distrib_left
thf(fact_4603_max__diff__distrib__left,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( ( minus_minus_real @ ( ord_max_real @ X @ Y ) @ Z2 )
      = ( ord_max_real @ ( minus_minus_real @ X @ Z2 ) @ ( minus_minus_real @ Y @ Z2 ) ) ) ).

% max_diff_distrib_left
thf(fact_4604_max__diff__distrib__left,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( ( minus_minus_rat @ ( ord_max_rat @ X @ Y ) @ Z2 )
      = ( ord_max_rat @ ( minus_minus_rat @ X @ Z2 ) @ ( minus_minus_rat @ Y @ Z2 ) ) ) ).

% max_diff_distrib_left
thf(fact_4605_max__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( minus_minus_int @ ( ord_max_int @ X @ Y ) @ Z2 )
      = ( ord_max_int @ ( minus_minus_int @ X @ Z2 ) @ ( minus_minus_int @ Y @ Z2 ) ) ) ).

% max_diff_distrib_left
thf(fact_4606_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).

% nat_add_max_right
thf(fact_4607_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).

% nat_add_max_left
thf(fact_4608_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
      = ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).

% nat_mult_max_left
thf(fact_4609_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
      = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).

% nat_mult_max_right
thf(fact_4610_Un__empty__right,axiom,
    ! [A2: set_real] :
      ( ( sup_sup_set_real @ A2 @ bot_bot_set_real )
      = A2 ) ).

% Un_empty_right
thf(fact_4611_Un__empty__right,axiom,
    ! [A2: set_o] :
      ( ( sup_sup_set_o @ A2 @ bot_bot_set_o )
      = A2 ) ).

% Un_empty_right
thf(fact_4612_Un__empty__right,axiom,
    ! [A2: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
      = A2 ) ).

% Un_empty_right
thf(fact_4613_Un__empty__right,axiom,
    ! [A2: set_int] :
      ( ( sup_sup_set_int @ A2 @ bot_bot_set_int )
      = A2 ) ).

% Un_empty_right
thf(fact_4614_Un__empty__left,axiom,
    ! [B2: set_real] :
      ( ( sup_sup_set_real @ bot_bot_set_real @ B2 )
      = B2 ) ).

% Un_empty_left
thf(fact_4615_Un__empty__left,axiom,
    ! [B2: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ B2 )
      = B2 ) ).

% Un_empty_left
thf(fact_4616_Un__empty__left,axiom,
    ! [B2: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ B2 )
      = B2 ) ).

% Un_empty_left
thf(fact_4617_Un__empty__left,axiom,
    ! [B2: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ B2 )
      = B2 ) ).

% Un_empty_left
thf(fact_4618_subset__Un__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( sup_sup_set_nat @ A6 @ B7 )
          = B7 ) ) ) ).

% subset_Un_eq
thf(fact_4619_subset__UnE,axiom,
    ! [C5: set_nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ C5 @ ( sup_sup_set_nat @ A2 @ B2 ) )
     => ~ ! [A7: set_nat] :
            ( ( ord_less_eq_set_nat @ A7 @ A2 )
           => ! [B9: set_nat] :
                ( ( ord_less_eq_set_nat @ B9 @ B2 )
               => ( C5
                 != ( sup_sup_set_nat @ A7 @ B9 ) ) ) ) ) ).

% subset_UnE
thf(fact_4620_Un__absorb2,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( sup_sup_set_nat @ A2 @ B2 )
        = A2 ) ) ).

% Un_absorb2
thf(fact_4621_Un__absorb1,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( sup_sup_set_nat @ A2 @ B2 )
        = B2 ) ) ).

% Un_absorb1
thf(fact_4622_Un__upper2,axiom,
    ! [B2: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).

% Un_upper2
thf(fact_4623_Un__upper1,axiom,
    ! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).

% Un_upper1
thf(fact_4624_Un__least,axiom,
    ! [A2: set_nat,C5: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C5 )
     => ( ( ord_less_eq_set_nat @ B2 @ C5 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C5 ) ) ) ).

% Un_least
thf(fact_4625_Un__mono,axiom,
    ! [A2: set_nat,C5: set_nat,B2: set_nat,D4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C5 )
     => ( ( ord_less_eq_set_nat @ B2 @ D4 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ C5 @ D4 ) ) ) ) ).

% Un_mono
thf(fact_4626_Un__Diff,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C5 )
      = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ C5 ) @ ( minus_minus_set_nat @ B2 @ C5 ) ) ) ).

% Un_Diff
thf(fact_4627_max__def__raw,axiom,
    ( ord_ma741700101516333627d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def_raw
thf(fact_4628_max__def__raw,axiom,
    ( ord_max_Code_integer
    = ( ^ [A3: code_integer,B3: code_integer] : ( if_Code_integer @ ( ord_le3102999989581377725nteger @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def_raw
thf(fact_4629_max__def__raw,axiom,
    ( ord_max_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def_raw
thf(fact_4630_max__def__raw,axiom,
    ( ord_max_rat
    = ( ^ [A3: rat,B3: rat] : ( if_rat @ ( ord_less_eq_rat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def_raw
thf(fact_4631_max__def__raw,axiom,
    ( ord_max_num
    = ( ^ [A3: num,B3: num] : ( if_num @ ( ord_less_eq_num @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def_raw
thf(fact_4632_max__def__raw,axiom,
    ( ord_max_nat
    = ( ^ [A3: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def_raw
thf(fact_4633_max__def__raw,axiom,
    ( ord_max_int
    = ( ^ [A3: int,B3: int] : ( if_int @ ( ord_less_eq_int @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).

% max_def_raw
thf(fact_4634_insert__def,axiom,
    ( insert_o
    = ( ^ [A3: $o] :
          ( sup_sup_set_o
          @ ( collect_o
            @ ^ [X6: $o] : X6 = A3 ) ) ) ) ).

% insert_def
thf(fact_4635_insert__def,axiom,
    ( insert_real
    = ( ^ [A3: real] :
          ( sup_sup_set_real
          @ ( collect_real
            @ ^ [X6: real] : X6 = A3 ) ) ) ) ).

% insert_def
thf(fact_4636_insert__def,axiom,
    ( insert_list_nat
    = ( ^ [A3: list_nat] :
          ( sup_sup_set_list_nat
          @ ( collect_list_nat
            @ ^ [X6: list_nat] : X6 = A3 ) ) ) ) ).

% insert_def
thf(fact_4637_insert__def,axiom,
    ( insert_set_nat
    = ( ^ [A3: set_nat] :
          ( sup_sup_set_set_nat
          @ ( collect_set_nat
            @ ^ [X6: set_nat] : X6 = A3 ) ) ) ) ).

% insert_def
thf(fact_4638_insert__def,axiom,
    ( insert_int
    = ( ^ [A3: int] :
          ( sup_sup_set_int
          @ ( collect_int
            @ ^ [X6: int] : X6 = A3 ) ) ) ) ).

% insert_def
thf(fact_4639_insert__def,axiom,
    ( insert_nat
    = ( ^ [A3: nat] :
          ( sup_sup_set_nat
          @ ( collect_nat
            @ ^ [X6: nat] : X6 = A3 ) ) ) ) ).

% insert_def
thf(fact_4640_nat__minus__add__max,axiom,
    ! [N: nat,M: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
      = ( ord_max_nat @ N @ M ) ) ).

% nat_minus_add_max
thf(fact_4641_singleton__Un__iff,axiom,
    ! [X: real,A2: set_real,B2: set_real] :
      ( ( ( insert_real @ X @ bot_bot_set_real )
        = ( sup_sup_set_real @ A2 @ B2 ) )
      = ( ( ( A2 = bot_bot_set_real )
          & ( B2
            = ( insert_real @ X @ bot_bot_set_real ) ) )
        | ( ( A2
            = ( insert_real @ X @ bot_bot_set_real ) )
          & ( B2 = bot_bot_set_real ) )
        | ( ( A2
            = ( insert_real @ X @ bot_bot_set_real ) )
          & ( B2
            = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_4642_singleton__Un__iff,axiom,
    ! [X: $o,A2: set_o,B2: set_o] :
      ( ( ( insert_o @ X @ bot_bot_set_o )
        = ( sup_sup_set_o @ A2 @ B2 ) )
      = ( ( ( A2 = bot_bot_set_o )
          & ( B2
            = ( insert_o @ X @ bot_bot_set_o ) ) )
        | ( ( A2
            = ( insert_o @ X @ bot_bot_set_o ) )
          & ( B2 = bot_bot_set_o ) )
        | ( ( A2
            = ( insert_o @ X @ bot_bot_set_o ) )
          & ( B2
            = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_4643_singleton__Un__iff,axiom,
    ! [X: nat,A2: set_nat,B2: set_nat] :
      ( ( ( insert_nat @ X @ bot_bot_set_nat )
        = ( sup_sup_set_nat @ A2 @ B2 ) )
      = ( ( ( A2 = bot_bot_set_nat )
          & ( B2
            = ( insert_nat @ X @ bot_bot_set_nat ) ) )
        | ( ( A2
            = ( insert_nat @ X @ bot_bot_set_nat ) )
          & ( B2 = bot_bot_set_nat ) )
        | ( ( A2
            = ( insert_nat @ X @ bot_bot_set_nat ) )
          & ( B2
            = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_4644_singleton__Un__iff,axiom,
    ! [X: int,A2: set_int,B2: set_int] :
      ( ( ( insert_int @ X @ bot_bot_set_int )
        = ( sup_sup_set_int @ A2 @ B2 ) )
      = ( ( ( A2 = bot_bot_set_int )
          & ( B2
            = ( insert_int @ X @ bot_bot_set_int ) ) )
        | ( ( A2
            = ( insert_int @ X @ bot_bot_set_int ) )
          & ( B2 = bot_bot_set_int ) )
        | ( ( A2
            = ( insert_int @ X @ bot_bot_set_int ) )
          & ( B2
            = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_4645_Un__singleton__iff,axiom,
    ! [A2: set_real,B2: set_real,X: real] :
      ( ( ( sup_sup_set_real @ A2 @ B2 )
        = ( insert_real @ X @ bot_bot_set_real ) )
      = ( ( ( A2 = bot_bot_set_real )
          & ( B2
            = ( insert_real @ X @ bot_bot_set_real ) ) )
        | ( ( A2
            = ( insert_real @ X @ bot_bot_set_real ) )
          & ( B2 = bot_bot_set_real ) )
        | ( ( A2
            = ( insert_real @ X @ bot_bot_set_real ) )
          & ( B2
            = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_4646_Un__singleton__iff,axiom,
    ! [A2: set_o,B2: set_o,X: $o] :
      ( ( ( sup_sup_set_o @ A2 @ B2 )
        = ( insert_o @ X @ bot_bot_set_o ) )
      = ( ( ( A2 = bot_bot_set_o )
          & ( B2
            = ( insert_o @ X @ bot_bot_set_o ) ) )
        | ( ( A2
            = ( insert_o @ X @ bot_bot_set_o ) )
          & ( B2 = bot_bot_set_o ) )
        | ( ( A2
            = ( insert_o @ X @ bot_bot_set_o ) )
          & ( B2
            = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_4647_Un__singleton__iff,axiom,
    ! [A2: set_nat,B2: set_nat,X: nat] :
      ( ( ( sup_sup_set_nat @ A2 @ B2 )
        = ( insert_nat @ X @ bot_bot_set_nat ) )
      = ( ( ( A2 = bot_bot_set_nat )
          & ( B2
            = ( insert_nat @ X @ bot_bot_set_nat ) ) )
        | ( ( A2
            = ( insert_nat @ X @ bot_bot_set_nat ) )
          & ( B2 = bot_bot_set_nat ) )
        | ( ( A2
            = ( insert_nat @ X @ bot_bot_set_nat ) )
          & ( B2
            = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_4648_Un__singleton__iff,axiom,
    ! [A2: set_int,B2: set_int,X: int] :
      ( ( ( sup_sup_set_int @ A2 @ B2 )
        = ( insert_int @ X @ bot_bot_set_int ) )
      = ( ( ( A2 = bot_bot_set_int )
          & ( B2
            = ( insert_int @ X @ bot_bot_set_int ) ) )
        | ( ( A2
            = ( insert_int @ X @ bot_bot_set_int ) )
          & ( B2 = bot_bot_set_int ) )
        | ( ( A2
            = ( insert_int @ X @ bot_bot_set_int ) )
          & ( B2
            = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_4649_insert__is__Un,axiom,
    ( insert_real
    = ( ^ [A3: real] : ( sup_sup_set_real @ ( insert_real @ A3 @ bot_bot_set_real ) ) ) ) ).

% insert_is_Un
thf(fact_4650_insert__is__Un,axiom,
    ( insert_o
    = ( ^ [A3: $o] : ( sup_sup_set_o @ ( insert_o @ A3 @ bot_bot_set_o ) ) ) ) ).

% insert_is_Un
thf(fact_4651_insert__is__Un,axiom,
    ( insert_nat
    = ( ^ [A3: nat] : ( sup_sup_set_nat @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ).

% insert_is_Un
thf(fact_4652_insert__is__Un,axiom,
    ( insert_int
    = ( ^ [A3: int] : ( sup_sup_set_int @ ( insert_int @ A3 @ bot_bot_set_int ) ) ) ) ).

% insert_is_Un
thf(fact_4653_Diff__partition,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
        = B2 ) ) ).

% Diff_partition
thf(fact_4654_Diff__subset__conv,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ C5 )
      = ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C5 ) ) ) ).

% Diff_subset_conv
thf(fact_4655_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I: int,J2: int] : ( if_set_int @ ( ord_less_int @ J2 @ I ) @ bot_bot_set_int @ ( insert_int @ I @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I @ one_one_int ) @ J2 ) ) ) ) ) ).

% simp_from_to
thf(fact_4656_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
      = ( if_VEBT_VEBT
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
          & ~ ( ( X = Mi )
              | ( X = Ma ) ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ X @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) ) ) ).

% vebt_insert.simps(5)
thf(fact_4657_vebt__delete_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y
                    = ( vEBT_Leaf @ $false @ B5 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A5: $o,B5: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A5 @ B5 ) )
               => ( ( Xa2
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y
                      = ( vEBT_Leaf @ A5 @ $false ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A5: $o,B5: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A5 @ B5 ) )
                 => ! [N2: nat] :
                      ( ( Xa2
                        = ( suc @ ( suc @ N2 ) ) )
                     => ( ( Y
                          = ( vEBT_Leaf @ A5 @ B5 ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList4 @ Summary3 ) )
                   => ( ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList4 @ Summary3 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList4 @ Summary3 ) @ Xa2 ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) )
                     => ( ( Y
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) @ Xa2 ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) )
                       => ( ( Y
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ Xa2 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) )
                         => ( ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa2 ) )
                               => ( Y
                                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) ) )
                              & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa2 ) )
                               => ( ( ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                   => ( Y
                                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) ) )
                                  & ( ~ ( ( Xa2 = Mi2 )
                                        & ( Xa2 = Ma2 ) )
                                   => ( Y
                                      = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                                        @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some_P7363390416028606310at_nat
                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa2 = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa2 != Mi2 )
                                                     => ( Xa2 = Ma2 ) ) )
                                                  @ ( if_nat
                                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                      = none_nat )
                                                    @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some_P7363390416028606310at_nat
                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa2 = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa2 != Mi2 )
                                                     => ( Xa2 = Ma2 ) ) )
                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList4 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ Summary3 ) )
                                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) ) ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
thf(fact_4658_vebt__insert_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( ( ( Xa2 = zero_zero_nat )
                   => ( Y
                      = ( vEBT_Leaf @ $true @ B5 ) ) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A5 @ $true ) ) )
                      & ( ( Xa2 != one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) )
               => ( ( Y
                    = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) @ Xa2 ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) )
                 => ( ( Y
                      = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ Xa2 ) ) ) )
             => ( ! [V2: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList4 @ Summary3 ) )
                   => ( ( Y
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList4 @ Summary3 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList4 @ Summary3 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) )
                     => ( ( Y
                          = ( if_VEBT_VEBT
                            @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                              & ~ ( ( Xa2 = Mi2 )
                                  | ( Xa2 = Ma2 ) ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary3 ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
thf(fact_4659_insert_H__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_set_vebt @ ( vEBT_VEBT_insert @ T @ X ) )
        = ( inf_inf_set_nat @ ( sup_sup_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).

% insert'_correct
thf(fact_4660_VEBT__internal_Oinsert_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_VEBT_insert @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( Y
                  = ( vEBT_vebt_insert @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList4 @ Summary3 ) )
               => ( ( ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
                     => ( Y
                        = ( vEBT_Node @ Info2 @ Deg2 @ TreeList4 @ Summary3 ) ) )
                    & ( ~ ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
                     => ( Y
                        = ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList4 @ Summary3 ) @ Xa2 ) ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList4 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ).

% VEBT_internal.insert'.pelims
thf(fact_4661_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( Y
                  = ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B5 )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ( ~ Y
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ( ~ Y
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) )
                     => ( ( Y
                          = ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
thf(fact_4662_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A5 )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B5 )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) @ Xa2 ) )
                       => ( ( Xa2 != Mi2 )
                         => ( ( Xa2 != Ma2 )
                           => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                              & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                               => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                  & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                                       => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
thf(fact_4663_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( Y
                  = ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B5 )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList4: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList4 @ S2 ) )
                 => ( ( Y
                      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList4 @ S2 ) @ Xa2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
thf(fact_4664_Int__iff,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( ( member_complex @ C @ ( inf_inf_set_complex @ A2 @ B2 ) )
      = ( ( member_complex @ C @ A2 )
        & ( member_complex @ C @ B2 ) ) ) ).

% Int_iff
thf(fact_4665_Int__iff,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( ( member_real @ C @ ( inf_inf_set_real @ A2 @ B2 ) )
      = ( ( member_real @ C @ A2 )
        & ( member_real @ C @ B2 ) ) ) ).

% Int_iff
thf(fact_4666_Int__iff,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
      = ( ( member_set_nat @ C @ A2 )
        & ( member_set_nat @ C @ B2 ) ) ) ).

% Int_iff
thf(fact_4667_Int__iff,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( ( member_int @ C @ ( inf_inf_set_int @ A2 @ B2 ) )
      = ( ( member_int @ C @ A2 )
        & ( member_int @ C @ B2 ) ) ) ).

% Int_iff
thf(fact_4668_Int__iff,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
      = ( ( member_nat @ C @ A2 )
        & ( member_nat @ C @ B2 ) ) ) ).

% Int_iff
thf(fact_4669_IntI,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( ( member_complex @ C @ A2 )
     => ( ( member_complex @ C @ B2 )
       => ( member_complex @ C @ ( inf_inf_set_complex @ A2 @ B2 ) ) ) ) ).

% IntI
thf(fact_4670_IntI,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( ( member_real @ C @ A2 )
     => ( ( member_real @ C @ B2 )
       => ( member_real @ C @ ( inf_inf_set_real @ A2 @ B2 ) ) ) ) ).

% IntI
thf(fact_4671_IntI,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ A2 )
     => ( ( member_set_nat @ C @ B2 )
       => ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).

% IntI
thf(fact_4672_IntI,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( ( member_int @ C @ A2 )
     => ( ( member_int @ C @ B2 )
       => ( member_int @ C @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ).

% IntI
thf(fact_4673_IntI,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ A2 )
     => ( ( member_nat @ C @ B2 )
       => ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).

% IntI
thf(fact_4674_max__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ Q2 @ zero_z5237406670263579293d_enat )
      = Q2 ) ).

% max_enat_simps(2)
thf(fact_4675_max__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q2 )
      = Q2 ) ).

% max_enat_simps(3)
thf(fact_4676_Int__subset__iff,axiom,
    ! [C5: set_nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ C5 @ ( inf_inf_set_nat @ A2 @ B2 ) )
      = ( ( ord_less_eq_set_nat @ C5 @ A2 )
        & ( ord_less_eq_set_nat @ C5 @ B2 ) ) ) ).

% Int_subset_iff
thf(fact_4677_Int__insert__right__if1,axiom,
    ! [A: $o,A2: set_o,B2: set_o] :
      ( ( member_o @ A @ A2 )
     => ( ( inf_inf_set_o @ A2 @ ( insert_o @ A @ B2 ) )
        = ( insert_o @ A @ ( inf_inf_set_o @ A2 @ B2 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_4678_Int__insert__right__if1,axiom,
    ! [A: complex,A2: set_complex,B2: set_complex] :
      ( ( member_complex @ A @ A2 )
     => ( ( inf_inf_set_complex @ A2 @ ( insert_complex @ A @ B2 ) )
        = ( insert_complex @ A @ ( inf_inf_set_complex @ A2 @ B2 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_4679_Int__insert__right__if1,axiom,
    ! [A: real,A2: set_real,B2: set_real] :
      ( ( member_real @ A @ A2 )
     => ( ( inf_inf_set_real @ A2 @ ( insert_real @ A @ B2 ) )
        = ( insert_real @ A @ ( inf_inf_set_real @ A2 @ B2 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_4680_Int__insert__right__if1,axiom,
    ! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ A @ A2 )
     => ( ( inf_inf_set_set_nat @ A2 @ ( insert_set_nat @ A @ B2 ) )
        = ( insert_set_nat @ A @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_4681_Int__insert__right__if1,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( ( member_int @ A @ A2 )
     => ( ( inf_inf_set_int @ A2 @ ( insert_int @ A @ B2 ) )
        = ( insert_int @ A @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_4682_Int__insert__right__if1,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
        = ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_4683_Int__insert__right__if0,axiom,
    ! [A: $o,A2: set_o,B2: set_o] :
      ( ~ ( member_o @ A @ A2 )
     => ( ( inf_inf_set_o @ A2 @ ( insert_o @ A @ B2 ) )
        = ( inf_inf_set_o @ A2 @ B2 ) ) ) ).

% Int_insert_right_if0
thf(fact_4684_Int__insert__right__if0,axiom,
    ! [A: complex,A2: set_complex,B2: set_complex] :
      ( ~ ( member_complex @ A @ A2 )
     => ( ( inf_inf_set_complex @ A2 @ ( insert_complex @ A @ B2 ) )
        = ( inf_inf_set_complex @ A2 @ B2 ) ) ) ).

% Int_insert_right_if0
thf(fact_4685_Int__insert__right__if0,axiom,
    ! [A: real,A2: set_real,B2: set_real] :
      ( ~ ( member_real @ A @ A2 )
     => ( ( inf_inf_set_real @ A2 @ ( insert_real @ A @ B2 ) )
        = ( inf_inf_set_real @ A2 @ B2 ) ) ) ).

% Int_insert_right_if0
thf(fact_4686_Int__insert__right__if0,axiom,
    ! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ ( member_set_nat @ A @ A2 )
     => ( ( inf_inf_set_set_nat @ A2 @ ( insert_set_nat @ A @ B2 ) )
        = ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ).

% Int_insert_right_if0
thf(fact_4687_Int__insert__right__if0,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( ~ ( member_int @ A @ A2 )
     => ( ( inf_inf_set_int @ A2 @ ( insert_int @ A @ B2 ) )
        = ( inf_inf_set_int @ A2 @ B2 ) ) ) ).

% Int_insert_right_if0
thf(fact_4688_Int__insert__right__if0,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( ~ ( member_nat @ A @ A2 )
     => ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
        = ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).

% Int_insert_right_if0
thf(fact_4689_insert__inter__insert,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( ( inf_inf_set_int @ ( insert_int @ A @ A2 ) @ ( insert_int @ A @ B2 ) )
      = ( insert_int @ A @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ).

% insert_inter_insert
thf(fact_4690_insert__inter__insert,axiom,
    ! [A: real,A2: set_real,B2: set_real] :
      ( ( inf_inf_set_real @ ( insert_real @ A @ A2 ) @ ( insert_real @ A @ B2 ) )
      = ( insert_real @ A @ ( inf_inf_set_real @ A2 @ B2 ) ) ) ).

% insert_inter_insert
thf(fact_4691_insert__inter__insert,axiom,
    ! [A: $o,A2: set_o,B2: set_o] :
      ( ( inf_inf_set_o @ ( insert_o @ A @ A2 ) @ ( insert_o @ A @ B2 ) )
      = ( insert_o @ A @ ( inf_inf_set_o @ A2 @ B2 ) ) ) ).

% insert_inter_insert
thf(fact_4692_insert__inter__insert,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ ( insert_nat @ A @ B2 ) )
      = ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).

% insert_inter_insert
thf(fact_4693_Int__insert__left__if1,axiom,
    ! [A: $o,C5: set_o,B2: set_o] :
      ( ( member_o @ A @ C5 )
     => ( ( inf_inf_set_o @ ( insert_o @ A @ B2 ) @ C5 )
        = ( insert_o @ A @ ( inf_inf_set_o @ B2 @ C5 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_4694_Int__insert__left__if1,axiom,
    ! [A: complex,C5: set_complex,B2: set_complex] :
      ( ( member_complex @ A @ C5 )
     => ( ( inf_inf_set_complex @ ( insert_complex @ A @ B2 ) @ C5 )
        = ( insert_complex @ A @ ( inf_inf_set_complex @ B2 @ C5 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_4695_Int__insert__left__if1,axiom,
    ! [A: real,C5: set_real,B2: set_real] :
      ( ( member_real @ A @ C5 )
     => ( ( inf_inf_set_real @ ( insert_real @ A @ B2 ) @ C5 )
        = ( insert_real @ A @ ( inf_inf_set_real @ B2 @ C5 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_4696_Int__insert__left__if1,axiom,
    ! [A: set_nat,C5: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ A @ C5 )
     => ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ B2 ) @ C5 )
        = ( insert_set_nat @ A @ ( inf_inf_set_set_nat @ B2 @ C5 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_4697_Int__insert__left__if1,axiom,
    ! [A: int,C5: set_int,B2: set_int] :
      ( ( member_int @ A @ C5 )
     => ( ( inf_inf_set_int @ ( insert_int @ A @ B2 ) @ C5 )
        = ( insert_int @ A @ ( inf_inf_set_int @ B2 @ C5 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_4698_Int__insert__left__if1,axiom,
    ! [A: nat,C5: set_nat,B2: set_nat] :
      ( ( member_nat @ A @ C5 )
     => ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C5 )
        = ( insert_nat @ A @ ( inf_inf_set_nat @ B2 @ C5 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_4699_Int__insert__left__if0,axiom,
    ! [A: $o,C5: set_o,B2: set_o] :
      ( ~ ( member_o @ A @ C5 )
     => ( ( inf_inf_set_o @ ( insert_o @ A @ B2 ) @ C5 )
        = ( inf_inf_set_o @ B2 @ C5 ) ) ) ).

% Int_insert_left_if0
thf(fact_4700_Int__insert__left__if0,axiom,
    ! [A: complex,C5: set_complex,B2: set_complex] :
      ( ~ ( member_complex @ A @ C5 )
     => ( ( inf_inf_set_complex @ ( insert_complex @ A @ B2 ) @ C5 )
        = ( inf_inf_set_complex @ B2 @ C5 ) ) ) ).

% Int_insert_left_if0
thf(fact_4701_Int__insert__left__if0,axiom,
    ! [A: real,C5: set_real,B2: set_real] :
      ( ~ ( member_real @ A @ C5 )
     => ( ( inf_inf_set_real @ ( insert_real @ A @ B2 ) @ C5 )
        = ( inf_inf_set_real @ B2 @ C5 ) ) ) ).

% Int_insert_left_if0
thf(fact_4702_Int__insert__left__if0,axiom,
    ! [A: set_nat,C5: set_set_nat,B2: set_set_nat] :
      ( ~ ( member_set_nat @ A @ C5 )
     => ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ B2 ) @ C5 )
        = ( inf_inf_set_set_nat @ B2 @ C5 ) ) ) ).

% Int_insert_left_if0
thf(fact_4703_Int__insert__left__if0,axiom,
    ! [A: int,C5: set_int,B2: set_int] :
      ( ~ ( member_int @ A @ C5 )
     => ( ( inf_inf_set_int @ ( insert_int @ A @ B2 ) @ C5 )
        = ( inf_inf_set_int @ B2 @ C5 ) ) ) ).

% Int_insert_left_if0
thf(fact_4704_Int__insert__left__if0,axiom,
    ! [A: nat,C5: set_nat,B2: set_nat] :
      ( ~ ( member_nat @ A @ C5 )
     => ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C5 )
        = ( inf_inf_set_nat @ B2 @ C5 ) ) ) ).

% Int_insert_left_if0
thf(fact_4705_Un__Int__eq_I1_J,axiom,
    ! [S3: set_nat,T3: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S3 @ T3 ) @ S3 )
      = S3 ) ).

% Un_Int_eq(1)
thf(fact_4706_Un__Int__eq_I2_J,axiom,
    ! [S3: set_nat,T3: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S3 @ T3 ) @ T3 )
      = T3 ) ).

% Un_Int_eq(2)
thf(fact_4707_Un__Int__eq_I3_J,axiom,
    ! [S3: set_nat,T3: set_nat] :
      ( ( inf_inf_set_nat @ S3 @ ( sup_sup_set_nat @ S3 @ T3 ) )
      = S3 ) ).

% Un_Int_eq(3)
thf(fact_4708_Un__Int__eq_I4_J,axiom,
    ! [T3: set_nat,S3: set_nat] :
      ( ( inf_inf_set_nat @ T3 @ ( sup_sup_set_nat @ S3 @ T3 ) )
      = T3 ) ).

% Un_Int_eq(4)
thf(fact_4709_Int__Un__eq_I1_J,axiom,
    ! [S3: set_nat,T3: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S3 @ T3 ) @ S3 )
      = S3 ) ).

% Int_Un_eq(1)
thf(fact_4710_Int__Un__eq_I2_J,axiom,
    ! [S3: set_nat,T3: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S3 @ T3 ) @ T3 )
      = T3 ) ).

% Int_Un_eq(2)
thf(fact_4711_Int__Un__eq_I3_J,axiom,
    ! [S3: set_nat,T3: set_nat] :
      ( ( sup_sup_set_nat @ S3 @ ( inf_inf_set_nat @ S3 @ T3 ) )
      = S3 ) ).

% Int_Un_eq(3)
thf(fact_4712_Int__Un__eq_I4_J,axiom,
    ! [T3: set_nat,S3: set_nat] :
      ( ( sup_sup_set_nat @ T3 @ ( inf_inf_set_nat @ S3 @ T3 ) )
      = T3 ) ).

% Int_Un_eq(4)
thf(fact_4713_disjoint__insert_I2_J,axiom,
    ! [A2: set_complex,B: complex,B2: set_complex] :
      ( ( bot_bot_set_complex
        = ( inf_inf_set_complex @ A2 @ ( insert_complex @ B @ B2 ) ) )
      = ( ~ ( member_complex @ B @ A2 )
        & ( bot_bot_set_complex
          = ( inf_inf_set_complex @ A2 @ B2 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_4714_disjoint__insert_I2_J,axiom,
    ! [A2: set_set_nat,B: set_nat,B2: set_set_nat] :
      ( ( bot_bot_set_set_nat
        = ( inf_inf_set_set_nat @ A2 @ ( insert_set_nat @ B @ B2 ) ) )
      = ( ~ ( member_set_nat @ B @ A2 )
        & ( bot_bot_set_set_nat
          = ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_4715_disjoint__insert_I2_J,axiom,
    ! [A2: set_real,B: real,B2: set_real] :
      ( ( bot_bot_set_real
        = ( inf_inf_set_real @ A2 @ ( insert_real @ B @ B2 ) ) )
      = ( ~ ( member_real @ B @ A2 )
        & ( bot_bot_set_real
          = ( inf_inf_set_real @ A2 @ B2 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_4716_disjoint__insert_I2_J,axiom,
    ! [A2: set_o,B: $o,B2: set_o] :
      ( ( bot_bot_set_o
        = ( inf_inf_set_o @ A2 @ ( insert_o @ B @ B2 ) ) )
      = ( ~ ( member_o @ B @ A2 )
        & ( bot_bot_set_o
          = ( inf_inf_set_o @ A2 @ B2 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_4717_disjoint__insert_I2_J,axiom,
    ! [A2: set_nat,B: nat,B2: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) )
      = ( ~ ( member_nat @ B @ A2 )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_4718_disjoint__insert_I2_J,axiom,
    ! [A2: set_int,B: int,B2: set_int] :
      ( ( bot_bot_set_int
        = ( inf_inf_set_int @ A2 @ ( insert_int @ B @ B2 ) ) )
      = ( ~ ( member_int @ B @ A2 )
        & ( bot_bot_set_int
          = ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_4719_disjoint__insert_I1_J,axiom,
    ! [B2: set_complex,A: complex,A2: set_complex] :
      ( ( ( inf_inf_set_complex @ B2 @ ( insert_complex @ A @ A2 ) )
        = bot_bot_set_complex )
      = ( ~ ( member_complex @ A @ B2 )
        & ( ( inf_inf_set_complex @ B2 @ A2 )
          = bot_bot_set_complex ) ) ) ).

% disjoint_insert(1)
thf(fact_4720_disjoint__insert_I1_J,axiom,
    ! [B2: set_set_nat,A: set_nat,A2: set_set_nat] :
      ( ( ( inf_inf_set_set_nat @ B2 @ ( insert_set_nat @ A @ A2 ) )
        = bot_bot_set_set_nat )
      = ( ~ ( member_set_nat @ A @ B2 )
        & ( ( inf_inf_set_set_nat @ B2 @ A2 )
          = bot_bot_set_set_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_4721_disjoint__insert_I1_J,axiom,
    ! [B2: set_real,A: real,A2: set_real] :
      ( ( ( inf_inf_set_real @ B2 @ ( insert_real @ A @ A2 ) )
        = bot_bot_set_real )
      = ( ~ ( member_real @ A @ B2 )
        & ( ( inf_inf_set_real @ B2 @ A2 )
          = bot_bot_set_real ) ) ) ).

% disjoint_insert(1)
thf(fact_4722_disjoint__insert_I1_J,axiom,
    ! [B2: set_o,A: $o,A2: set_o] :
      ( ( ( inf_inf_set_o @ B2 @ ( insert_o @ A @ A2 ) )
        = bot_bot_set_o )
      = ( ~ ( member_o @ A @ B2 )
        & ( ( inf_inf_set_o @ B2 @ A2 )
          = bot_bot_set_o ) ) ) ).

% disjoint_insert(1)
thf(fact_4723_disjoint__insert_I1_J,axiom,
    ! [B2: set_nat,A: nat,A2: set_nat] :
      ( ( ( inf_inf_set_nat @ B2 @ ( insert_nat @ A @ A2 ) )
        = bot_bot_set_nat )
      = ( ~ ( member_nat @ A @ B2 )
        & ( ( inf_inf_set_nat @ B2 @ A2 )
          = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_4724_disjoint__insert_I1_J,axiom,
    ! [B2: set_int,A: int,A2: set_int] :
      ( ( ( inf_inf_set_int @ B2 @ ( insert_int @ A @ A2 ) )
        = bot_bot_set_int )
      = ( ~ ( member_int @ A @ B2 )
        & ( ( inf_inf_set_int @ B2 @ A2 )
          = bot_bot_set_int ) ) ) ).

% disjoint_insert(1)
thf(fact_4725_insert__disjoint_I2_J,axiom,
    ! [A: complex,A2: set_complex,B2: set_complex] :
      ( ( bot_bot_set_complex
        = ( inf_inf_set_complex @ ( insert_complex @ A @ A2 ) @ B2 ) )
      = ( ~ ( member_complex @ A @ B2 )
        & ( bot_bot_set_complex
          = ( inf_inf_set_complex @ A2 @ B2 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_4726_insert__disjoint_I2_J,axiom,
    ! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( bot_bot_set_set_nat
        = ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ A2 ) @ B2 ) )
      = ( ~ ( member_set_nat @ A @ B2 )
        & ( bot_bot_set_set_nat
          = ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_4727_insert__disjoint_I2_J,axiom,
    ! [A: real,A2: set_real,B2: set_real] :
      ( ( bot_bot_set_real
        = ( inf_inf_set_real @ ( insert_real @ A @ A2 ) @ B2 ) )
      = ( ~ ( member_real @ A @ B2 )
        & ( bot_bot_set_real
          = ( inf_inf_set_real @ A2 @ B2 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_4728_insert__disjoint_I2_J,axiom,
    ! [A: $o,A2: set_o,B2: set_o] :
      ( ( bot_bot_set_o
        = ( inf_inf_set_o @ ( insert_o @ A @ A2 ) @ B2 ) )
      = ( ~ ( member_o @ A @ B2 )
        & ( bot_bot_set_o
          = ( inf_inf_set_o @ A2 @ B2 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_4729_insert__disjoint_I2_J,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B2 ) )
      = ( ~ ( member_nat @ A @ B2 )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_4730_insert__disjoint_I2_J,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( ( bot_bot_set_int
        = ( inf_inf_set_int @ ( insert_int @ A @ A2 ) @ B2 ) )
      = ( ~ ( member_int @ A @ B2 )
        & ( bot_bot_set_int
          = ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_4731_insert__disjoint_I1_J,axiom,
    ! [A: complex,A2: set_complex,B2: set_complex] :
      ( ( ( inf_inf_set_complex @ ( insert_complex @ A @ A2 ) @ B2 )
        = bot_bot_set_complex )
      = ( ~ ( member_complex @ A @ B2 )
        & ( ( inf_inf_set_complex @ A2 @ B2 )
          = bot_bot_set_complex ) ) ) ).

% insert_disjoint(1)
thf(fact_4732_insert__disjoint_I1_J,axiom,
    ! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ A2 ) @ B2 )
        = bot_bot_set_set_nat )
      = ( ~ ( member_set_nat @ A @ B2 )
        & ( ( inf_inf_set_set_nat @ A2 @ B2 )
          = bot_bot_set_set_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_4733_insert__disjoint_I1_J,axiom,
    ! [A: real,A2: set_real,B2: set_real] :
      ( ( ( inf_inf_set_real @ ( insert_real @ A @ A2 ) @ B2 )
        = bot_bot_set_real )
      = ( ~ ( member_real @ A @ B2 )
        & ( ( inf_inf_set_real @ A2 @ B2 )
          = bot_bot_set_real ) ) ) ).

% insert_disjoint(1)
thf(fact_4734_insert__disjoint_I1_J,axiom,
    ! [A: $o,A2: set_o,B2: set_o] :
      ( ( ( inf_inf_set_o @ ( insert_o @ A @ A2 ) @ B2 )
        = bot_bot_set_o )
      = ( ~ ( member_o @ A @ B2 )
        & ( ( inf_inf_set_o @ A2 @ B2 )
          = bot_bot_set_o ) ) ) ).

% insert_disjoint(1)
thf(fact_4735_insert__disjoint_I1_J,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B2 )
        = bot_bot_set_nat )
      = ( ~ ( member_nat @ A @ B2 )
        & ( ( inf_inf_set_nat @ A2 @ B2 )
          = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_4736_insert__disjoint_I1_J,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( ( ( inf_inf_set_int @ ( insert_int @ A @ A2 ) @ B2 )
        = bot_bot_set_int )
      = ( ~ ( member_int @ A @ B2 )
        & ( ( inf_inf_set_int @ A2 @ B2 )
          = bot_bot_set_int ) ) ) ).

% insert_disjoint(1)
thf(fact_4737_Diff__disjoint,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( inf_inf_set_real @ A2 @ ( minus_minus_set_real @ B2 @ A2 ) )
      = bot_bot_set_real ) ).

% Diff_disjoint
thf(fact_4738_Diff__disjoint,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( inf_inf_set_o @ A2 @ ( minus_minus_set_o @ B2 @ A2 ) )
      = bot_bot_set_o ) ).

% Diff_disjoint
thf(fact_4739_Diff__disjoint,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( inf_inf_set_int @ A2 @ ( minus_minus_set_int @ B2 @ A2 ) )
      = bot_bot_set_int ) ).

% Diff_disjoint
thf(fact_4740_Diff__disjoint,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( inf_inf_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
      = bot_bot_set_nat ) ).

% Diff_disjoint
thf(fact_4741_Int__left__commute,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C5 ) )
      = ( inf_inf_set_nat @ B2 @ ( inf_inf_set_nat @ A2 @ C5 ) ) ) ).

% Int_left_commute
thf(fact_4742_Int__left__absorb,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
      = ( inf_inf_set_nat @ A2 @ B2 ) ) ).

% Int_left_absorb
thf(fact_4743_Int__commute,axiom,
    ( inf_inf_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] : ( inf_inf_set_nat @ B7 @ A6 ) ) ) ).

% Int_commute
thf(fact_4744_Int__absorb,axiom,
    ! [A2: set_nat] :
      ( ( inf_inf_set_nat @ A2 @ A2 )
      = A2 ) ).

% Int_absorb
thf(fact_4745_Int__assoc,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C5 )
      = ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C5 ) ) ) ).

% Int_assoc
thf(fact_4746_IntD2,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( ( member_complex @ C @ ( inf_inf_set_complex @ A2 @ B2 ) )
     => ( member_complex @ C @ B2 ) ) ).

% IntD2
thf(fact_4747_IntD2,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( ( member_real @ C @ ( inf_inf_set_real @ A2 @ B2 ) )
     => ( member_real @ C @ B2 ) ) ).

% IntD2
thf(fact_4748_IntD2,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
     => ( member_set_nat @ C @ B2 ) ) ).

% IntD2
thf(fact_4749_IntD2,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( ( member_int @ C @ ( inf_inf_set_int @ A2 @ B2 ) )
     => ( member_int @ C @ B2 ) ) ).

% IntD2
thf(fact_4750_IntD2,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
     => ( member_nat @ C @ B2 ) ) ).

% IntD2
thf(fact_4751_IntD1,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( ( member_complex @ C @ ( inf_inf_set_complex @ A2 @ B2 ) )
     => ( member_complex @ C @ A2 ) ) ).

% IntD1
thf(fact_4752_IntD1,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( ( member_real @ C @ ( inf_inf_set_real @ A2 @ B2 ) )
     => ( member_real @ C @ A2 ) ) ).

% IntD1
thf(fact_4753_IntD1,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
     => ( member_set_nat @ C @ A2 ) ) ).

% IntD1
thf(fact_4754_IntD1,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( ( member_int @ C @ ( inf_inf_set_int @ A2 @ B2 ) )
     => ( member_int @ C @ A2 ) ) ).

% IntD1
thf(fact_4755_IntD1,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
     => ( member_nat @ C @ A2 ) ) ).

% IntD1
thf(fact_4756_IntE,axiom,
    ! [C: complex,A2: set_complex,B2: set_complex] :
      ( ( member_complex @ C @ ( inf_inf_set_complex @ A2 @ B2 ) )
     => ~ ( ( member_complex @ C @ A2 )
         => ~ ( member_complex @ C @ B2 ) ) ) ).

% IntE
thf(fact_4757_IntE,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( ( member_real @ C @ ( inf_inf_set_real @ A2 @ B2 ) )
     => ~ ( ( member_real @ C @ A2 )
         => ~ ( member_real @ C @ B2 ) ) ) ).

% IntE
thf(fact_4758_IntE,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
     => ~ ( ( member_set_nat @ C @ A2 )
         => ~ ( member_set_nat @ C @ B2 ) ) ) ).

% IntE
thf(fact_4759_IntE,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( ( member_int @ C @ ( inf_inf_set_int @ A2 @ B2 ) )
     => ~ ( ( member_int @ C @ A2 )
         => ~ ( member_int @ C @ B2 ) ) ) ).

% IntE
thf(fact_4760_IntE,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
     => ~ ( ( member_nat @ C @ A2 )
         => ~ ( member_nat @ C @ B2 ) ) ) ).

% IntE
thf(fact_4761_Int__def,axiom,
    ( inf_inf_set_complex
    = ( ^ [A6: set_complex,B7: set_complex] :
          ( collect_complex
          @ ^ [X6: complex] :
              ( ( member_complex @ X6 @ A6 )
              & ( member_complex @ X6 @ B7 ) ) ) ) ) ).

% Int_def
thf(fact_4762_Int__def,axiom,
    ( inf_inf_set_real
    = ( ^ [A6: set_real,B7: set_real] :
          ( collect_real
          @ ^ [X6: real] :
              ( ( member_real @ X6 @ A6 )
              & ( member_real @ X6 @ B7 ) ) ) ) ) ).

% Int_def
thf(fact_4763_Int__def,axiom,
    ( inf_inf_set_list_nat
    = ( ^ [A6: set_list_nat,B7: set_list_nat] :
          ( collect_list_nat
          @ ^ [X6: list_nat] :
              ( ( member_list_nat @ X6 @ A6 )
              & ( member_list_nat @ X6 @ B7 ) ) ) ) ) ).

% Int_def
thf(fact_4764_Int__def,axiom,
    ( inf_inf_set_set_nat
    = ( ^ [A6: set_set_nat,B7: set_set_nat] :
          ( collect_set_nat
          @ ^ [X6: set_nat] :
              ( ( member_set_nat @ X6 @ A6 )
              & ( member_set_nat @ X6 @ B7 ) ) ) ) ) ).

% Int_def
thf(fact_4765_Int__def,axiom,
    ( inf_inf_set_int
    = ( ^ [A6: set_int,B7: set_int] :
          ( collect_int
          @ ^ [X6: int] :
              ( ( member_int @ X6 @ A6 )
              & ( member_int @ X6 @ B7 ) ) ) ) ) ).

% Int_def
thf(fact_4766_Int__def,axiom,
    ( inf_inf_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( collect_nat
          @ ^ [X6: nat] :
              ( ( member_nat @ X6 @ A6 )
              & ( member_nat @ X6 @ B7 ) ) ) ) ) ).

% Int_def
thf(fact_4767_Int__Collect,axiom,
    ! [X: complex,A2: set_complex,P: complex > $o] :
      ( ( member_complex @ X @ ( inf_inf_set_complex @ A2 @ ( collect_complex @ P ) ) )
      = ( ( member_complex @ X @ A2 )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_4768_Int__Collect,axiom,
    ! [X: real,A2: set_real,P: real > $o] :
      ( ( member_real @ X @ ( inf_inf_set_real @ A2 @ ( collect_real @ P ) ) )
      = ( ( member_real @ X @ A2 )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_4769_Int__Collect,axiom,
    ! [X: list_nat,A2: set_list_nat,P: list_nat > $o] :
      ( ( member_list_nat @ X @ ( inf_inf_set_list_nat @ A2 @ ( collect_list_nat @ P ) ) )
      = ( ( member_list_nat @ X @ A2 )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_4770_Int__Collect,axiom,
    ! [X: set_nat,A2: set_set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ X @ ( inf_inf_set_set_nat @ A2 @ ( collect_set_nat @ P ) ) )
      = ( ( member_set_nat @ X @ A2 )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_4771_Int__Collect,axiom,
    ! [X: int,A2: set_int,P: int > $o] :
      ( ( member_int @ X @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) )
      = ( ( member_int @ X @ A2 )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_4772_Int__Collect,axiom,
    ! [X: nat,A2: set_nat,P: nat > $o] :
      ( ( member_nat @ X @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) )
      = ( ( member_nat @ X @ A2 )
        & ( P @ X ) ) ) ).

% Int_Collect
thf(fact_4773_Collect__conj__eq,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ( collect_real
        @ ^ [X6: real] :
            ( ( P @ X6 )
            & ( Q @ X6 ) ) )
      = ( inf_inf_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_4774_Collect__conj__eq,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X6: list_nat] :
            ( ( P @ X6 )
            & ( Q @ X6 ) ) )
      = ( inf_inf_set_list_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_4775_Collect__conj__eq,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X6: set_nat] :
            ( ( P @ X6 )
            & ( Q @ X6 ) ) )
      = ( inf_inf_set_set_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_4776_Collect__conj__eq,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( collect_int
        @ ^ [X6: int] :
            ( ( P @ X6 )
            & ( Q @ X6 ) ) )
      = ( inf_inf_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_4777_Collect__conj__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X6: nat] :
            ( ( P @ X6 )
            & ( Q @ X6 ) ) )
      = ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_4778_Int__emptyI,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A2 )
         => ~ ( member_complex @ X5 @ B2 ) )
     => ( ( inf_inf_set_complex @ A2 @ B2 )
        = bot_bot_set_complex ) ) ).

% Int_emptyI
thf(fact_4779_Int__emptyI,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ! [X5: set_nat] :
          ( ( member_set_nat @ X5 @ A2 )
         => ~ ( member_set_nat @ X5 @ B2 ) )
     => ( ( inf_inf_set_set_nat @ A2 @ B2 )
        = bot_bot_set_set_nat ) ) ).

% Int_emptyI
thf(fact_4780_Int__emptyI,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A2 )
         => ~ ( member_real @ X5 @ B2 ) )
     => ( ( inf_inf_set_real @ A2 @ B2 )
        = bot_bot_set_real ) ) ).

% Int_emptyI
thf(fact_4781_Int__emptyI,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A2 )
         => ~ ( member_o @ X5 @ B2 ) )
     => ( ( inf_inf_set_o @ A2 @ B2 )
        = bot_bot_set_o ) ) ).

% Int_emptyI
thf(fact_4782_Int__emptyI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ~ ( member_nat @ X5 @ B2 ) )
     => ( ( inf_inf_set_nat @ A2 @ B2 )
        = bot_bot_set_nat ) ) ).

% Int_emptyI
thf(fact_4783_Int__emptyI,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ~ ( member_int @ X5 @ B2 ) )
     => ( ( inf_inf_set_int @ A2 @ B2 )
        = bot_bot_set_int ) ) ).

% Int_emptyI
thf(fact_4784_disjoint__iff,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ( ( inf_inf_set_complex @ A2 @ B2 )
        = bot_bot_set_complex )
      = ( ! [X6: complex] :
            ( ( member_complex @ X6 @ A2 )
           => ~ ( member_complex @ X6 @ B2 ) ) ) ) ).

% disjoint_iff
thf(fact_4785_disjoint__iff,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ( ( inf_inf_set_set_nat @ A2 @ B2 )
        = bot_bot_set_set_nat )
      = ( ! [X6: set_nat] :
            ( ( member_set_nat @ X6 @ A2 )
           => ~ ( member_set_nat @ X6 @ B2 ) ) ) ) ).

% disjoint_iff
thf(fact_4786_disjoint__iff,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( ( inf_inf_set_real @ A2 @ B2 )
        = bot_bot_set_real )
      = ( ! [X6: real] :
            ( ( member_real @ X6 @ A2 )
           => ~ ( member_real @ X6 @ B2 ) ) ) ) ).

% disjoint_iff
thf(fact_4787_disjoint__iff,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( ( inf_inf_set_o @ A2 @ B2 )
        = bot_bot_set_o )
      = ( ! [X6: $o] :
            ( ( member_o @ X6 @ A2 )
           => ~ ( member_o @ X6 @ B2 ) ) ) ) ).

% disjoint_iff
thf(fact_4788_disjoint__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ( inf_inf_set_nat @ A2 @ B2 )
        = bot_bot_set_nat )
      = ( ! [X6: nat] :
            ( ( member_nat @ X6 @ A2 )
           => ~ ( member_nat @ X6 @ B2 ) ) ) ) ).

% disjoint_iff
thf(fact_4789_disjoint__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ( inf_inf_set_int @ A2 @ B2 )
        = bot_bot_set_int )
      = ( ! [X6: int] :
            ( ( member_int @ X6 @ A2 )
           => ~ ( member_int @ X6 @ B2 ) ) ) ) ).

% disjoint_iff
thf(fact_4790_Int__empty__left,axiom,
    ! [B2: set_real] :
      ( ( inf_inf_set_real @ bot_bot_set_real @ B2 )
      = bot_bot_set_real ) ).

% Int_empty_left
thf(fact_4791_Int__empty__left,axiom,
    ! [B2: set_o] :
      ( ( inf_inf_set_o @ bot_bot_set_o @ B2 )
      = bot_bot_set_o ) ).

% Int_empty_left
thf(fact_4792_Int__empty__left,axiom,
    ! [B2: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ B2 )
      = bot_bot_set_nat ) ).

% Int_empty_left
thf(fact_4793_Int__empty__left,axiom,
    ! [B2: set_int] :
      ( ( inf_inf_set_int @ bot_bot_set_int @ B2 )
      = bot_bot_set_int ) ).

% Int_empty_left
thf(fact_4794_Int__empty__right,axiom,
    ! [A2: set_real] :
      ( ( inf_inf_set_real @ A2 @ bot_bot_set_real )
      = bot_bot_set_real ) ).

% Int_empty_right
thf(fact_4795_Int__empty__right,axiom,
    ! [A2: set_o] :
      ( ( inf_inf_set_o @ A2 @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% Int_empty_right
thf(fact_4796_Int__empty__right,axiom,
    ! [A2: set_nat] :
      ( ( inf_inf_set_nat @ A2 @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% Int_empty_right
thf(fact_4797_Int__empty__right,axiom,
    ! [A2: set_int] :
      ( ( inf_inf_set_int @ A2 @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% Int_empty_right
thf(fact_4798_disjoint__iff__not__equal,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( ( inf_inf_set_real @ A2 @ B2 )
        = bot_bot_set_real )
      = ( ! [X6: real] :
            ( ( member_real @ X6 @ A2 )
           => ! [Y6: real] :
                ( ( member_real @ Y6 @ B2 )
               => ( X6 != Y6 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_4799_disjoint__iff__not__equal,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( ( inf_inf_set_o @ A2 @ B2 )
        = bot_bot_set_o )
      = ( ! [X6: $o] :
            ( ( member_o @ X6 @ A2 )
           => ! [Y6: $o] :
                ( ( member_o @ Y6 @ B2 )
               => ( X6 = ~ Y6 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_4800_disjoint__iff__not__equal,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ( inf_inf_set_nat @ A2 @ B2 )
        = bot_bot_set_nat )
      = ( ! [X6: nat] :
            ( ( member_nat @ X6 @ A2 )
           => ! [Y6: nat] :
                ( ( member_nat @ Y6 @ B2 )
               => ( X6 != Y6 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_4801_disjoint__iff__not__equal,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ( inf_inf_set_int @ A2 @ B2 )
        = bot_bot_set_int )
      = ( ! [X6: int] :
            ( ( member_int @ X6 @ A2 )
           => ! [Y6: int] :
                ( ( member_int @ Y6 @ B2 )
               => ( X6 != Y6 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_4802_Int__mono,axiom,
    ! [A2: set_nat,C5: set_nat,B2: set_nat,D4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C5 )
     => ( ( ord_less_eq_set_nat @ B2 @ D4 )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ C5 @ D4 ) ) ) ) ).

% Int_mono
thf(fact_4803_Int__lower1,axiom,
    ! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ A2 ) ).

% Int_lower1
thf(fact_4804_Int__lower2,axiom,
    ! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ B2 ) ).

% Int_lower2
thf(fact_4805_Int__absorb1,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( inf_inf_set_nat @ A2 @ B2 )
        = B2 ) ) ).

% Int_absorb1
thf(fact_4806_Int__absorb2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( inf_inf_set_nat @ A2 @ B2 )
        = A2 ) ) ).

% Int_absorb2
thf(fact_4807_Int__greatest,axiom,
    ! [C5: set_nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ C5 @ A2 )
     => ( ( ord_less_eq_set_nat @ C5 @ B2 )
       => ( ord_less_eq_set_nat @ C5 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).

% Int_greatest
thf(fact_4808_Int__Collect__mono,axiom,
    ! [A2: set_complex,B2: set_complex,P: complex > $o,Q: complex > $o] :
      ( ( ord_le211207098394363844omplex @ A2 @ B2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A2 )
           => ( ( P @ X5 )
             => ( Q @ X5 ) ) )
       => ( ord_le211207098394363844omplex @ ( inf_inf_set_complex @ A2 @ ( collect_complex @ P ) ) @ ( inf_inf_set_complex @ B2 @ ( collect_complex @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_4809_Int__Collect__mono,axiom,
    ! [A2: set_real,B2: set_real,P: real > $o,Q: real > $o] :
      ( ( ord_less_eq_set_real @ A2 @ B2 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ A2 )
           => ( ( P @ X5 )
             => ( Q @ X5 ) ) )
       => ( ord_less_eq_set_real @ ( inf_inf_set_real @ A2 @ ( collect_real @ P ) ) @ ( inf_inf_set_real @ B2 @ ( collect_real @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_4810_Int__Collect__mono,axiom,
    ! [A2: set_list_nat,B2: set_list_nat,P: list_nat > $o,Q: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
     => ( ! [X5: list_nat] :
            ( ( member_list_nat @ X5 @ A2 )
           => ( ( P @ X5 )
             => ( Q @ X5 ) ) )
       => ( ord_le6045566169113846134st_nat @ ( inf_inf_set_list_nat @ A2 @ ( collect_list_nat @ P ) ) @ ( inf_inf_set_list_nat @ B2 @ ( collect_list_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_4811_Int__Collect__mono,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
     => ( ! [X5: set_nat] :
            ( ( member_set_nat @ X5 @ A2 )
           => ( ( P @ X5 )
             => ( Q @ X5 ) ) )
       => ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ ( collect_set_nat @ P ) ) @ ( inf_inf_set_set_nat @ B2 @ ( collect_set_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_4812_Int__Collect__mono,axiom,
    ! [A2: set_int,B2: set_int,P: int > $o,Q: int > $o] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A2 )
           => ( ( P @ X5 )
             => ( Q @ X5 ) ) )
       => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) @ ( inf_inf_set_int @ B2 @ ( collect_int @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_4813_Int__Collect__mono,axiom,
    ! [A2: set_nat,B2: set_nat,P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ A2 )
           => ( ( P @ X5 )
             => ( Q @ X5 ) ) )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B2 @ ( collect_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_4814_Int__insert__right,axiom,
    ! [A: $o,A2: set_o,B2: set_o] :
      ( ( ( member_o @ A @ A2 )
       => ( ( inf_inf_set_o @ A2 @ ( insert_o @ A @ B2 ) )
          = ( insert_o @ A @ ( inf_inf_set_o @ A2 @ B2 ) ) ) )
      & ( ~ ( member_o @ A @ A2 )
       => ( ( inf_inf_set_o @ A2 @ ( insert_o @ A @ B2 ) )
          = ( inf_inf_set_o @ A2 @ B2 ) ) ) ) ).

% Int_insert_right
thf(fact_4815_Int__insert__right,axiom,
    ! [A: complex,A2: set_complex,B2: set_complex] :
      ( ( ( member_complex @ A @ A2 )
       => ( ( inf_inf_set_complex @ A2 @ ( insert_complex @ A @ B2 ) )
          = ( insert_complex @ A @ ( inf_inf_set_complex @ A2 @ B2 ) ) ) )
      & ( ~ ( member_complex @ A @ A2 )
       => ( ( inf_inf_set_complex @ A2 @ ( insert_complex @ A @ B2 ) )
          = ( inf_inf_set_complex @ A2 @ B2 ) ) ) ) ).

% Int_insert_right
thf(fact_4816_Int__insert__right,axiom,
    ! [A: real,A2: set_real,B2: set_real] :
      ( ( ( member_real @ A @ A2 )
       => ( ( inf_inf_set_real @ A2 @ ( insert_real @ A @ B2 ) )
          = ( insert_real @ A @ ( inf_inf_set_real @ A2 @ B2 ) ) ) )
      & ( ~ ( member_real @ A @ A2 )
       => ( ( inf_inf_set_real @ A2 @ ( insert_real @ A @ B2 ) )
          = ( inf_inf_set_real @ A2 @ B2 ) ) ) ) ).

% Int_insert_right
thf(fact_4817_Int__insert__right,axiom,
    ! [A: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( ( member_set_nat @ A @ A2 )
       => ( ( inf_inf_set_set_nat @ A2 @ ( insert_set_nat @ A @ B2 ) )
          = ( insert_set_nat @ A @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) )
      & ( ~ ( member_set_nat @ A @ A2 )
       => ( ( inf_inf_set_set_nat @ A2 @ ( insert_set_nat @ A @ B2 ) )
          = ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).

% Int_insert_right
thf(fact_4818_Int__insert__right,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( ( ( member_int @ A @ A2 )
       => ( ( inf_inf_set_int @ A2 @ ( insert_int @ A @ B2 ) )
          = ( insert_int @ A @ ( inf_inf_set_int @ A2 @ B2 ) ) ) )
      & ( ~ ( member_int @ A @ A2 )
       => ( ( inf_inf_set_int @ A2 @ ( insert_int @ A @ B2 ) )
          = ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ).

% Int_insert_right
thf(fact_4819_Int__insert__right,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( ( ( member_nat @ A @ A2 )
       => ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
          = ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) )
      & ( ~ ( member_nat @ A @ A2 )
       => ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
          = ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).

% Int_insert_right
thf(fact_4820_Int__insert__left,axiom,
    ! [A: $o,C5: set_o,B2: set_o] :
      ( ( ( member_o @ A @ C5 )
       => ( ( inf_inf_set_o @ ( insert_o @ A @ B2 ) @ C5 )
          = ( insert_o @ A @ ( inf_inf_set_o @ B2 @ C5 ) ) ) )
      & ( ~ ( member_o @ A @ C5 )
       => ( ( inf_inf_set_o @ ( insert_o @ A @ B2 ) @ C5 )
          = ( inf_inf_set_o @ B2 @ C5 ) ) ) ) ).

% Int_insert_left
thf(fact_4821_Int__insert__left,axiom,
    ! [A: complex,C5: set_complex,B2: set_complex] :
      ( ( ( member_complex @ A @ C5 )
       => ( ( inf_inf_set_complex @ ( insert_complex @ A @ B2 ) @ C5 )
          = ( insert_complex @ A @ ( inf_inf_set_complex @ B2 @ C5 ) ) ) )
      & ( ~ ( member_complex @ A @ C5 )
       => ( ( inf_inf_set_complex @ ( insert_complex @ A @ B2 ) @ C5 )
          = ( inf_inf_set_complex @ B2 @ C5 ) ) ) ) ).

% Int_insert_left
thf(fact_4822_Int__insert__left,axiom,
    ! [A: real,C5: set_real,B2: set_real] :
      ( ( ( member_real @ A @ C5 )
       => ( ( inf_inf_set_real @ ( insert_real @ A @ B2 ) @ C5 )
          = ( insert_real @ A @ ( inf_inf_set_real @ B2 @ C5 ) ) ) )
      & ( ~ ( member_real @ A @ C5 )
       => ( ( inf_inf_set_real @ ( insert_real @ A @ B2 ) @ C5 )
          = ( inf_inf_set_real @ B2 @ C5 ) ) ) ) ).

% Int_insert_left
thf(fact_4823_Int__insert__left,axiom,
    ! [A: set_nat,C5: set_set_nat,B2: set_set_nat] :
      ( ( ( member_set_nat @ A @ C5 )
       => ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ B2 ) @ C5 )
          = ( insert_set_nat @ A @ ( inf_inf_set_set_nat @ B2 @ C5 ) ) ) )
      & ( ~ ( member_set_nat @ A @ C5 )
       => ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ B2 ) @ C5 )
          = ( inf_inf_set_set_nat @ B2 @ C5 ) ) ) ) ).

% Int_insert_left
thf(fact_4824_Int__insert__left,axiom,
    ! [A: int,C5: set_int,B2: set_int] :
      ( ( ( member_int @ A @ C5 )
       => ( ( inf_inf_set_int @ ( insert_int @ A @ B2 ) @ C5 )
          = ( insert_int @ A @ ( inf_inf_set_int @ B2 @ C5 ) ) ) )
      & ( ~ ( member_int @ A @ C5 )
       => ( ( inf_inf_set_int @ ( insert_int @ A @ B2 ) @ C5 )
          = ( inf_inf_set_int @ B2 @ C5 ) ) ) ) ).

% Int_insert_left
thf(fact_4825_Int__insert__left,axiom,
    ! [A: nat,C5: set_nat,B2: set_nat] :
      ( ( ( member_nat @ A @ C5 )
       => ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C5 )
          = ( insert_nat @ A @ ( inf_inf_set_nat @ B2 @ C5 ) ) ) )
      & ( ~ ( member_nat @ A @ C5 )
       => ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C5 )
          = ( inf_inf_set_nat @ B2 @ C5 ) ) ) ) ).

% Int_insert_left
thf(fact_4826_Un__Int__distrib2,axiom,
    ! [B2: set_nat,C5: set_nat,A2: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B2 @ C5 ) @ A2 )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ B2 @ A2 ) @ ( sup_sup_set_nat @ C5 @ A2 ) ) ) ).

% Un_Int_distrib2
thf(fact_4827_Int__Un__distrib2,axiom,
    ! [B2: set_nat,C5: set_nat,A2: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B2 @ C5 ) @ A2 )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ B2 @ A2 ) @ ( inf_inf_set_nat @ C5 @ A2 ) ) ) ).

% Int_Un_distrib2
thf(fact_4828_Un__Int__distrib,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C5 ) )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ A2 @ C5 ) ) ) ).

% Un_Int_distrib
thf(fact_4829_Int__Un__distrib,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( ( inf_inf_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C5 ) )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ A2 @ C5 ) ) ) ).

% Int_Un_distrib
thf(fact_4830_Un__Int__crazy,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ B2 @ C5 ) ) @ ( inf_inf_set_nat @ C5 @ A2 ) )
      = ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ B2 @ C5 ) ) @ ( sup_sup_set_nat @ C5 @ A2 ) ) ) ).

% Un_Int_crazy
thf(fact_4831_Diff__Int__distrib2,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ C5 )
      = ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ C5 ) @ ( inf_inf_set_nat @ B2 @ C5 ) ) ) ).

% Diff_Int_distrib2
thf(fact_4832_Diff__Int__distrib,axiom,
    ! [C5: set_nat,A2: set_nat,B2: set_nat] :
      ( ( inf_inf_set_nat @ C5 @ ( minus_minus_set_nat @ A2 @ B2 ) )
      = ( minus_minus_set_nat @ ( inf_inf_set_nat @ C5 @ A2 ) @ ( inf_inf_set_nat @ C5 @ B2 ) ) ) ).

% Diff_Int_distrib
thf(fact_4833_Diff__Diff__Int,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ ( minus_minus_set_nat @ A2 @ B2 ) )
      = ( inf_inf_set_nat @ A2 @ B2 ) ) ).

% Diff_Diff_Int
thf(fact_4834_Diff__Int2,axiom,
    ! [A2: set_nat,C5: set_nat,B2: set_nat] :
      ( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ C5 ) @ ( inf_inf_set_nat @ B2 @ C5 ) )
      = ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ C5 ) @ B2 ) ) ).

% Diff_Int2
thf(fact_4835_Int__Diff,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C5 )
      = ( inf_inf_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C5 ) ) ) ).

% Int_Diff
thf(fact_4836_Int__Diff__disjoint,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( inf_inf_set_real @ ( inf_inf_set_real @ A2 @ B2 ) @ ( minus_minus_set_real @ A2 @ B2 ) )
      = bot_bot_set_real ) ).

% Int_Diff_disjoint
thf(fact_4837_Int__Diff__disjoint,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( inf_inf_set_o @ ( inf_inf_set_o @ A2 @ B2 ) @ ( minus_minus_set_o @ A2 @ B2 ) )
      = bot_bot_set_o ) ).

% Int_Diff_disjoint
thf(fact_4838_Int__Diff__disjoint,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( inf_inf_set_int @ ( inf_inf_set_int @ A2 @ B2 ) @ ( minus_minus_set_int @ A2 @ B2 ) )
      = bot_bot_set_int ) ).

% Int_Diff_disjoint
thf(fact_4839_Int__Diff__disjoint,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ A2 @ B2 ) )
      = bot_bot_set_nat ) ).

% Int_Diff_disjoint
thf(fact_4840_Diff__triv,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( ( inf_inf_set_real @ A2 @ B2 )
        = bot_bot_set_real )
     => ( ( minus_minus_set_real @ A2 @ B2 )
        = A2 ) ) ).

% Diff_triv
thf(fact_4841_Diff__triv,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( ( inf_inf_set_o @ A2 @ B2 )
        = bot_bot_set_o )
     => ( ( minus_minus_set_o @ A2 @ B2 )
        = A2 ) ) ).

% Diff_triv
thf(fact_4842_Diff__triv,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ( inf_inf_set_int @ A2 @ B2 )
        = bot_bot_set_int )
     => ( ( minus_minus_set_int @ A2 @ B2 )
        = A2 ) ) ).

% Diff_triv
thf(fact_4843_Diff__triv,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ( inf_inf_set_nat @ A2 @ B2 )
        = bot_bot_set_nat )
     => ( ( minus_minus_set_nat @ A2 @ B2 )
        = A2 ) ) ).

% Diff_triv
thf(fact_4844_Un__Int__assoc__eq,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C5 )
        = ( inf_inf_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C5 ) ) )
      = ( ord_less_eq_set_nat @ C5 @ A2 ) ) ).

% Un_Int_assoc_eq
thf(fact_4845_Un__Diff__Int,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ A2 @ B2 ) )
      = A2 ) ).

% Un_Diff_Int
thf(fact_4846_Int__Diff__Un,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ A2 @ B2 ) )
      = A2 ) ).

% Int_Diff_Un
thf(fact_4847_Diff__Int,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C5 ) )
      = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ A2 @ C5 ) ) ) ).

% Diff_Int
thf(fact_4848_Diff__Un,axiom,
    ! [A2: set_nat,B2: set_nat,C5: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C5 ) )
      = ( inf_inf_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ A2 @ C5 ) ) ) ).

% Diff_Un
thf(fact_4849_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) )
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B5 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList4 @ Summary3 ) @ Xa2 ) )
                 => ~ ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
thf(fact_4850_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A5 )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B5 )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList4: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList4 @ S2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList4 @ S2 ) @ Xa2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
thf(fact_4851_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) )
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B5 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList4: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList4 @ S2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList4 @ S2 ) @ Xa2 ) )
                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
thf(fact_4852_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
                   => ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList4: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList4 @ Vc2 ) )
                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList4 @ Vc2 ) @ Xa2 ) )
                     => ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) ) ) ) ) )
               => ~ ! [V2: nat,TreeList4: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList4 @ Vd2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList4 @ Vd2 ) @ Xa2 ) )
                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
thf(fact_4853_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ~ Y
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
                 => ( ( Y
                      = ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList4: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList4 @ Vc2 ) )
                   => ( ( Y
                        = ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) ) ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList4 @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [V2: nat,TreeList4: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList4 @ Vd2 ) )
                     => ( ( Y
                          = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList4 @ Vd2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
thf(fact_4854_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
               => ~ ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList4: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList4 @ Vc2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList4 @ Vc2 ) @ Xa2 ) )
                 => ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 )
                      | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) ) ) ) ) )
           => ~ ! [V2: nat,TreeList4: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList4 @ Vd2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList4 @ Vd2 ) @ Xa2 ) )
                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList4 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
thf(fact_4855_max__bot2,axiom,
    ! [X: set_real] :
      ( ( ord_max_set_real @ X @ bot_bot_set_real )
      = X ) ).

% max_bot2
thf(fact_4856_max__bot2,axiom,
    ! [X: set_o] :
      ( ( ord_max_set_o @ X @ bot_bot_set_o )
      = X ) ).

% max_bot2
thf(fact_4857_max__bot2,axiom,
    ! [X: set_nat] :
      ( ( ord_max_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% max_bot2
thf(fact_4858_max__bot2,axiom,
    ! [X: set_int] :
      ( ( ord_max_set_int @ X @ bot_bot_set_int )
      = X ) ).

% max_bot2
thf(fact_4859_max__bot2,axiom,
    ! [X: nat] :
      ( ( ord_max_nat @ X @ bot_bot_nat )
      = X ) ).

% max_bot2
thf(fact_4860_max__bot2,axiom,
    ! [X: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ X @ bot_bo4199563552545308370d_enat )
      = X ) ).

% max_bot2
thf(fact_4861_max__bot,axiom,
    ! [X: set_real] :
      ( ( ord_max_set_real @ bot_bot_set_real @ X )
      = X ) ).

% max_bot
thf(fact_4862_max__bot,axiom,
    ! [X: set_o] :
      ( ( ord_max_set_o @ bot_bot_set_o @ X )
      = X ) ).

% max_bot
thf(fact_4863_max__bot,axiom,
    ! [X: set_nat] :
      ( ( ord_max_set_nat @ bot_bot_set_nat @ X )
      = X ) ).

% max_bot
thf(fact_4864_max__bot,axiom,
    ! [X: set_int] :
      ( ( ord_max_set_int @ bot_bot_set_int @ X )
      = X ) ).

% max_bot
thf(fact_4865_max__bot,axiom,
    ! [X: nat] :
      ( ( ord_max_nat @ bot_bot_nat @ X )
      = X ) ).

% max_bot
thf(fact_4866_max__bot,axiom,
    ! [X: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ bot_bo4199563552545308370d_enat @ X )
      = X ) ).

% max_bot
thf(fact_4867_max_Oabsorb3,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( ord_max_Code_integer @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4868_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_max_real @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4869_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4870_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4871_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4872_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4873_max_Oabsorb3,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_4874_max_Oabsorb4,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_max_Code_integer @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_4875_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_max_real @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_4876_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_4877_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_4878_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_4879_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_4880_max_Oabsorb4,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_4881_inf__bot__left,axiom,
    ! [X: extended_enat] :
      ( ( inf_in1870772243966228564d_enat @ bot_bo4199563552545308370d_enat @ X )
      = bot_bo4199563552545308370d_enat ) ).

% inf_bot_left
thf(fact_4882_inf__bot__left,axiom,
    ! [X: set_real] :
      ( ( inf_inf_set_real @ bot_bot_set_real @ X )
      = bot_bot_set_real ) ).

% inf_bot_left
thf(fact_4883_inf__bot__left,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ bot_bot_set_o @ X )
      = bot_bot_set_o ) ).

% inf_bot_left
thf(fact_4884_inf__bot__left,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
      = bot_bot_set_nat ) ).

% inf_bot_left
thf(fact_4885_inf__bot__left,axiom,
    ! [X: set_int] :
      ( ( inf_inf_set_int @ bot_bot_set_int @ X )
      = bot_bot_set_int ) ).

% inf_bot_left
thf(fact_4886_inf__bot__right,axiom,
    ! [X: extended_enat] :
      ( ( inf_in1870772243966228564d_enat @ X @ bot_bo4199563552545308370d_enat )
      = bot_bo4199563552545308370d_enat ) ).

% inf_bot_right
thf(fact_4887_inf__bot__right,axiom,
    ! [X: set_real] :
      ( ( inf_inf_set_real @ X @ bot_bot_set_real )
      = bot_bot_set_real ) ).

% inf_bot_right
thf(fact_4888_inf__bot__right,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ X @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% inf_bot_right
thf(fact_4889_inf__bot__right,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% inf_bot_right
thf(fact_4890_inf__bot__right,axiom,
    ! [X: set_int] :
      ( ( inf_inf_set_int @ X @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% inf_bot_right
thf(fact_4891_sup__bot__left,axiom,
    ! [X: extended_enat] :
      ( ( sup_su3973961784419623482d_enat @ bot_bo4199563552545308370d_enat @ X )
      = X ) ).

% sup_bot_left
thf(fact_4892_sup__bot__left,axiom,
    ! [X: set_real] :
      ( ( sup_sup_set_real @ bot_bot_set_real @ X )
      = X ) ).

% sup_bot_left
thf(fact_4893_sup__bot__left,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ X )
      = X ) ).

% sup_bot_left
thf(fact_4894_sup__bot__left,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_4895_sup__bot__left,axiom,
    ! [X: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ X )
      = X ) ).

% sup_bot_left
thf(fact_4896_sup__bot__right,axiom,
    ! [X: extended_enat] :
      ( ( sup_su3973961784419623482d_enat @ X @ bot_bo4199563552545308370d_enat )
      = X ) ).

% sup_bot_right
thf(fact_4897_sup__bot__right,axiom,
    ! [X: set_real] :
      ( ( sup_sup_set_real @ X @ bot_bot_set_real )
      = X ) ).

% sup_bot_right
thf(fact_4898_sup__bot__right,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ X @ bot_bot_set_o )
      = X ) ).

% sup_bot_right
thf(fact_4899_sup__bot__right,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% sup_bot_right
thf(fact_4900_sup__bot__right,axiom,
    ! [X: set_int] :
      ( ( sup_sup_set_int @ X @ bot_bot_set_int )
      = X ) ).

% sup_bot_right
thf(fact_4901_bot__eq__sup__iff,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( bot_bo4199563552545308370d_enat
        = ( sup_su3973961784419623482d_enat @ X @ Y ) )
      = ( ( X = bot_bo4199563552545308370d_enat )
        & ( Y = bot_bo4199563552545308370d_enat ) ) ) ).

% bot_eq_sup_iff
thf(fact_4902_bot__eq__sup__iff,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( bot_bot_set_real
        = ( sup_sup_set_real @ X @ Y ) )
      = ( ( X = bot_bot_set_real )
        & ( Y = bot_bot_set_real ) ) ) ).

% bot_eq_sup_iff
thf(fact_4903_bot__eq__sup__iff,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( bot_bot_set_o
        = ( sup_sup_set_o @ X @ Y ) )
      = ( ( X = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% bot_eq_sup_iff
thf(fact_4904_bot__eq__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( bot_bot_set_nat
        = ( sup_sup_set_nat @ X @ Y ) )
      = ( ( X = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% bot_eq_sup_iff
thf(fact_4905_bot__eq__sup__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( bot_bot_set_int
        = ( sup_sup_set_int @ X @ Y ) )
      = ( ( X = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% bot_eq_sup_iff
thf(fact_4906_sup__eq__bot__iff,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ( sup_su3973961784419623482d_enat @ X @ Y )
        = bot_bo4199563552545308370d_enat )
      = ( ( X = bot_bo4199563552545308370d_enat )
        & ( Y = bot_bo4199563552545308370d_enat ) ) ) ).

% sup_eq_bot_iff
thf(fact_4907_sup__eq__bot__iff,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( ( sup_sup_set_real @ X @ Y )
        = bot_bot_set_real )
      = ( ( X = bot_bot_set_real )
        & ( Y = bot_bot_set_real ) ) ) ).

% sup_eq_bot_iff
thf(fact_4908_sup__eq__bot__iff,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( ( sup_sup_set_o @ X @ Y )
        = bot_bot_set_o )
      = ( ( X = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% sup_eq_bot_iff
thf(fact_4909_sup__eq__bot__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( sup_sup_set_nat @ X @ Y )
        = bot_bot_set_nat )
      = ( ( X = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% sup_eq_bot_iff
thf(fact_4910_sup__eq__bot__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ( sup_sup_set_int @ X @ Y )
        = bot_bot_set_int )
      = ( ( X = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% sup_eq_bot_iff
thf(fact_4911_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ( sup_su3973961784419623482d_enat @ A @ B )
        = bot_bo4199563552545308370d_enat )
      = ( ( A = bot_bo4199563552545308370d_enat )
        & ( B = bot_bo4199563552545308370d_enat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_4912_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_real,B: set_real] :
      ( ( ( sup_sup_set_real @ A @ B )
        = bot_bot_set_real )
      = ( ( A = bot_bot_set_real )
        & ( B = bot_bot_set_real ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_4913_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ( sup_sup_set_o @ A @ B )
        = bot_bot_set_o )
      = ( ( A = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_4914_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( sup_sup_set_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_4915_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( sup_sup_set_int @ A @ B )
        = bot_bot_set_int )
      = ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_4916_sup__bot_Oleft__neutral,axiom,
    ! [A: extended_enat] :
      ( ( sup_su3973961784419623482d_enat @ bot_bo4199563552545308370d_enat @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_4917_sup__bot_Oleft__neutral,axiom,
    ! [A: set_real] :
      ( ( sup_sup_set_real @ bot_bot_set_real @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_4918_sup__bot_Oleft__neutral,axiom,
    ! [A: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_4919_sup__bot_Oleft__neutral,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_4920_sup__bot_Oleft__neutral,axiom,
    ! [A: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_4921_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( bot_bo4199563552545308370d_enat
        = ( sup_su3973961784419623482d_enat @ A @ B ) )
      = ( ( A = bot_bo4199563552545308370d_enat )
        & ( B = bot_bo4199563552545308370d_enat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_4922_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_real,B: set_real] :
      ( ( bot_bot_set_real
        = ( sup_sup_set_real @ A @ B ) )
      = ( ( A = bot_bot_set_real )
        & ( B = bot_bot_set_real ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_4923_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( bot_bot_set_o
        = ( sup_sup_set_o @ A @ B ) )
      = ( ( A = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_4924_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_nat
        = ( sup_sup_set_nat @ A @ B ) )
      = ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_4925_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_int
        = ( sup_sup_set_int @ A @ B ) )
      = ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_4926_sup__bot_Oright__neutral,axiom,
    ! [A: extended_enat] :
      ( ( sup_su3973961784419623482d_enat @ A @ bot_bo4199563552545308370d_enat )
      = A ) ).

% sup_bot.right_neutral
thf(fact_4927_sup__bot_Oright__neutral,axiom,
    ! [A: set_real] :
      ( ( sup_sup_set_real @ A @ bot_bot_set_real )
      = A ) ).

% sup_bot.right_neutral
thf(fact_4928_sup__bot_Oright__neutral,axiom,
    ! [A: set_o] :
      ( ( sup_sup_set_o @ A @ bot_bot_set_o )
      = A ) ).

% sup_bot.right_neutral
thf(fact_4929_sup__bot_Oright__neutral,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
      = A ) ).

% sup_bot.right_neutral
thf(fact_4930_sup__bot_Oright__neutral,axiom,
    ! [A: set_int] :
      ( ( sup_sup_set_int @ A @ bot_bot_set_int )
      = A ) ).

% sup_bot.right_neutral
thf(fact_4931_max__less__iff__conj,axiom,
    ! [X: code_integer,Y: code_integer,Z2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( ord_max_Code_integer @ X @ Y ) @ Z2 )
      = ( ( ord_le6747313008572928689nteger @ X @ Z2 )
        & ( ord_le6747313008572928689nteger @ Y @ Z2 ) ) ) ).

% max_less_iff_conj
thf(fact_4932_max__less__iff__conj,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( ( ord_less_real @ ( ord_max_real @ X @ Y ) @ Z2 )
      = ( ( ord_less_real @ X @ Z2 )
        & ( ord_less_real @ Y @ Z2 ) ) ) ).

% max_less_iff_conj
thf(fact_4933_max__less__iff__conj,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ X @ Y ) @ Z2 )
      = ( ( ord_less_rat @ X @ Z2 )
        & ( ord_less_rat @ Y @ Z2 ) ) ) ).

% max_less_iff_conj
thf(fact_4934_max__less__iff__conj,axiom,
    ! [X: num,Y: num,Z2: num] :
      ( ( ord_less_num @ ( ord_max_num @ X @ Y ) @ Z2 )
      = ( ( ord_less_num @ X @ Z2 )
        & ( ord_less_num @ Y @ Z2 ) ) ) ).

% max_less_iff_conj
thf(fact_4935_max__less__iff__conj,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z2 )
      = ( ( ord_less_nat @ X @ Z2 )
        & ( ord_less_nat @ Y @ Z2 ) ) ) ).

% max_less_iff_conj
thf(fact_4936_max__less__iff__conj,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( ord_less_int @ ( ord_max_int @ X @ Y ) @ Z2 )
      = ( ( ord_less_int @ X @ Z2 )
        & ( ord_less_int @ Y @ Z2 ) ) ) ).

% max_less_iff_conj
thf(fact_4937_max__less__iff__conj,axiom,
    ! [X: extended_enat,Y: extended_enat,Z2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ X @ Y ) @ Z2 )
      = ( ( ord_le72135733267957522d_enat @ X @ Z2 )
        & ( ord_le72135733267957522d_enat @ Y @ Z2 ) ) ) ).

% max_less_iff_conj
thf(fact_4938_bot__set__def,axiom,
    ( bot_bot_set_list_nat
    = ( collect_list_nat @ bot_bot_list_nat_o ) ) ).

% bot_set_def
thf(fact_4939_bot__set__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat @ bot_bot_set_nat_o ) ) ).

% bot_set_def
thf(fact_4940_bot__set__def,axiom,
    ( bot_bot_set_real
    = ( collect_real @ bot_bot_real_o ) ) ).

% bot_set_def
thf(fact_4941_bot__set__def,axiom,
    ( bot_bot_set_o
    = ( collect_o @ bot_bot_o_o ) ) ).

% bot_set_def
thf(fact_4942_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_4943_bot__set__def,axiom,
    ( bot_bot_set_int
    = ( collect_int @ bot_bot_int_o ) ) ).

% bot_set_def
thf(fact_4944_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_4945_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
thf(fact_4946_lt__ex,axiom,
    ! [X: real] :
    ? [Y4: real] : ( ord_less_real @ Y4 @ X ) ).

% lt_ex
thf(fact_4947_lt__ex,axiom,
    ! [X: rat] :
    ? [Y4: rat] : ( ord_less_rat @ Y4 @ X ) ).

% lt_ex
thf(fact_4948_lt__ex,axiom,
    ! [X: int] :
    ? [Y4: int] : ( ord_less_int @ Y4 @ X ) ).

% lt_ex
thf(fact_4949_gt__ex,axiom,
    ! [X: real] :
    ? [X_12: real] : ( ord_less_real @ X @ X_12 ) ).

% gt_ex
thf(fact_4950_gt__ex,axiom,
    ! [X: rat] :
    ? [X_12: rat] : ( ord_less_rat @ X @ X_12 ) ).

% gt_ex
thf(fact_4951_gt__ex,axiom,
    ! [X: nat] :
    ? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).

% gt_ex
thf(fact_4952_gt__ex,axiom,
    ! [X: int] :
    ? [X_12: int] : ( ord_less_int @ X @ X_12 ) ).

% gt_ex
thf(fact_4953_dense,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ? [Z4: real] :
          ( ( ord_less_real @ X @ Z4 )
          & ( ord_less_real @ Z4 @ Y ) ) ) ).

% dense
thf(fact_4954_dense,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ? [Z4: rat] :
          ( ( ord_less_rat @ X @ Z4 )
          & ( ord_less_rat @ Z4 @ Y ) ) ) ).

% dense
thf(fact_4955_less__imp__neq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_4956_less__imp__neq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_4957_less__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_4958_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_4959_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_4960_less__imp__neq,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_4961_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order.asym
thf(fact_4962_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order.asym
thf(fact_4963_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order.asym
thf(fact_4964_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_4965_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order.asym
thf(fact_4966_order_Oasym,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ~ ( ord_le72135733267957522d_enat @ B @ A ) ) ).

% order.asym
thf(fact_4967_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_4968_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_4969_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_4970_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_4971_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_4972_ord__eq__less__trans,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( A = B )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_4973_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_4974_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_4975_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_4976_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_4977_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_4978_ord__less__eq__trans,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( B = C )
       => ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_4979_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X5: nat] :
          ( ! [Y5: nat] :
              ( ( ord_less_nat @ Y5 @ X5 )
             => ( P @ Y5 ) )
         => ( P @ X5 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_4980_less__induct,axiom,
    ! [P: extended_enat > $o,A: extended_enat] :
      ( ! [X5: extended_enat] :
          ( ! [Y5: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ Y5 @ X5 )
             => ( P @ Y5 ) )
         => ( P @ X5 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_4981_antisym__conv3,axiom,
    ! [Y: real,X: real] :
      ( ~ ( ord_less_real @ Y @ X )
     => ( ( ~ ( ord_less_real @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_4982_antisym__conv3,axiom,
    ! [Y: rat,X: rat] :
      ( ~ ( ord_less_rat @ Y @ X )
     => ( ( ~ ( ord_less_rat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_4983_antisym__conv3,axiom,
    ! [Y: num,X: num] :
      ( ~ ( ord_less_num @ Y @ X )
     => ( ( ~ ( ord_less_num @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_4984_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_4985_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_int @ Y @ X )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_4986_antisym__conv3,axiom,
    ! [Y: extended_enat,X: extended_enat] :
      ( ~ ( ord_le72135733267957522d_enat @ Y @ X )
     => ( ( ~ ( ord_le72135733267957522d_enat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_4987_linorder__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_4988_linorder__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_4989_linorder__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_num @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_4990_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_4991_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_4992_linorder__cases,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ~ ( ord_le72135733267957522d_enat @ X @ Y )
     => ( ( X != Y )
       => ( ord_le72135733267957522d_enat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_4993_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ~ ( ord_less_real @ A @ B ) ) ).

% dual_order.asym
thf(fact_4994_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ~ ( ord_less_rat @ A @ B ) ) ).

% dual_order.asym
thf(fact_4995_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ~ ( ord_less_num @ A @ B ) ) ).

% dual_order.asym
thf(fact_4996_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_4997_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ~ ( ord_less_int @ A @ B ) ) ).

% dual_order.asym
thf(fact_4998_dual__order_Oasym,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ~ ( ord_le72135733267957522d_enat @ A @ B ) ) ).

% dual_order.asym
thf(fact_4999_dual__order_Oirrefl,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% dual_order.irrefl
thf(fact_5000_dual__order_Oirrefl,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% dual_order.irrefl
thf(fact_5001_dual__order_Oirrefl,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% dual_order.irrefl
thf(fact_5002_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_5003_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_5004_dual__order_Oirrefl,axiom,
    ! [A: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ A @ A ) ).

% dual_order.irrefl
thf(fact_5005_exists__least__iff,axiom,
    ( ( ^ [P3: nat > $o] :
        ? [X7: nat] : ( P3 @ X7 ) )
    = ( ^ [P4: nat > $o] :
        ? [N3: nat] :
          ( ( P4 @ N3 )
          & ! [M4: nat] :
              ( ( ord_less_nat @ M4 @ N3 )
             => ~ ( P4 @ M4 ) ) ) ) ) ).

% exists_least_iff
thf(fact_5006_exists__least__iff,axiom,
    ( ( ^ [P3: extended_enat > $o] :
        ? [X7: extended_enat] : ( P3 @ X7 ) )
    = ( ^ [P4: extended_enat > $o] :
        ? [N3: extended_enat] :
          ( ( P4 @ N3 )
          & ! [M4: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ M4 @ N3 )
             => ~ ( P4 @ M4 ) ) ) ) ) ).

% exists_least_iff
thf(fact_5007_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_5008_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_5009_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_5010_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_5011_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_5012_linorder__less__wlog,axiom,
    ! [P: extended_enat > extended_enat > $o,A: extended_enat,B: extended_enat] :
      ( ! [A5: extended_enat,B5: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: extended_enat] : ( P @ A5 @ A5 )
       => ( ! [A5: extended_enat,B5: extended_enat] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_5013_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_5014_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_5015_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_5016_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_5017_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_5018_order_Ostrict__trans,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_5019_not__less__iff__gr__or__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_real @ X @ Y ) )
      = ( ( ord_less_real @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_5020_not__less__iff__gr__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_rat @ X @ Y ) )
      = ( ( ord_less_rat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_5021_not__less__iff__gr__or__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_num @ X @ Y ) )
      = ( ( ord_less_num @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_5022_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_5023_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ( ord_less_int @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_5024_not__less__iff__gr__or__eq,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ~ ( ord_le72135733267957522d_enat @ X @ Y ) )
      = ( ( ord_le72135733267957522d_enat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_5025_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_5026_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_5027_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_5028_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_5029_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_5030_dual__order_Ostrict__trans,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( ( ord_le72135733267957522d_enat @ C @ B )
       => ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_5031_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_5032_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_5033_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_5034_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_5035_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_5036_order_Ostrict__implies__not__eq,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_5037_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_5038_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_5039_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_5040_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_5041_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_5042_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_5043_linorder__neqE,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ ( ord_less_real @ X @ Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_5044_linorder__neqE,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ ( ord_less_rat @ X @ Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_5045_linorder__neqE,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
     => ( ~ ( ord_less_num @ X @ Y )
       => ( ord_less_num @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_5046_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_5047_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_5048_linorder__neqE,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( X != Y )
     => ( ~ ( ord_le72135733267957522d_enat @ X @ Y )
       => ( ord_le72135733267957522d_enat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_5049_order__less__asym,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_asym
thf(fact_5050_order__less__asym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_asym
thf(fact_5051_order__less__asym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_asym
thf(fact_5052_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_5053_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_asym
thf(fact_5054_order__less__asym,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ~ ( ord_le72135733267957522d_enat @ Y @ X ) ) ).

% order_less_asym
thf(fact_5055_linorder__neq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
      = ( ( ord_less_real @ X @ Y )
        | ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_5056_linorder__neq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
      = ( ( ord_less_rat @ X @ Y )
        | ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_5057_linorder__neq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
      = ( ( ord_less_num @ X @ Y )
        | ( ord_less_num @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_5058_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      = ( ( ord_less_nat @ X @ Y )
        | ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_5059_linorder__neq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      = ( ( ord_less_int @ X @ Y )
        | ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_5060_linorder__neq__iff,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( X != Y )
      = ( ( ord_le72135733267957522d_enat @ X @ Y )
        | ( ord_le72135733267957522d_enat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_5061_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order_less_asym'
thf(fact_5062_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order_less_asym'
thf(fact_5063_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order_less_asym'
thf(fact_5064_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_5065_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order_less_asym'
thf(fact_5066_order__less__asym_H,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ~ ( ord_le72135733267957522d_enat @ B @ A ) ) ).

% order_less_asym'
thf(fact_5067_order__less__trans,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_real @ Y @ Z2 )
       => ( ord_less_real @ X @ Z2 ) ) ) ).

% order_less_trans
thf(fact_5068_order__less__trans,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ Z2 )
       => ( ord_less_rat @ X @ Z2 ) ) ) ).

% order_less_trans
thf(fact_5069_order__less__trans,axiom,
    ! [X: num,Y: num,Z2: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_num @ Y @ Z2 )
       => ( ord_less_num @ X @ Z2 ) ) ) ).

% order_less_trans
thf(fact_5070_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z2 )
       => ( ord_less_nat @ X @ Z2 ) ) ) ).

% order_less_trans
thf(fact_5071_order__less__trans,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z2 )
       => ( ord_less_int @ X @ Z2 ) ) ) ).

% order_less_trans
thf(fact_5072_order__less__trans,axiom,
    ! [X: extended_enat,Y: extended_enat,Z2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ( ( ord_le72135733267957522d_enat @ Y @ Z2 )
       => ( ord_le72135733267957522d_enat @ X @ Z2 ) ) ) ).

% order_less_trans
thf(fact_5073_ord__eq__less__subst,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_5074_ord__eq__less__subst,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_5075_ord__eq__less__subst,axiom,
    ! [A: num,F: real > num,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_5076_ord__eq__less__subst,axiom,
    ! [A: nat,F: real > nat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_5077_ord__eq__less__subst,axiom,
    ! [A: int,F: real > int,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_5078_ord__eq__less__subst,axiom,
    ! [A: extended_enat,F: real > extended_enat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_5079_ord__eq__less__subst,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_5080_ord__eq__less__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_5081_ord__eq__less__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_5082_ord__eq__less__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_5083_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_5084_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_5085_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_5086_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_5087_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_5088_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > extended_enat,C: extended_enat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_5089_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_5090_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_5091_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_5092_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_5093_order__less__irrefl,axiom,
    ! [X: real] :
      ~ ( ord_less_real @ X @ X ) ).

% order_less_irrefl
thf(fact_5094_order__less__irrefl,axiom,
    ! [X: rat] :
      ~ ( ord_less_rat @ X @ X ) ).

% order_less_irrefl
thf(fact_5095_order__less__irrefl,axiom,
    ! [X: num] :
      ~ ( ord_less_num @ X @ X ) ).

% order_less_irrefl
thf(fact_5096_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_5097_order__less__irrefl,axiom,
    ! [X: int] :
      ~ ( ord_less_int @ X @ X ) ).

% order_less_irrefl
thf(fact_5098_order__less__irrefl,axiom,
    ! [X: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ X @ X ) ).

% order_less_irrefl
thf(fact_5099_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,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_5100_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,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_5101_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,Y4: num] :
              ( ( ord_less_num @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_5102_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,Y4: nat] :
              ( ( ord_less_nat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_5103_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,Y4: int] :
              ( ( ord_less_int @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_5104_order__less__subst1,axiom,
    ! [A: real,F: extended_enat > real,B: extended_enat,C: extended_enat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ! [X5: extended_enat,Y4: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_5105_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,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_5106_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,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_5107_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,Y4: num] :
              ( ( ord_less_num @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_5108_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,Y4: nat] :
              ( ( ord_less_nat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_5109_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,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_5110_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,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_5111_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,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_5112_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,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_5113_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,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_5114_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > extended_enat,C: extended_enat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_5115_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,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_5116_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,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_5117_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,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_5118_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,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_5119_order__less__not__sym,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_5120_order__less__not__sym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_5121_order__less__not__sym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_5122_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_5123_order__less__not__sym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_5124_order__less__not__sym,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ~ ( ord_le72135733267957522d_enat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_5125_order__less__imp__triv,axiom,
    ! [X: real,Y: real,P: $o] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_real @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_5126_order__less__imp__triv,axiom,
    ! [X: rat,Y: rat,P: $o] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_5127_order__less__imp__triv,axiom,
    ! [X: num,Y: num,P: $o] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_num @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_5128_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_5129_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P: $o] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_5130_order__less__imp__triv,axiom,
    ! [X: extended_enat,Y: extended_enat,P: $o] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ( ( ord_le72135733267957522d_enat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_5131_linorder__less__linear,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
      | ( X = Y )
      | ( ord_less_real @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_5132_linorder__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
      | ( X = Y )
      | ( ord_less_rat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_5133_linorder__less__linear,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
      | ( X = Y )
      | ( ord_less_num @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_5134_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
      | ( X = Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_5135_linorder__less__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
      | ( X = Y )
      | ( ord_less_int @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_5136_linorder__less__linear,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
      | ( X = Y )
      | ( ord_le72135733267957522d_enat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_5137_order__less__imp__not__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_5138_order__less__imp__not__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_5139_order__less__imp__not__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_5140_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_5141_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_5142_order__less__imp__not__eq,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_5143_order__less__imp__not__eq2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_5144_order__less__imp__not__eq2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_5145_order__less__imp__not__eq2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_5146_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_5147_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_5148_order__less__imp__not__eq2,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_5149_order__less__imp__not__less,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_5150_order__less__imp__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_5151_order__less__imp__not__less,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_5152_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_5153_order__less__imp__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_5154_order__less__imp__not__less,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ~ ( ord_le72135733267957522d_enat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_5155_leD,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ Y @ X )
     => ~ ( ord_less_real @ X @ Y ) ) ).

% leD
thf(fact_5156_leD,axiom,
    ! [Y: extended_enat,X: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Y @ X )
     => ~ ( ord_le72135733267957522d_enat @ X @ Y ) ) ).

% leD
thf(fact_5157_leD,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ~ ( ord_less_set_nat @ X @ Y ) ) ).

% leD
thf(fact_5158_leD,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ~ ( ord_less_rat @ X @ Y ) ) ).

% leD
thf(fact_5159_leD,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ~ ( ord_less_num @ X @ Y ) ) ).

% leD
thf(fact_5160_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_5161_leD,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ~ ( ord_less_int @ X @ Y ) ) ).

% leD
thf(fact_5162_leI,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ord_less_eq_real @ Y @ X ) ) ).

% leI
thf(fact_5163_leI,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ~ ( ord_le72135733267957522d_enat @ X @ Y )
     => ( ord_le2932123472753598470d_enat @ Y @ X ) ) ).

% leI
thf(fact_5164_leI,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ord_less_eq_rat @ Y @ X ) ) ).

% leI
thf(fact_5165_leI,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ord_less_eq_num @ Y @ X ) ) ).

% leI
thf(fact_5166_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_5167_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% leI
thf(fact_5168_nless__le,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( ord_less_real @ A @ B ) )
      = ( ~ ( ord_less_eq_real @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_5169_nless__le,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ~ ( ord_le72135733267957522d_enat @ A @ B ) )
      = ( ~ ( ord_le2932123472753598470d_enat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_5170_nless__le,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ~ ( ord_less_set_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_5171_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( ord_less_rat @ A @ B ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_5172_nless__le,axiom,
    ! [A: num,B: num] :
      ( ( ~ ( ord_less_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_5173_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_5174_nless__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_5175_antisym__conv1,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ( ord_less_eq_real @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_5176_antisym__conv1,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ~ ( ord_le72135733267957522d_enat @ X @ Y )
     => ( ( ord_le2932123472753598470d_enat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_5177_antisym__conv1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ~ ( ord_less_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_5178_antisym__conv1,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_5179_antisym__conv1,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ( ord_less_eq_num @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_5180_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_5181_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_5182_antisym__conv2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ~ ( ord_less_real @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_5183_antisym__conv2,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X @ Y )
     => ( ( ~ ( ord_le72135733267957522d_enat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_5184_antisym__conv2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ~ ( ord_less_set_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_5185_antisym__conv2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ~ ( ord_less_rat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_5186_antisym__conv2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ~ ( ord_less_num @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_5187_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_5188_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_5189_dense__ge,axiom,
    ! [Z2: real,Y: real] :
      ( ! [X5: real] :
          ( ( ord_less_real @ Z2 @ X5 )
         => ( ord_less_eq_real @ Y @ X5 ) )
     => ( ord_less_eq_real @ Y @ Z2 ) ) ).

% dense_ge
thf(fact_5190_dense__ge,axiom,
    ! [Z2: rat,Y: rat] :
      ( ! [X5: rat] :
          ( ( ord_less_rat @ Z2 @ X5 )
         => ( ord_less_eq_rat @ Y @ X5 ) )
     => ( ord_less_eq_rat @ Y @ Z2 ) ) ).

% dense_ge
thf(fact_5191_dense__le,axiom,
    ! [Y: real,Z2: real] :
      ( ! [X5: real] :
          ( ( ord_less_real @ X5 @ Y )
         => ( ord_less_eq_real @ X5 @ Z2 ) )
     => ( ord_less_eq_real @ Y @ Z2 ) ) ).

% dense_le
thf(fact_5192_dense__le,axiom,
    ! [Y: rat,Z2: rat] :
      ( ! [X5: rat] :
          ( ( ord_less_rat @ X5 @ Y )
         => ( ord_less_eq_rat @ X5 @ Z2 ) )
     => ( ord_less_eq_rat @ Y @ Z2 ) ) ).

% dense_le
thf(fact_5193_less__le__not__le,axiom,
    ( ord_less_real
    = ( ^ [X6: real,Y6: real] :
          ( ( ord_less_eq_real @ X6 @ Y6 )
          & ~ ( ord_less_eq_real @ Y6 @ X6 ) ) ) ) ).

% less_le_not_le
thf(fact_5194_less__le__not__le,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [X6: extended_enat,Y6: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ X6 @ Y6 )
          & ~ ( ord_le2932123472753598470d_enat @ Y6 @ X6 ) ) ) ) ).

% less_le_not_le
thf(fact_5195_less__le__not__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X6: set_nat,Y6: set_nat] :
          ( ( ord_less_eq_set_nat @ X6 @ Y6 )
          & ~ ( ord_less_eq_set_nat @ Y6 @ X6 ) ) ) ) ).

% less_le_not_le
thf(fact_5196_less__le__not__le,axiom,
    ( ord_less_rat
    = ( ^ [X6: rat,Y6: rat] :
          ( ( ord_less_eq_rat @ X6 @ Y6 )
          & ~ ( ord_less_eq_rat @ Y6 @ X6 ) ) ) ) ).

% less_le_not_le
thf(fact_5197_less__le__not__le,axiom,
    ( ord_less_num
    = ( ^ [X6: num,Y6: num] :
          ( ( ord_less_eq_num @ X6 @ Y6 )
          & ~ ( ord_less_eq_num @ Y6 @ X6 ) ) ) ) ).

% less_le_not_le
thf(fact_5198_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X6: nat,Y6: nat] :
          ( ( ord_less_eq_nat @ X6 @ Y6 )
          & ~ ( ord_less_eq_nat @ Y6 @ X6 ) ) ) ) ).

% less_le_not_le
thf(fact_5199_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X6: int,Y6: int] :
          ( ( ord_less_eq_int @ X6 @ Y6 )
          & ~ ( ord_less_eq_int @ Y6 @ X6 ) ) ) ) ).

% less_le_not_le
thf(fact_5200_not__le__imp__less,axiom,
    ! [Y: real,X: real] :
      ( ~ ( ord_less_eq_real @ Y @ X )
     => ( ord_less_real @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_5201_not__le__imp__less,axiom,
    ! [Y: extended_enat,X: extended_enat] :
      ( ~ ( ord_le2932123472753598470d_enat @ Y @ X )
     => ( ord_le72135733267957522d_enat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_5202_not__le__imp__less,axiom,
    ! [Y: rat,X: rat] :
      ( ~ ( ord_less_eq_rat @ Y @ X )
     => ( ord_less_rat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_5203_not__le__imp__less,axiom,
    ! [Y: num,X: num] :
      ( ~ ( ord_less_eq_num @ Y @ X )
     => ( ord_less_num @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_5204_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X )
     => ( ord_less_nat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_5205_not__le__imp__less,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_eq_int @ Y @ X )
     => ( ord_less_int @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_5206_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [A3: real,B3: real] :
          ( ( ord_less_real @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_5207_order_Oorder__iff__strict,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_5208_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_set_nat @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_5209_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [A3: rat,B3: rat] :
          ( ( ord_less_rat @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_5210_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [A3: num,B3: num] :
          ( ( ord_less_num @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_5211_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_nat @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_5212_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_int @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_5213_order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [A3: real,B3: real] :
          ( ( ord_less_eq_real @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_5214_order_Ostrict__iff__order,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_5215_order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_5216_order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [A3: rat,B3: rat] :
          ( ( ord_less_eq_rat @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_5217_order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [A3: num,B3: num] :
          ( ( ord_less_eq_num @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_5218_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_5219_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_eq_int @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_5220_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_5221_order_Ostrict__trans1,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_le72135733267957522d_enat @ B @ C )
       => ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_5222_order_Ostrict__trans1,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_set_nat @ B @ C )
       => ( ord_less_set_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_5223_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_5224_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_5225_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_5226_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_5227_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_5228_order_Ostrict__trans2,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ B @ C )
       => ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_5229_order_Ostrict__trans2,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ord_less_set_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_5230_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_5231_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_5232_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_5233_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_5234_order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [A3: real,B3: real] :
          ( ( ord_less_eq_real @ A3 @ B3 )
          & ~ ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_5235_order_Ostrict__iff__not,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ A3 @ B3 )
          & ~ ( ord_le2932123472753598470d_enat @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_5236_order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( ord_less_eq_set_nat @ A3 @ B3 )
          & ~ ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_5237_order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [A3: rat,B3: rat] :
          ( ( ord_less_eq_rat @ A3 @ B3 )
          & ~ ( ord_less_eq_rat @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_5238_order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [A3: num,B3: num] :
          ( ( ord_less_eq_num @ A3 @ B3 )
          & ~ ( ord_less_eq_num @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_5239_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
          & ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_5240_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_eq_int @ A3 @ B3 )
          & ~ ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_5241_dense__ge__bounded,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( ord_less_real @ Z2 @ X )
     => ( ! [W2: real] :
            ( ( ord_less_real @ Z2 @ W2 )
           => ( ( ord_less_real @ W2 @ X )
             => ( ord_less_eq_real @ Y @ W2 ) ) )
       => ( ord_less_eq_real @ Y @ Z2 ) ) ) ).

% dense_ge_bounded
thf(fact_5242_dense__ge__bounded,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ Z2 @ X )
     => ( ! [W2: rat] :
            ( ( ord_less_rat @ Z2 @ W2 )
           => ( ( ord_less_rat @ W2 @ X )
             => ( ord_less_eq_rat @ Y @ W2 ) ) )
       => ( ord_less_eq_rat @ Y @ Z2 ) ) ) ).

% dense_ge_bounded
thf(fact_5243_dense__le__bounded,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ! [W2: real] :
            ( ( ord_less_real @ X @ W2 )
           => ( ( ord_less_real @ W2 @ Y )
             => ( ord_less_eq_real @ W2 @ Z2 ) ) )
       => ( ord_less_eq_real @ Y @ Z2 ) ) ) ).

% dense_le_bounded
thf(fact_5244_dense__le__bounded,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ! [W2: rat] :
            ( ( ord_less_rat @ X @ W2 )
           => ( ( ord_less_rat @ W2 @ Y )
             => ( ord_less_eq_rat @ W2 @ Z2 ) ) )
       => ( ord_less_eq_rat @ Y @ Z2 ) ) ) ).

% dense_le_bounded
thf(fact_5245_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [B3: real,A3: real] :
          ( ( ord_less_real @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_5246_dual__order_Oorder__iff__strict,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B3: extended_enat,A3: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_5247_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B3: set_nat,A3: set_nat] :
          ( ( ord_less_set_nat @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_5248_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [B3: rat,A3: rat] :
          ( ( ord_less_rat @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_5249_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [B3: num,A3: num] :
          ( ( ord_less_num @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_5250_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_less_nat @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_5251_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B3: int,A3: int] :
          ( ( ord_less_int @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_5252_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [B3: real,A3: real] :
          ( ( ord_less_eq_real @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_5253_dual__order_Ostrict__iff__order,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [B3: extended_enat,A3: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_5254_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [B3: set_nat,A3: set_nat] :
          ( ( ord_less_eq_set_nat @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_5255_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [B3: rat,A3: rat] :
          ( ( ord_less_eq_rat @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_5256_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [B3: num,A3: num] :
          ( ( ord_less_eq_num @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_5257_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_5258_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B3: int,A3: int] :
          ( ( ord_less_eq_int @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_5259_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_5260_dual__order_Ostrict__trans1,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A )
     => ( ( ord_le72135733267957522d_enat @ C @ B )
       => ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_5261_dual__order_Ostrict__trans1,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_set_nat @ C @ B )
       => ( ord_less_set_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_5262_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_5263_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_5264_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_5265_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_5266_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_5267_dual__order_Ostrict__trans2,axiom,
    ! [B: extended_enat,A: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( ( ord_le2932123472753598470d_enat @ C @ B )
       => ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_5268_dual__order_Ostrict__trans2,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C @ B )
       => ( ord_less_set_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_5269_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_5270_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_5271_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_5272_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_5273_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [B3: real,A3: real] :
          ( ( ord_less_eq_real @ B3 @ A3 )
          & ~ ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_5274_dual__order_Ostrict__iff__not,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [B3: extended_enat,A3: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ B3 @ A3 )
          & ~ ( ord_le2932123472753598470d_enat @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_5275_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [B3: set_nat,A3: set_nat] :
          ( ( ord_less_eq_set_nat @ B3 @ A3 )
          & ~ ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_5276_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [B3: rat,A3: rat] :
          ( ( ord_less_eq_rat @ B3 @ A3 )
          & ~ ( ord_less_eq_rat @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_5277_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [B3: num,A3: num] :
          ( ( ord_less_eq_num @ B3 @ A3 )
          & ~ ( ord_less_eq_num @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_5278_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A3 )
          & ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_5279_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B3: int,A3: int] :
          ( ( ord_less_eq_int @ B3 @ A3 )
          & ~ ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_5280_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_5281_order_Ostrict__implies__order,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ord_le2932123472753598470d_enat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_5282_order_Ostrict__implies__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_5283_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_5284_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_5285_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_5286_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_5287_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_5288_dual__order_Ostrict__implies__order,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( ord_le2932123472753598470d_enat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_5289_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ord_less_eq_set_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_5290_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_5291_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_5292_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_5293_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_5294_order__le__less,axiom,
    ( ord_less_eq_real
    = ( ^ [X6: real,Y6: real] :
          ( ( ord_less_real @ X6 @ Y6 )
          | ( X6 = Y6 ) ) ) ) ).

% order_le_less
thf(fact_5295_order__le__less,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [X6: extended_enat,Y6: extended_enat] :
          ( ( ord_le72135733267957522d_enat @ X6 @ Y6 )
          | ( X6 = Y6 ) ) ) ) ).

% order_le_less
thf(fact_5296_order__le__less,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X6: set_nat,Y6: set_nat] :
          ( ( ord_less_set_nat @ X6 @ Y6 )
          | ( X6 = Y6 ) ) ) ) ).

% order_le_less
thf(fact_5297_order__le__less,axiom,
    ( ord_less_eq_rat
    = ( ^ [X6: rat,Y6: rat] :
          ( ( ord_less_rat @ X6 @ Y6 )
          | ( X6 = Y6 ) ) ) ) ).

% order_le_less
thf(fact_5298_order__le__less,axiom,
    ( ord_less_eq_num
    = ( ^ [X6: num,Y6: num] :
          ( ( ord_less_num @ X6 @ Y6 )
          | ( X6 = Y6 ) ) ) ) ).

% order_le_less
thf(fact_5299_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X6: nat,Y6: nat] :
          ( ( ord_less_nat @ X6 @ Y6 )
          | ( X6 = Y6 ) ) ) ) ).

% order_le_less
thf(fact_5300_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X6: int,Y6: int] :
          ( ( ord_less_int @ X6 @ Y6 )
          | ( X6 = Y6 ) ) ) ) ).

% order_le_less
thf(fact_5301_order__less__le,axiom,
    ( ord_less_real
    = ( ^ [X6: real,Y6: real] :
          ( ( ord_less_eq_real @ X6 @ Y6 )
          & ( X6 != Y6 ) ) ) ) ).

% order_less_le
thf(fact_5302_order__less__le,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [X6: extended_enat,Y6: extended_enat] :
          ( ( ord_le2932123472753598470d_enat @ X6 @ Y6 )
          & ( X6 != Y6 ) ) ) ) ).

% order_less_le
thf(fact_5303_order__less__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X6: set_nat,Y6: set_nat] :
          ( ( ord_less_eq_set_nat @ X6 @ Y6 )
          & ( X6 != Y6 ) ) ) ) ).

% order_less_le
thf(fact_5304_order__less__le,axiom,
    ( ord_less_rat
    = ( ^ [X6: rat,Y6: rat] :
          ( ( ord_less_eq_rat @ X6 @ Y6 )
          & ( X6 != Y6 ) ) ) ) ).

% order_less_le
thf(fact_5305_order__less__le,axiom,
    ( ord_less_num
    = ( ^ [X6: num,Y6: num] :
          ( ( ord_less_eq_num @ X6 @ Y6 )
          & ( X6 != Y6 ) ) ) ) ).

% order_less_le
thf(fact_5306_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X6: nat,Y6: nat] :
          ( ( ord_less_eq_nat @ X6 @ Y6 )
          & ( X6 != Y6 ) ) ) ) ).

% order_less_le
thf(fact_5307_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X6: int,Y6: int] :
          ( ( ord_less_eq_int @ X6 @ Y6 )
          & ( X6 != Y6 ) ) ) ) ).

% order_less_le
thf(fact_5308_linorder__not__le,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_eq_real @ X @ Y ) )
      = ( ord_less_real @ Y @ X ) ) ).

% linorder_not_le
thf(fact_5309_linorder__not__le,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ~ ( ord_le2932123472753598470d_enat @ X @ Y ) )
      = ( ord_le72135733267957522d_enat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_5310_linorder__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_eq_rat @ X @ Y ) )
      = ( ord_less_rat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_5311_linorder__not__le,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_eq_num @ X @ Y ) )
      = ( ord_less_num @ Y @ X ) ) ).

% linorder_not_le
thf(fact_5312_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_5313_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_eq_int @ X @ Y ) )
      = ( ord_less_int @ Y @ X ) ) ).

% linorder_not_le
thf(fact_5314_linorder__not__less,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_real @ X @ Y ) )
      = ( ord_less_eq_real @ Y @ X ) ) ).

% linorder_not_less
thf(fact_5315_linorder__not__less,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ~ ( ord_le72135733267957522d_enat @ X @ Y ) )
      = ( ord_le2932123472753598470d_enat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_5316_linorder__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_rat @ X @ Y ) )
      = ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_5317_linorder__not__less,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_num @ X @ Y ) )
      = ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_not_less
thf(fact_5318_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_5319_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_not_less
thf(fact_5320_order__less__imp__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_5321_order__less__imp__le,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ( ord_le2932123472753598470d_enat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_5322_order__less__imp__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_set_nat @ X @ Y )
     => ( ord_less_eq_set_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_5323_order__less__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_5324_order__less__imp__le,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ord_less_eq_num @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_5325_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_5326_order__less__imp__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_5327_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_5328_order__le__neq__trans,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( A != B )
       => ( ord_le72135733267957522d_enat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_5329_order__le__neq__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_5330_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_5331_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_5332_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_5333_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_5334_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_5335_order__neq__le__trans,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A != B )
     => ( ( ord_le2932123472753598470d_enat @ A @ B )
       => ( ord_le72135733267957522d_enat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_5336_order__neq__le__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A != B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( ord_less_set_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_5337_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_5338_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_5339_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_5340_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_5341_order__le__less__trans,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_real @ Y @ Z2 )
       => ( ord_less_real @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_5342_order__le__less__trans,axiom,
    ! [X: extended_enat,Y: extended_enat,Z2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X @ Y )
     => ( ( ord_le72135733267957522d_enat @ Y @ Z2 )
       => ( ord_le72135733267957522d_enat @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_5343_order__le__less__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z2: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_set_nat @ Y @ Z2 )
       => ( ord_less_set_nat @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_5344_order__le__less__trans,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ Z2 )
       => ( ord_less_rat @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_5345_order__le__less__trans,axiom,
    ! [X: num,Y: num,Z2: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_num @ Y @ Z2 )
       => ( ord_less_num @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_5346_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z2 )
       => ( ord_less_nat @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_5347_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z2 )
       => ( ord_less_int @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_5348_order__less__le__trans,axiom,
    ! [X: real,Y: real,Z2: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_eq_real @ Y @ Z2 )
       => ( ord_less_real @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_5349_order__less__le__trans,axiom,
    ! [X: extended_enat,Y: extended_enat,Z2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ Y )
     => ( ( ord_le2932123472753598470d_enat @ Y @ Z2 )
       => ( ord_le72135733267957522d_enat @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_5350_order__less__le__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z2: set_nat] :
      ( ( ord_less_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ Z2 )
       => ( ord_less_set_nat @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_5351_order__less__le__trans,axiom,
    ! [X: rat,Y: rat,Z2: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ Z2 )
       => ( ord_less_rat @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_5352_order__less__le__trans,axiom,
    ! [X: num,Y: num,Z2: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ Z2 )
       => ( ord_less_num @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_5353_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z2 )
       => ( ord_less_nat @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_5354_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z2 )
       => ( ord_less_int @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_5355_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,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_5356_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: real > extended_enat,B: real,C: real] :
      ( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_5357_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,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_5358_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: rat > extended_enat,B: rat,C: rat] :
      ( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_5359_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,Y4: num] :
              ( ( ord_less_num @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_5360_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: num > extended_enat,B: num,C: num] :
      ( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_num @ X5 @ Y4 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_5361_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,Y4: nat] :
              ( ( ord_less_nat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_5362_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
      ( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_nat @ X5 @ Y4 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_5363_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,Y4: int] :
              ( ( ord_less_int @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_5364_order__le__less__subst1,axiom,
    ! [A: extended_enat,F: int > extended_enat,B: int,C: int] :
      ( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X5: int,Y4: int] :
              ( ( ord_less_int @ X5 @ Y4 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_5365_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,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_5366_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > extended_enat,C: extended_enat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_5367_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,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_5368_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,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_5369_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,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_5370_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,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_5371_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,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_5372_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > extended_enat,C: extended_enat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_5373_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,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_5374_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,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_5375_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,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_5376_order__less__le__subst1,axiom,
    ! [A: extended_enat,F: rat > extended_enat,B: rat,C: rat] :
      ( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_5377_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,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_5378_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,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_5379_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,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_5380_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,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_5381_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,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_5382_order__less__le__subst1,axiom,
    ! [A: extended_enat,F: num > extended_enat,B: num,C: num] :
      ( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_le2932123472753598470d_enat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_5383_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,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_5384_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,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_5385_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,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_5386_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: real > extended_enat,C: extended_enat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_5387_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,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_5388_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: rat > extended_enat,C: extended_enat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_5389_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,Y4: num] :
              ( ( ord_less_num @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_5390_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: num > extended_enat,C: extended_enat] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_num @ X5 @ Y4 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_5391_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,Y4: nat] :
              ( ( ord_less_nat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_5392_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_nat @ X5 @ Y4 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_5393_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,Y4: int] :
              ( ( ord_less_int @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_5394_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > extended_enat,C: extended_enat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
       => ( ! [X5: int,Y4: int] :
              ( ( ord_less_int @ X5 @ Y4 )
             => ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_5395_linorder__le__less__linear,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
      | ( ord_less_real @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_5396_linorder__le__less__linear,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X @ Y )
      | ( ord_le72135733267957522d_enat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_5397_linorder__le__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
      | ( ord_less_rat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_5398_linorder__le__less__linear,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
      | ( ord_less_num @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_5399_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_5400_linorder__le__less__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_int @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_5401_order__le__imp__less__or__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_real @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_5402_order__le__imp__less__or__eq,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X @ Y )
     => ( ( ord_le72135733267957522d_enat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_5403_order__le__imp__less__or__eq,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_set_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_5404_order__le__imp__less__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_rat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_5405_order__le__imp__less__or__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_num @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_5406_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_5407_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_5408_bot_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
     => ( A = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
thf(fact_5409_bot_Oextremum__uniqueI,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
     => ( A = bot_bot_set_o ) ) ).

% bot.extremum_uniqueI
thf(fact_5410_bot_Oextremum__uniqueI,axiom,
    ! [A: set_int] :
      ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
     => ( A = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
thf(fact_5411_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_5412_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_5413_bot_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
      = ( A = bot_bot_set_real ) ) ).

% bot.extremum_unique
thf(fact_5414_bot_Oextremum__unique,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
      = ( A = bot_bot_set_o ) ) ).

% bot.extremum_unique
thf(fact_5415_bot_Oextremum__unique,axiom,
    ! [A: set_int] :
      ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
      = ( A = bot_bot_set_int ) ) ).

% bot.extremum_unique
thf(fact_5416_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
      = ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
thf(fact_5417_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_5418_bot_Oextremum,axiom,
    ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).

% bot.extremum
thf(fact_5419_bot_Oextremum,axiom,
    ! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).

% bot.extremum
thf(fact_5420_bot_Oextremum,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).

% bot.extremum
thf(fact_5421_bot_Oextremum,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% bot.extremum
thf(fact_5422_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_5423_bot_Oextremum__strict,axiom,
    ! [A: set_real] :
      ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).

% bot.extremum_strict
thf(fact_5424_bot_Oextremum__strict,axiom,
    ! [A: set_o] :
      ~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).

% bot.extremum_strict
thf(fact_5425_bot_Oextremum__strict,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_5426_bot_Oextremum__strict,axiom,
    ! [A: set_int] :
      ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).

% bot.extremum_strict
thf(fact_5427_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_5428_bot_Oextremum__strict,axiom,
    ! [A: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ A @ bot_bo4199563552545308370d_enat ) ).

% bot.extremum_strict
thf(fact_5429_bot_Onot__eq__extremum,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
      = ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).

% bot.not_eq_extremum
thf(fact_5430_bot_Onot__eq__extremum,axiom,
    ! [A: set_o] :
      ( ( A != bot_bot_set_o )
      = ( ord_less_set_o @ bot_bot_set_o @ A ) ) ).

% bot.not_eq_extremum
thf(fact_5431_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
      = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_5432_bot_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
      = ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).

% bot.not_eq_extremum
thf(fact_5433_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_5434_bot_Onot__eq__extremum,axiom,
    ! [A: extended_enat] :
      ( ( A != bot_bo4199563552545308370d_enat )
      = ( ord_le72135733267957522d_enat @ bot_bo4199563552545308370d_enat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_5435_less__infI1,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ X )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_5436_less__infI1,axiom,
    ! [A: real,X: real,B: real] :
      ( ( ord_less_real @ A @ X )
     => ( ord_less_real @ ( inf_inf_real @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_5437_less__infI1,axiom,
    ! [A: rat,X: rat,B: rat] :
      ( ( ord_less_rat @ A @ X )
     => ( ord_less_rat @ ( inf_inf_rat @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_5438_less__infI1,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( ( ord_less_nat @ A @ X )
     => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_5439_less__infI1,axiom,
    ! [A: int,X: int,B: int] :
      ( ( ord_less_int @ A @ X )
     => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_5440_less__infI1,axiom,
    ! [A: extended_enat,X: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ X )
     => ( ord_le72135733267957522d_enat @ ( inf_in1870772243966228564d_enat @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_5441_less__infI2,axiom,
    ! [B: set_nat,X: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ X )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_5442_less__infI2,axiom,
    ! [B: real,X: real,A: real] :
      ( ( ord_less_real @ B @ X )
     => ( ord_less_real @ ( inf_inf_real @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_5443_less__infI2,axiom,
    ! [B: rat,X: rat,A: rat] :
      ( ( ord_less_rat @ B @ X )
     => ( ord_less_rat @ ( inf_inf_rat @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_5444_less__infI2,axiom,
    ! [B: nat,X: nat,A: nat] :
      ( ( ord_less_nat @ B @ X )
     => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_5445_less__infI2,axiom,
    ! [B: int,X: int,A: int] :
      ( ( ord_less_int @ B @ X )
     => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_5446_less__infI2,axiom,
    ! [B: extended_enat,X: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ X )
     => ( ord_le72135733267957522d_enat @ ( inf_in1870772243966228564d_enat @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_5447_inf_Oabsorb3,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ( inf_inf_set_nat @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_5448_inf_Oabsorb3,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( inf_inf_real @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_5449_inf_Oabsorb3,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( inf_inf_rat @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_5450_inf_Oabsorb3,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( inf_inf_nat @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_5451_inf_Oabsorb3,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( inf_inf_int @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_5452_inf_Oabsorb3,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( inf_in1870772243966228564d_enat @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_5453_inf_Oabsorb4,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ( inf_inf_set_nat @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_5454_inf_Oabsorb4,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( inf_inf_real @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_5455_inf_Oabsorb4,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( inf_inf_rat @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_5456_inf_Oabsorb4,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( inf_inf_nat @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_5457_inf_Oabsorb4,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( inf_inf_int @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_5458_inf_Oabsorb4,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( ( inf_in1870772243966228564d_enat @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_5459_inf_Ostrict__boundedE,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
     => ~ ( ( ord_less_set_nat @ A @ B )
         => ~ ( ord_less_set_nat @ A @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_5460_inf_Ostrict__boundedE,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( inf_inf_real @ B @ C ) )
     => ~ ( ( ord_less_real @ A @ B )
         => ~ ( ord_less_real @ A @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_5461_inf_Ostrict__boundedE,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( inf_inf_rat @ B @ C ) )
     => ~ ( ( ord_less_rat @ A @ B )
         => ~ ( ord_less_rat @ A @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_5462_inf_Ostrict__boundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ ( inf_inf_nat @ B @ C ) )
     => ~ ( ( ord_less_nat @ A @ B )
         => ~ ( ord_less_nat @ A @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_5463_inf_Ostrict__boundedE,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ ( inf_inf_int @ B @ C ) )
     => ~ ( ( ord_less_int @ A @ B )
         => ~ ( ord_less_int @ A @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_5464_inf_Ostrict__boundedE,axiom,
    ! [A: extended_enat,B: extended_enat,C: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ ( inf_in1870772243966228564d_enat @ B @ C ) )
     => ~ ( ( ord_le72135733267957522d_enat @ A @ B )
         => ~ ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).

% inf.strict_boundedE
thf(fact_5465_inf_Ostrict__order__iff,axiom,
    ( ord_less_set_nat
    = ( ^ [A3: set_nat,B3: set_nat] :
          ( ( A3
            = ( inf_inf_set_nat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_5466_inf_Ostrict__order__iff,axiom,
    ( ord_less_real
    = ( ^ [A3: real,B3: real] :
          ( ( A3
            = ( inf_inf_real @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_5467_inf_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [A3: rat,B3: rat] :
          ( ( A3
            = ( inf_inf_rat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_5468_inf_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( A3
            = ( inf_inf_nat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_5469_inf_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B3: int] :
          ( ( A3
            = ( inf_inf_int @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_5470_inf_Ostrict__order__iff,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [A3: extended_enat,B3: extended_enat] :
          ( ( A3
            = ( inf_in1870772243966228564d_enat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_5471_inf_Ostrict__coboundedI1,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ C )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_5472_inf_Ostrict__coboundedI1,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ A @ C )
     => ( ord_less_real @ ( inf_inf_real @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_5473_inf_Ostrict__coboundedI1,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ A @ C )
     => ( ord_less_rat @ ( inf_inf_rat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_5474_inf_Ostrict__coboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ A @ C )
     => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_5475_inf_Ostrict__coboundedI1,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ A @ C )
     => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_5476_inf_Ostrict__coboundedI1,axiom,
    ! [A: extended_enat,C: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ C )
     => ( ord_le72135733267957522d_enat @ ( inf_in1870772243966228564d_enat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI1
thf(fact_5477_inf_Ostrict__coboundedI2,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ C )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_5478_inf_Ostrict__coboundedI2,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_real @ B @ C )
     => ( ord_less_real @ ( inf_inf_real @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_5479_inf_Ostrict__coboundedI2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ B @ C )
     => ( ord_less_rat @ ( inf_inf_rat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_5480_inf_Ostrict__coboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ B @ C )
     => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_5481_inf_Ostrict__coboundedI2,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_int @ B @ C )
     => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_5482_inf_Ostrict__coboundedI2,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ C )
     => ( ord_le72135733267957522d_enat @ ( inf_in1870772243966228564d_enat @ A @ B ) @ C ) ) ).

% inf.strict_coboundedI2
thf(fact_5483_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ C @ B )
     => ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_5484_sup_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ B )
     => ( ord_less_real @ C @ ( sup_sup_real @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_5485_sup_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ B )
     => ( ord_less_rat @ C @ ( sup_sup_rat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_5486_sup_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ C @ B )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_5487_sup_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_int @ C @ B )
     => ( ord_less_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_5488_sup_Ostrict__coboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ C @ B )
     => ( ord_le72135733267957522d_enat @ C @ ( sup_su3973961784419623482d_enat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_5489_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ C @ A )
     => ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_5490_sup_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ A )
     => ( ord_less_real @ C @ ( sup_sup_real @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_5491_sup_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ A )
     => ( ord_less_rat @ C @ ( sup_sup_rat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_5492_sup_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ C @ A )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_5493_sup_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ A )
     => ( ord_less_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_5494_sup_Ostrict__coboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ C @ A )
     => ( ord_le72135733267957522d_enat @ C @ ( sup_su3973961784419623482d_enat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_5495_sup_Ostrict__order__iff,axiom,
    ( ord_less_set_nat
    = ( ^ [B3: set_nat,A3: set_nat] :
          ( ( A3
            = ( sup_sup_set_nat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_5496_sup_Ostrict__order__iff,axiom,
    ( ord_less_real
    = ( ^ [B3: real,A3: real] :
          ( ( A3
            = ( sup_sup_real @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_5497_sup_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B3: rat,A3: rat] :
          ( ( A3
            = ( sup_sup_rat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_5498_sup_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( A3
            = ( sup_sup_nat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_5499_sup_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B3: int,A3: int] :
          ( ( A3
            = ( sup_sup_int @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_5500_sup_Ostrict__order__iff,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [B3: extended_enat,A3: extended_enat] :
          ( ( A3
            = ( sup_su3973961784419623482d_enat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_5501_sup_Ostrict__boundedE,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_set_nat @ B @ A )
         => ~ ( ord_less_set_nat @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_5502_sup_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_real @ ( sup_sup_real @ B @ C ) @ A )
     => ~ ( ( ord_less_real @ B @ A )
         => ~ ( ord_less_real @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_5503_sup_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( sup_sup_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_rat @ B @ A )
         => ~ ( ord_less_rat @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_5504_sup_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_nat @ B @ A )
         => ~ ( ord_less_nat @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_5505_sup_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_int @ ( sup_sup_int @ B @ C ) @ A )
     => ~ ( ( ord_less_int @ B @ A )
         => ~ ( ord_less_int @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_5506_sup_Ostrict__boundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( sup_su3973961784419623482d_enat @ B @ C ) @ A )
     => ~ ( ( ord_le72135733267957522d_enat @ B @ A )
         => ~ ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_5507_sup_Oabsorb4,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ( sup_sup_set_nat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_5508_sup_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( sup_sup_real @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_5509_sup_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( sup_sup_rat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_5510_sup_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( sup_sup_nat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_5511_sup_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( sup_sup_int @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_5512_sup_Oabsorb4,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( sup_su3973961784419623482d_enat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_5513_sup_Oabsorb3,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ( sup_sup_set_nat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_5514_sup_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( sup_sup_real @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_5515_sup_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( sup_sup_rat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_5516_sup_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( sup_sup_nat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_5517_sup_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( sup_sup_int @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_5518_sup_Oabsorb3,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( ( sup_su3973961784419623482d_enat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_5519_less__supI2,axiom,
    ! [X: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ X @ B )
     => ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% less_supI2
thf(fact_5520_less__supI2,axiom,
    ! [X: real,B: real,A: real] :
      ( ( ord_less_real @ X @ B )
     => ( ord_less_real @ X @ ( sup_sup_real @ A @ B ) ) ) ).

% less_supI2
thf(fact_5521_less__supI2,axiom,
    ! [X: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ X @ B )
     => ( ord_less_rat @ X @ ( sup_sup_rat @ A @ B ) ) ) ).

% less_supI2
thf(fact_5522_less__supI2,axiom,
    ! [X: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ X @ B )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).

% less_supI2
thf(fact_5523_less__supI2,axiom,
    ! [X: int,B: int,A: int] :
      ( ( ord_less_int @ X @ B )
     => ( ord_less_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).

% less_supI2
thf(fact_5524_less__supI2,axiom,
    ! [X: extended_enat,B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ B )
     => ( ord_le72135733267957522d_enat @ X @ ( sup_su3973961784419623482d_enat @ A @ B ) ) ) ).

% less_supI2
thf(fact_5525_less__supI1,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ X @ A )
     => ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% less_supI1
thf(fact_5526_less__supI1,axiom,
    ! [X: real,A: real,B: real] :
      ( ( ord_less_real @ X @ A )
     => ( ord_less_real @ X @ ( sup_sup_real @ A @ B ) ) ) ).

% less_supI1
thf(fact_5527_less__supI1,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ X @ A )
     => ( ord_less_rat @ X @ ( sup_sup_rat @ A @ B ) ) ) ).

% less_supI1
thf(fact_5528_less__supI1,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ X @ A )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).

% less_supI1
thf(fact_5529_less__supI1,axiom,
    ! [X: int,A: int,B: int] :
      ( ( ord_less_int @ X @ A )
     => ( ord_less_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).

% less_supI1
thf(fact_5530_less__supI1,axiom,
    ! [X: extended_enat,A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ X @ A )
     => ( ord_le72135733267957522d_enat @ X @ ( sup_su3973961784419623482d_enat @ A @ B ) ) ) ).

% less_supI1
thf(fact_5531_max_Ostrict__coboundedI2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ B )
     => ( ord_le6747313008572928689nteger @ C @ ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_5532_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ B )
     => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_5533_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ B )
     => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_5534_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( ( ord_less_num @ C @ B )
     => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_5535_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ C @ B )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_5536_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_int @ C @ B )
     => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_5537_max_Ostrict__coboundedI2,axiom,
    ! [C: extended_enat,B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ C @ B )
     => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_5538_max_Ostrict__coboundedI1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ A )
     => ( ord_le6747313008572928689nteger @ C @ ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_5539_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ A )
     => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_5540_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ A )
     => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_5541_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( ( ord_less_num @ C @ A )
     => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_5542_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ C @ A )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_5543_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ A )
     => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_5544_max_Ostrict__coboundedI1,axiom,
    ! [C: extended_enat,A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ C @ A )
     => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_5545_max_Ostrict__order__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [B3: code_integer,A3: code_integer] :
          ( ( A3
            = ( ord_max_Code_integer @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% max.strict_order_iff
thf(fact_5546_max_Ostrict__order__iff,axiom,
    ( ord_less_real
    = ( ^ [B3: real,A3: real] :
          ( ( A3
            = ( ord_max_real @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% max.strict_order_iff
thf(fact_5547_max_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B3: rat,A3: rat] :
          ( ( A3
            = ( ord_max_rat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% max.strict_order_iff
thf(fact_5548_max_Ostrict__order__iff,axiom,
    ( ord_less_num
    = ( ^ [B3: num,A3: num] :
          ( ( A3
            = ( ord_max_num @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% max.strict_order_iff
thf(fact_5549_max_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( A3
            = ( ord_max_nat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% max.strict_order_iff
thf(fact_5550_max_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B3: int,A3: int] :
          ( ( A3
            = ( ord_max_int @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% max.strict_order_iff
thf(fact_5551_max_Ostrict__order__iff,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [B3: extended_enat,A3: extended_enat] :
          ( ( A3
            = ( ord_ma741700101516333627d_enat @ A3 @ B3 ) )
          & ( A3 != B3 ) ) ) ) ).

% max.strict_order_iff
thf(fact_5552_max_Ostrict__boundedE,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( ord_max_Code_integer @ B @ C ) @ A )
     => ~ ( ( ord_le6747313008572928689nteger @ B @ A )
         => ~ ( ord_le6747313008572928689nteger @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_5553_max_Ostrict__boundedE,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_real @ ( ord_max_real @ B @ C ) @ A )
     => ~ ( ( ord_less_real @ B @ A )
         => ~ ( ord_less_real @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_5554_max_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_rat @ B @ A )
         => ~ ( ord_less_rat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_5555_max_Ostrict__boundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_num @ ( ord_max_num @ B @ C ) @ A )
     => ~ ( ( ord_less_num @ B @ A )
         => ~ ( ord_less_num @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_5556_max_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_nat @ B @ A )
         => ~ ( ord_less_nat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_5557_max_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_int @ ( ord_max_int @ B @ C ) @ A )
     => ~ ( ( ord_less_int @ B @ A )
         => ~ ( ord_less_int @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_5558_max_Ostrict__boundedE,axiom,
    ! [B: extended_enat,C: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
     => ~ ( ( ord_le72135733267957522d_enat @ B @ A )
         => ~ ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).

% max.strict_boundedE
thf(fact_5559_less__max__iff__disj,axiom,
    ! [Z2: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z2 @ ( ord_max_Code_integer @ X @ Y ) )
      = ( ( ord_le6747313008572928689nteger @ Z2 @ X )
        | ( ord_le6747313008572928689nteger @ Z2 @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_5560_less__max__iff__disj,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( ord_less_real @ Z2 @ ( ord_max_real @ X @ Y ) )
      = ( ( ord_less_real @ Z2 @ X )
        | ( ord_less_real @ Z2 @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_5561_less__max__iff__disj,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ Z2 @ ( ord_max_rat @ X @ Y ) )
      = ( ( ord_less_rat @ Z2 @ X )
        | ( ord_less_rat @ Z2 @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_5562_less__max__iff__disj,axiom,
    ! [Z2: num,X: num,Y: num] :
      ( ( ord_less_num @ Z2 @ ( ord_max_num @ X @ Y ) )
      = ( ( ord_less_num @ Z2 @ X )
        | ( ord_less_num @ Z2 @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_5563_less__max__iff__disj,axiom,
    ! [Z2: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ Z2 @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Z2 @ X )
        | ( ord_less_nat @ Z2 @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_5564_less__max__iff__disj,axiom,
    ! [Z2: int,X: int,Y: int] :
      ( ( ord_less_int @ Z2 @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_int @ Z2 @ X )
        | ( ord_less_int @ Z2 @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_5565_less__max__iff__disj,axiom,
    ! [Z2: extended_enat,X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z2 @ ( ord_ma741700101516333627d_enat @ X @ Y ) )
      = ( ( ord_le72135733267957522d_enat @ Z2 @ X )
        | ( ord_le72135733267957522d_enat @ Z2 @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_5566_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_real] :
      ( ( inf_inf_set_real @ bot_bot_set_real @ X )
      = bot_bot_set_real ) ).

% boolean_algebra.conj_zero_left
thf(fact_5567_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ bot_bot_set_o @ X )
      = bot_bot_set_o ) ).

% boolean_algebra.conj_zero_left
thf(fact_5568_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_zero_left
thf(fact_5569_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_int] :
      ( ( inf_inf_set_int @ bot_bot_set_int @ X )
      = bot_bot_set_int ) ).

% boolean_algebra.conj_zero_left
thf(fact_5570_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_real] :
      ( ( inf_inf_set_real @ X @ bot_bot_set_real )
      = bot_bot_set_real ) ).

% boolean_algebra.conj_zero_right
thf(fact_5571_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ X @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% boolean_algebra.conj_zero_right
thf(fact_5572_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_zero_right
thf(fact_5573_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_int] :
      ( ( inf_inf_set_int @ X @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% boolean_algebra.conj_zero_right
thf(fact_5574_Bolzano,axiom,
    ! [A: real,B: real,P: real > real > $o] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [A5: real,B5: real,C2: real] :
            ( ( P @ A5 @ B5 )
           => ( ( P @ B5 @ C2 )
             => ( ( ord_less_eq_real @ A5 @ B5 )
               => ( ( ord_less_eq_real @ B5 @ C2 )
                 => ( P @ A5 @ C2 ) ) ) ) )
       => ( ! [X5: real] :
              ( ( ord_less_eq_real @ A @ X5 )
             => ( ( ord_less_eq_real @ X5 @ B )
               => ? [D5: real] :
                    ( ( ord_less_real @ zero_zero_real @ D5 )
                    & ! [A5: real,B5: real] :
                        ( ( ( ord_less_eq_real @ A5 @ X5 )
                          & ( ord_less_eq_real @ X5 @ B5 )
                          & ( ord_less_real @ ( minus_minus_real @ B5 @ A5 ) @ D5 ) )
                       => ( P @ A5 @ B5 ) ) ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Bolzano
thf(fact_5575_set__union,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( union_VEBT_VEBT @ Xs2 @ Ys2 ) )
      = ( sup_su6272177626956685416T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ ( set_VEBT_VEBT2 @ Ys2 ) ) ) ).

% set_union
thf(fact_5576_set__union,axiom,
    ! [Xs2: list_int,Ys2: list_int] :
      ( ( set_int2 @ ( union_int @ Xs2 @ Ys2 ) )
      = ( sup_sup_set_int @ ( set_int2 @ Xs2 ) @ ( set_int2 @ Ys2 ) ) ) ).

% set_union
thf(fact_5577_set__union,axiom,
    ! [Xs2: list_nat,Ys2: list_nat] :
      ( ( set_nat2 @ ( union_nat @ Xs2 @ Ys2 ) )
      = ( sup_sup_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ Ys2 ) ) ) ).

% set_union
thf(fact_5578_divides__aux__eq,axiom,
    ! [Q2: nat,R2: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
      = ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_5579_divides__aux__eq,axiom,
    ! [Q2: int,R2: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q2 @ R2 ) )
      = ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_5580_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
thf(fact_5581_inf__set__def,axiom,
    ( inf_inf_set_complex
    = ( ^ [A6: set_complex,B7: set_complex] :
          ( collect_complex
          @ ( inf_inf_complex_o
            @ ^ [X6: complex] : ( member_complex @ X6 @ A6 )
            @ ^ [X6: complex] : ( member_complex @ X6 @ B7 ) ) ) ) ) ).

% inf_set_def
thf(fact_5582_inf__set__def,axiom,
    ( inf_inf_set_real
    = ( ^ [A6: set_real,B7: set_real] :
          ( collect_real
          @ ( inf_inf_real_o
            @ ^ [X6: real] : ( member_real @ X6 @ A6 )
            @ ^ [X6: real] : ( member_real @ X6 @ B7 ) ) ) ) ) ).

% inf_set_def
thf(fact_5583_inf__set__def,axiom,
    ( inf_inf_set_list_nat
    = ( ^ [A6: set_list_nat,B7: set_list_nat] :
          ( collect_list_nat
          @ ( inf_inf_list_nat_o
            @ ^ [X6: list_nat] : ( member_list_nat @ X6 @ A6 )
            @ ^ [X6: list_nat] : ( member_list_nat @ X6 @ B7 ) ) ) ) ) ).

% inf_set_def
thf(fact_5584_inf__set__def,axiom,
    ( inf_inf_set_set_nat
    = ( ^ [A6: set_set_nat,B7: set_set_nat] :
          ( collect_set_nat
          @ ( inf_inf_set_nat_o
            @ ^ [X6: set_nat] : ( member_set_nat @ X6 @ A6 )
            @ ^ [X6: set_nat] : ( member_set_nat @ X6 @ B7 ) ) ) ) ) ).

% inf_set_def
thf(fact_5585_inf__set__def,axiom,
    ( inf_inf_set_int
    = ( ^ [A6: set_int,B7: set_int] :
          ( collect_int
          @ ( inf_inf_int_o
            @ ^ [X6: int] : ( member_int @ X6 @ A6 )
            @ ^ [X6: int] : ( member_int @ X6 @ B7 ) ) ) ) ) ).

% inf_set_def
thf(fact_5586_inf__set__def,axiom,
    ( inf_inf_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( collect_nat
          @ ( inf_inf_nat_o
            @ ^ [X6: nat] : ( member_nat @ X6 @ A6 )
            @ ^ [X6: nat] : ( member_nat @ X6 @ B7 ) ) ) ) ) ).

% inf_set_def
thf(fact_5587_sup__set__def,axiom,
    ( sup_sup_set_complex
    = ( ^ [A6: set_complex,B7: set_complex] :
          ( collect_complex
          @ ( sup_sup_complex_o
            @ ^ [X6: complex] : ( member_complex @ X6 @ A6 )
            @ ^ [X6: complex] : ( member_complex @ X6 @ B7 ) ) ) ) ) ).

% sup_set_def
thf(fact_5588_sup__set__def,axiom,
    ( sup_sup_set_real
    = ( ^ [A6: set_real,B7: set_real] :
          ( collect_real
          @ ( sup_sup_real_o
            @ ^ [X6: real] : ( member_real @ X6 @ A6 )
            @ ^ [X6: real] : ( member_real @ X6 @ B7 ) ) ) ) ) ).

% sup_set_def
thf(fact_5589_sup__set__def,axiom,
    ( sup_sup_set_list_nat
    = ( ^ [A6: set_list_nat,B7: set_list_nat] :
          ( collect_list_nat
          @ ( sup_sup_list_nat_o
            @ ^ [X6: list_nat] : ( member_list_nat @ X6 @ A6 )
            @ ^ [X6: list_nat] : ( member_list_nat @ X6 @ B7 ) ) ) ) ) ).

% sup_set_def
thf(fact_5590_sup__set__def,axiom,
    ( sup_sup_set_set_nat
    = ( ^ [A6: set_set_nat,B7: set_set_nat] :
          ( collect_set_nat
          @ ( sup_sup_set_nat_o
            @ ^ [X6: set_nat] : ( member_set_nat @ X6 @ A6 )
            @ ^ [X6: set_nat] : ( member_set_nat @ X6 @ B7 ) ) ) ) ) ).

% sup_set_def
thf(fact_5591_sup__set__def,axiom,
    ( sup_sup_set_int
    = ( ^ [A6: set_int,B7: set_int] :
          ( collect_int
          @ ( sup_sup_int_o
            @ ^ [X6: int] : ( member_int @ X6 @ A6 )
            @ ^ [X6: int] : ( member_int @ X6 @ B7 ) ) ) ) ) ).

% sup_set_def
thf(fact_5592_sup__set__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( collect_nat
          @ ( sup_sup_nat_o
            @ ^ [X6: nat] : ( member_nat @ X6 @ A6 )
            @ ^ [X6: nat] : ( member_nat @ X6 @ B7 ) ) ) ) ) ).

% sup_set_def
thf(fact_5593_sup__enat__def,axiom,
    sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).

% sup_enat_def
thf(fact_5594_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_real] :
      ( ( sup_sup_set_real @ X @ bot_bot_set_real )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_5595_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ X @ bot_bot_set_o )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_5596_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_5597_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_int] :
      ( ( sup_sup_set_int @ X @ bot_bot_set_int )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_5598_diff__shunt__var,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( ( minus_minus_set_real @ X @ Y )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_5599_diff__shunt__var,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( ( minus_minus_set_o @ X @ Y )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_5600_diff__shunt__var,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ( minus_minus_set_int @ X @ Y )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_5601_diff__shunt__var,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( minus_minus_set_nat @ X @ Y )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_5602_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R2: int] :
      ( ( ord_less_eq_int @ B @ zero_zero_int )
     => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q2 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_5603_product__nth,axiom,
    ! [N: nat,Xs2: list_num,Ys2: list_num] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_num @ Xs2 ) @ ( size_size_list_num @ Ys2 ) ) )
     => ( ( nth_Pr6456567536196504476um_num @ ( product_num_num @ Xs2 @ Ys2 ) @ N )
        = ( product_Pair_num_num @ ( nth_num @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_num @ Ys2 ) ) ) @ ( nth_num @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_num @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_5604_product__nth,axiom,
    ! [N: nat,Xs2: list_Code_integer,Ys2: list_o] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s3445333598471063425nteger @ Xs2 ) @ ( size_size_list_o @ Ys2 ) ) )
     => ( ( nth_Pr8522763379788166057eger_o @ ( produc3607205314601156340eger_o @ Xs2 @ Ys2 ) @ N )
        = ( produc6677183202524767010eger_o @ ( nth_Code_integer @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) @ ( nth_o @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_5605_product__nth,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
     => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys2 ) @ N )
        = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_5606_product__nth,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,Ys2: list_o] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys2 ) ) )
     => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys2 ) @ N )
        = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) @ ( nth_o @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_5607_product__nth,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,Ys2: list_nat] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) ) )
     => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys2 ) @ N )
        = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) @ ( nth_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_5608_product__nth,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,Ys2: list_int] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys2 ) ) )
     => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys2 ) @ N )
        = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) @ ( nth_int @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_5609_product__nth,axiom,
    ! [N: nat,Xs2: list_o,Ys2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
     => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys2 ) @ N )
        = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_5610_product__nth,axiom,
    ! [N: nat,Xs2: list_o,Ys2: list_o] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys2 ) ) )
     => ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs2 @ Ys2 ) @ N )
        = ( product_Pair_o_o @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) @ ( nth_o @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_5611_product__nth,axiom,
    ! [N: nat,Xs2: list_o,Ys2: list_nat] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) ) )
     => ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs2 @ Ys2 ) @ N )
        = ( product_Pair_o_nat @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) @ ( nth_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_5612_product__nth,axiom,
    ! [N: nat,Xs2: list_o,Ys2: list_int] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys2 ) ) )
     => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs2 @ Ys2 ) @ N )
        = ( product_Pair_o_int @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) @ ( nth_int @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ) ).

% product_nth
thf(fact_5613_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( vEBT_Leaf @ $false @ $false ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Va: nat] :
                ( ( X
                  = ( suc @ ( suc @ Va ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                     => ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                     => ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
thf(fact_5614_triangle__def,axiom,
    ( nat_triangle
    = ( ^ [N3: nat] : ( divide_divide_nat @ ( times_times_nat @ N3 @ ( suc @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% triangle_def
thf(fact_5615_obtain__set__succ,axiom,
    ! [X: nat,Z2: nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_nat @ X @ Z2 )
     => ( ( vEBT_VEBT_max_in_set @ A2 @ Z2 )
       => ( ( finite_finite_nat @ B2 )
         => ( ( A2 = B2 )
           => ? [X_12: nat] : ( vEBT_is_succ_in_set @ A2 @ X @ X_12 ) ) ) ) ) ).

% obtain_set_succ
thf(fact_5616_obtain__set__pred,axiom,
    ! [Z2: nat,X: nat,A2: set_nat] :
      ( ( ord_less_nat @ Z2 @ X )
     => ( ( vEBT_VEBT_min_in_set @ A2 @ Z2 )
       => ( ( finite_finite_nat @ A2 )
         => ? [X_12: nat] : ( vEBT_is_pred_in_set @ A2 @ X @ X_12 ) ) ) ) ).

% obtain_set_pred
thf(fact_5617_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q2 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_5618_intind,axiom,
    ! [I3: nat,N: nat,P: int > $o,X: int] :
      ( ( ord_less_nat @ I3 @ N )
     => ( ( P @ X )
       => ( P @ ( nth_int @ ( replicate_int @ N @ X ) @ I3 ) ) ) ) ).

% intind
thf(fact_5619_intind,axiom,
    ! [I3: nat,N: nat,P: nat > $o,X: nat] :
      ( ( ord_less_nat @ I3 @ N )
     => ( ( P @ X )
       => ( P @ ( nth_nat @ ( replicate_nat @ N @ X ) @ I3 ) ) ) ) ).

% intind
thf(fact_5620_intind,axiom,
    ! [I3: nat,N: nat,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I3 @ N )
     => ( ( P @ X )
       => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I3 ) ) ) ) ).

% intind
thf(fact_5621_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_finite
thf(fact_5622_pred__none__empty,axiom,
    ! [Xs2: set_nat,A: nat] :
      ( ~ ? [X_12: nat] : ( vEBT_is_pred_in_set @ Xs2 @ A @ X_12 )
     => ( ( finite_finite_nat @ Xs2 )
       => ~ ? [X3: nat] :
              ( ( member_nat @ X3 @ Xs2 )
              & ( ord_less_nat @ X3 @ A ) ) ) ) ).

% pred_none_empty
thf(fact_5623_succ__none__empty,axiom,
    ! [Xs2: set_nat,A: nat] :
      ( ~ ? [X_12: nat] : ( vEBT_is_succ_in_set @ Xs2 @ A @ X_12 )
     => ( ( finite_finite_nat @ Xs2 )
       => ~ ? [X3: nat] :
              ( ( member_nat @ X3 @ Xs2 )
              & ( ord_less_nat @ A @ X3 ) ) ) ) ).

% succ_none_empty
thf(fact_5624_List_Ofinite__set,axiom,
    ! [Xs2: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_5625_List_Ofinite__set,axiom,
    ! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_5626_List_Ofinite__set,axiom,
    ! [Xs2: list_int] : ( finite_finite_int @ ( set_int2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_5627_List_Ofinite__set,axiom,
    ! [Xs2: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_5628_replicate__eq__replicate,axiom,
    ! [M: nat,X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( ( replicate_VEBT_VEBT @ M @ X )
        = ( replicate_VEBT_VEBT @ N @ Y ) )
      = ( ( M = N )
        & ( ( M != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
thf(fact_5629_length__replicate,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_5630_length__replicate,axiom,
    ! [N: nat,X: $o] :
      ( ( size_size_list_o @ ( replicate_o @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_5631_length__replicate,axiom,
    ! [N: nat,X: nat] :
      ( ( size_size_list_nat @ ( replicate_nat @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_5632_length__replicate,axiom,
    ! [N: nat,X: int] :
      ( ( size_size_list_int @ ( replicate_int @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_5633_triangle__0,axiom,
    ( ( nat_triangle @ zero_zero_nat )
    = zero_zero_nat ) ).

% triangle_0
thf(fact_5634_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) )
      = ( ord_less_rat @ A @ B ) ) ).

% infinite_Icc_iff
thf(fact_5635_infinite__Icc__iff,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) )
      = ( ord_less_real @ A @ B ) ) ).

% infinite_Icc_iff
thf(fact_5636_Ball__set__replicate,axiom,
    ! [N: nat,A: int,P: int > $o] :
      ( ( ! [X6: int] :
            ( ( member_int @ X6 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
           => ( P @ X6 ) ) )
      = ( ( P @ A )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_5637_Ball__set__replicate,axiom,
    ! [N: nat,A: nat,P: nat > $o] :
      ( ( ! [X6: nat] :
            ( ( member_nat @ X6 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
           => ( P @ X6 ) ) )
      = ( ( P @ A )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_5638_Ball__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X6: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
           => ( P @ X6 ) ) )
      = ( ( P @ A )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_5639_Bex__set__replicate,axiom,
    ! [N: nat,A: int,P: int > $o] :
      ( ( ? [X6: int] :
            ( ( member_int @ X6 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
            & ( P @ X6 ) ) )
      = ( ( P @ A )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_5640_Bex__set__replicate,axiom,
    ! [N: nat,A: nat,P: nat > $o] :
      ( ( ? [X6: nat] :
            ( ( member_nat @ X6 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
            & ( P @ X6 ) ) )
      = ( ( P @ A )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_5641_Bex__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ? [X6: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
            & ( P @ X6 ) ) )
      = ( ( P @ A )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_5642_in__set__replicate,axiom,
    ! [X: complex,N: nat,Y: complex] :
      ( ( member_complex @ X @ ( set_complex2 @ ( replicate_complex @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_5643_in__set__replicate,axiom,
    ! [X: real,N: nat,Y: real] :
      ( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_5644_in__set__replicate,axiom,
    ! [X: set_nat,N: nat,Y: set_nat] :
      ( ( member_set_nat @ X @ ( set_set_nat2 @ ( replicate_set_nat @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_5645_in__set__replicate,axiom,
    ! [X: int,N: nat,Y: int] :
      ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_5646_in__set__replicate,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_5647_in__set__replicate,axiom,
    ! [X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_5648_nth__replicate,axiom,
    ! [I3: nat,N: nat,X: int] :
      ( ( ord_less_nat @ I3 @ N )
     => ( ( nth_int @ ( replicate_int @ N @ X ) @ I3 )
        = X ) ) ).

% nth_replicate
thf(fact_5649_nth__replicate,axiom,
    ! [I3: nat,N: nat,X: nat] :
      ( ( ord_less_nat @ I3 @ N )
     => ( ( nth_nat @ ( replicate_nat @ N @ X ) @ I3 )
        = X ) ) ).

% nth_replicate
thf(fact_5650_nth__replicate,axiom,
    ! [I3: nat,N: nat,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I3 @ N )
     => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I3 )
        = X ) ) ).

% nth_replicate
thf(fact_5651_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys2 ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ).

% length_product
thf(fact_5652_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys2: list_o] :
      ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys2 ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys2 ) ) ) ).

% length_product
thf(fact_5653_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys2: list_nat] :
      ( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys2 ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) ) ) ).

% length_product
thf(fact_5654_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys2: list_int] :
      ( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys2 ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys2 ) ) ) ).

% length_product
thf(fact_5655_length__product,axiom,
    ! [Xs2: list_o,Ys2: list_VEBT_VEBT] :
      ( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys2 ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ).

% length_product
thf(fact_5656_length__product,axiom,
    ! [Xs2: list_o,Ys2: list_o] :
      ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs2 @ Ys2 ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys2 ) ) ) ).

% length_product
thf(fact_5657_length__product,axiom,
    ! [Xs2: list_o,Ys2: list_nat] :
      ( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs2 @ Ys2 ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) ) ) ).

% length_product
thf(fact_5658_length__product,axiom,
    ! [Xs2: list_o,Ys2: list_int] :
      ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs2 @ Ys2 ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys2 ) ) ) ).

% length_product
thf(fact_5659_length__product,axiom,
    ! [Xs2: list_nat,Ys2: list_VEBT_VEBT] :
      ( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs2 @ Ys2 ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ).

% length_product
thf(fact_5660_length__product,axiom,
    ! [Xs2: list_nat,Ys2: list_o] :
      ( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs2 @ Ys2 ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_o @ Ys2 ) ) ) ).

% length_product
thf(fact_5661_triangle__Suc,axiom,
    ! [N: nat] :
      ( ( nat_triangle @ ( suc @ N ) )
      = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).

% triangle_Suc
thf(fact_5662_set__replicate,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( N != zero_zero_nat )
     => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X ) )
        = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% set_replicate
thf(fact_5663_set__replicate,axiom,
    ! [N: nat,X: real] :
      ( ( N != zero_zero_nat )
     => ( ( set_real2 @ ( replicate_real @ N @ X ) )
        = ( insert_real @ X @ bot_bot_set_real ) ) ) ).

% set_replicate
thf(fact_5664_set__replicate,axiom,
    ! [N: nat,X: $o] :
      ( ( N != zero_zero_nat )
     => ( ( set_o2 @ ( replicate_o @ N @ X ) )
        = ( insert_o @ X @ bot_bot_set_o ) ) ) ).

% set_replicate
thf(fact_5665_set__replicate,axiom,
    ! [N: nat,X: nat] :
      ( ( N != zero_zero_nat )
     => ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
        = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% set_replicate
thf(fact_5666_set__replicate,axiom,
    ! [N: nat,X: int] :
      ( ( N != zero_zero_nat )
     => ( ( set_int2 @ ( replicate_int @ N @ X ) )
        = ( insert_int @ X @ bot_bot_set_int ) ) ) ).

% set_replicate
thf(fact_5667_bounded__nat__set__is__finite,axiom,
    ! [N5: set_nat,N: nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ N5 )
         => ( ord_less_nat @ X5 @ N ) )
     => ( finite_finite_nat @ N5 ) ) ).

% bounded_nat_set_is_finite
thf(fact_5668_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N6: set_nat] :
        ? [M4: nat] :
        ! [X6: nat] :
          ( ( member_nat @ X6 @ N6 )
         => ( ord_less_nat @ X6 @ M4 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_5669_finite__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ? [Xs3: list_VEBT_VEBT] :
          ( ( set_VEBT_VEBT2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_5670_finite__list,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ? [Xs3: list_nat] :
          ( ( set_nat2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_5671_finite__list,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ? [Xs3: list_int] :
          ( ( set_int2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_5672_finite__list,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ? [Xs3: list_complex] :
          ( ( set_complex2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_5673_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I3: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K3: nat] :
            ( ( P @ K3 )
            & ( ord_less_nat @ K3 @ I3 ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_5674_finite__lists__length__eq,axiom,
    ! [A2: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A2 )
              & ( ( size_s3451745648224563538omplex @ Xs )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_5675_finite__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
              & ( ( size_s6755466524823107622T_VEBT @ Xs )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_5676_finite__lists__length__eq,axiom,
    ! [A2: set_o,N: nat] :
      ( ( finite_finite_o @ A2 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A2 )
              & ( ( size_size_list_o @ Xs )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_5677_finite__lists__length__eq,axiom,
    ! [A2: set_int,N: nat] :
      ( ( finite_finite_int @ A2 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
              & ( ( size_size_list_int @ Xs )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_5678_finite__lists__length__eq,axiom,
    ! [A2: set_nat,N: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
              & ( ( size_size_list_nat @ Xs )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_5679_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_5680_mod__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R2 ) )
     => ( ( modulo_modulo_int @ K @ L )
        = R2 ) ) ).

% mod_int_unique
thf(fact_5681_replicate__eqI,axiom,
    ! [Xs2: list_complex,N: nat,X: complex] :
      ( ( ( size_s3451745648224563538omplex @ Xs2 )
        = N )
     => ( ! [Y4: complex] :
            ( ( member_complex @ Y4 @ ( set_complex2 @ Xs2 ) )
           => ( Y4 = X ) )
       => ( Xs2
          = ( replicate_complex @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_5682_replicate__eqI,axiom,
    ! [Xs2: list_real,N: nat,X: real] :
      ( ( ( size_size_list_real @ Xs2 )
        = N )
     => ( ! [Y4: real] :
            ( ( member_real @ Y4 @ ( set_real2 @ Xs2 ) )
           => ( Y4 = X ) )
       => ( Xs2
          = ( replicate_real @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_5683_replicate__eqI,axiom,
    ! [Xs2: list_set_nat,N: nat,X: set_nat] :
      ( ( ( size_s3254054031482475050et_nat @ Xs2 )
        = N )
     => ( ! [Y4: set_nat] :
            ( ( member_set_nat @ Y4 @ ( set_set_nat2 @ Xs2 ) )
           => ( Y4 = X ) )
       => ( Xs2
          = ( replicate_set_nat @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_5684_replicate__eqI,axiom,
    ! [Xs2: list_VEBT_VEBT,N: nat,X: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = N )
     => ( ! [Y4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ Y4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( Y4 = X ) )
       => ( Xs2
          = ( replicate_VEBT_VEBT @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_5685_replicate__eqI,axiom,
    ! [Xs2: list_o,N: nat,X: $o] :
      ( ( ( size_size_list_o @ Xs2 )
        = N )
     => ( ! [Y4: $o] :
            ( ( member_o @ Y4 @ ( set_o2 @ Xs2 ) )
           => ( Y4 = X ) )
       => ( Xs2
          = ( replicate_o @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_5686_replicate__eqI,axiom,
    ! [Xs2: list_nat,N: nat,X: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = N )
     => ( ! [Y4: nat] :
            ( ( member_nat @ Y4 @ ( set_nat2 @ Xs2 ) )
           => ( Y4 = X ) )
       => ( Xs2
          = ( replicate_nat @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_5687_replicate__eqI,axiom,
    ! [Xs2: list_int,N: nat,X: int] :
      ( ( ( size_size_list_int @ Xs2 )
        = N )
     => ( ! [Y4: int] :
            ( ( member_int @ Y4 @ ( set_int2 @ Xs2 ) )
           => ( Y4 = X ) )
       => ( Xs2
          = ( replicate_int @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_5688_replicate__length__same,axiom,
    ! [Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( X5 = X ) )
     => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_5689_replicate__length__same,axiom,
    ! [Xs2: list_o,X: $o] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ ( set_o2 @ Xs2 ) )
         => ( X5 = X ) )
     => ( ( replicate_o @ ( size_size_list_o @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_5690_replicate__length__same,axiom,
    ! [Xs2: list_nat,X: nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
         => ( X5 = X ) )
     => ( ( replicate_nat @ ( size_size_list_nat @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_5691_replicate__length__same,axiom,
    ! [Xs2: list_int,X: int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ ( set_int2 @ Xs2 ) )
         => ( X5 = X ) )
     => ( ( replicate_int @ ( size_size_list_int @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_5692_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_5693_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_5694_finite__lists__length__le,axiom,
    ! [A2: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_5695_finite__lists__length__le,axiom,
    ! [A2: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_5696_finite__lists__length__le,axiom,
    ! [A2: set_o,N: nat] :
      ( ( finite_finite_o @ A2 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_5697_finite__lists__length__le,axiom,
    ! [A2: set_int,N: nat] :
      ( ( finite_finite_int @ A2 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_5698_finite__lists__length__le,axiom,
    ! [A2: set_nat,N: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_5699_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q2: int] :
      ( ( L != zero_zero_int )
     => ( ( K
          = ( times_times_int @ Q2 @ L ) )
       => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ zero_zero_int ) ) ) ) ).

% eucl_rel_int_dividesI
thf(fact_5700_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_5701_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M ) ) ) ) ).

% finite_divisors_nat
thf(fact_5702_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N5: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N5 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
     => ( finite_finite_nat @ N5 ) ) ).

% subset_eq_atLeast0_atMost_finite
thf(fact_5703_set__replicate__Suc,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ ( suc @ N ) @ X ) )
      = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ).

% set_replicate_Suc
thf(fact_5704_set__replicate__Suc,axiom,
    ! [N: nat,X: real] :
      ( ( set_real2 @ ( replicate_real @ ( suc @ N ) @ X ) )
      = ( insert_real @ X @ bot_bot_set_real ) ) ).

% set_replicate_Suc
thf(fact_5705_set__replicate__Suc,axiom,
    ! [N: nat,X: $o] :
      ( ( set_o2 @ ( replicate_o @ ( suc @ N ) @ X ) )
      = ( insert_o @ X @ bot_bot_set_o ) ) ).

% set_replicate_Suc
thf(fact_5706_set__replicate__Suc,axiom,
    ! [N: nat,X: nat] :
      ( ( set_nat2 @ ( replicate_nat @ ( suc @ N ) @ X ) )
      = ( insert_nat @ X @ bot_bot_set_nat ) ) ).

% set_replicate_Suc
thf(fact_5707_set__replicate__Suc,axiom,
    ! [N: nat,X: int] :
      ( ( set_int2 @ ( replicate_int @ ( suc @ N ) @ X ) )
      = ( insert_int @ X @ bot_bot_set_int ) ) ).

% set_replicate_Suc
thf(fact_5708_set__replicate__conv__if,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X ) )
          = bot_bo8194388402131092736T_VEBT ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X ) )
          = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% set_replicate_conv_if
thf(fact_5709_set__replicate__conv__if,axiom,
    ! [N: nat,X: real] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_real2 @ ( replicate_real @ N @ X ) )
          = bot_bot_set_real ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_real2 @ ( replicate_real @ N @ X ) )
          = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).

% set_replicate_conv_if
thf(fact_5710_set__replicate__conv__if,axiom,
    ! [N: nat,X: $o] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_o2 @ ( replicate_o @ N @ X ) )
          = bot_bot_set_o ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_o2 @ ( replicate_o @ N @ X ) )
          = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).

% set_replicate_conv_if
thf(fact_5711_set__replicate__conv__if,axiom,
    ! [N: nat,X: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
          = bot_bot_set_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
          = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% set_replicate_conv_if
thf(fact_5712_set__replicate__conv__if,axiom,
    ! [N: nat,X: int] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_int2 @ ( replicate_int @ N @ X ) )
          = bot_bot_set_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_int2 @ ( replicate_int @ N @ X ) )
          = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).

% set_replicate_conv_if
thf(fact_5713_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R2 ) )
      = ( ( K
          = ( plus_plus_int @ ( times_times_int @ L @ Q2 ) @ R2 ) )
        & ( ( ord_less_int @ zero_zero_int @ L )
         => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
            & ( ord_less_int @ R2 @ L ) ) )
        & ( ~ ( ord_less_int @ zero_zero_int @ L )
         => ( ( ( ord_less_int @ L @ zero_zero_int )
             => ( ( ord_less_int @ L @ R2 )
                & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
            & ( ~ ( ord_less_int @ L @ zero_zero_int )
             => ( Q2 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_5714_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.simps(3)
thf(fact_5715_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_5716_finite__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Z3: real] :
              ( ( power_power_real @ Z3 @ N )
              = one_one_real ) ) ) ) ).

% finite_roots_unity
thf(fact_5717_finite__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z3: complex] :
              ( ( power_power_complex @ Z3 @ N )
              = one_one_complex ) ) ) ) ).

% finite_roots_unity
thf(fact_5718_finite__induct__select,axiom,
    ! [S3: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [T4: set_complex] :
              ( ( ord_less_set_complex @ T4 @ S3 )
             => ( ( P @ T4 )
               => ? [X3: complex] :
                    ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ S3 @ T4 ) )
                    & ( P @ ( insert_complex @ X3 @ T4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_5719_finite__induct__select,axiom,
    ! [S3: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ S3 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [T4: set_real] :
              ( ( ord_less_set_real @ T4 @ S3 )
             => ( ( P @ T4 )
               => ? [X3: real] :
                    ( ( member_real @ X3 @ ( minus_minus_set_real @ S3 @ T4 ) )
                    & ( P @ ( insert_real @ X3 @ T4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_5720_finite__induct__select,axiom,
    ! [S3: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ S3 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [T4: set_o] :
              ( ( ord_less_set_o @ T4 @ S3 )
             => ( ( P @ T4 )
               => ? [X3: $o] :
                    ( ( member_o @ X3 @ ( minus_minus_set_o @ S3 @ T4 ) )
                    & ( P @ ( insert_o @ X3 @ T4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_5721_finite__induct__select,axiom,
    ! [S3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ S3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [T4: set_int] :
              ( ( ord_less_set_int @ T4 @ S3 )
             => ( ( P @ T4 )
               => ? [X3: int] :
                    ( ( member_int @ X3 @ ( minus_minus_set_int @ S3 @ T4 ) )
                    & ( P @ ( insert_int @ X3 @ T4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_5722_finite__induct__select,axiom,
    ! [S3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ S3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [T4: set_nat] :
              ( ( ord_less_set_nat @ T4 @ S3 )
             => ( ( P @ T4 )
               => ? [X3: nat] :
                    ( ( member_nat @ X3 @ ( minus_minus_set_nat @ S3 @ T4 ) )
                    & ( P @ ( insert_nat @ X3 @ T4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_5723_remove__induct,axiom,
    ! [P: set_set_nat > $o,B2: set_set_nat] :
      ( ( P @ bot_bot_set_set_nat )
     => ( ( ~ ( finite1152437895449049373et_nat @ B2 )
         => ( P @ B2 ) )
       => ( ! [A8: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A8 )
             => ( ( A8 != bot_bot_set_set_nat )
               => ( ( ord_le6893508408891458716et_nat @ A8 @ B2 )
                 => ( ! [X3: set_nat] :
                        ( ( member_set_nat @ X3 @ A8 )
                       => ( P @ ( minus_2163939370556025621et_nat @ A8 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_5724_remove__induct,axiom,
    ! [P: set_complex > $o,B2: set_complex] :
      ( ( P @ bot_bot_set_complex )
     => ( ( ~ ( finite3207457112153483333omplex @ B2 )
         => ( P @ B2 ) )
       => ( ! [A8: set_complex] :
              ( ( finite3207457112153483333omplex @ A8 )
             => ( ( A8 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A8 @ B2 )
                 => ( ! [X3: complex] :
                        ( ( member_complex @ X3 @ A8 )
                       => ( P @ ( minus_811609699411566653omplex @ A8 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_5725_remove__induct,axiom,
    ! [P: set_real > $o,B2: set_real] :
      ( ( P @ bot_bot_set_real )
     => ( ( ~ ( finite_finite_real @ B2 )
         => ( P @ B2 ) )
       => ( ! [A8: set_real] :
              ( ( finite_finite_real @ A8 )
             => ( ( A8 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A8 @ B2 )
                 => ( ! [X3: real] :
                        ( ( member_real @ X3 @ A8 )
                       => ( P @ ( minus_minus_set_real @ A8 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_5726_remove__induct,axiom,
    ! [P: set_o > $o,B2: set_o] :
      ( ( P @ bot_bot_set_o )
     => ( ( ~ ( finite_finite_o @ B2 )
         => ( P @ B2 ) )
       => ( ! [A8: set_o] :
              ( ( finite_finite_o @ A8 )
             => ( ( A8 != bot_bot_set_o )
               => ( ( ord_less_eq_set_o @ A8 @ B2 )
                 => ( ! [X3: $o] :
                        ( ( member_o @ X3 @ A8 )
                       => ( P @ ( minus_minus_set_o @ A8 @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_5727_remove__induct,axiom,
    ! [P: set_int > $o,B2: set_int] :
      ( ( P @ bot_bot_set_int )
     => ( ( ~ ( finite_finite_int @ B2 )
         => ( P @ B2 ) )
       => ( ! [A8: set_int] :
              ( ( finite_finite_int @ A8 )
             => ( ( A8 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A8 @ B2 )
                 => ( ! [X3: int] :
                        ( ( member_int @ X3 @ A8 )
                       => ( P @ ( minus_minus_set_int @ A8 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_5728_remove__induct,axiom,
    ! [P: set_nat > $o,B2: set_nat] :
      ( ( P @ bot_bot_set_nat )
     => ( ( ~ ( finite_finite_nat @ B2 )
         => ( P @ B2 ) )
       => ( ! [A8: set_nat] :
              ( ( finite_finite_nat @ A8 )
             => ( ( A8 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A8 @ B2 )
                 => ( ! [X3: nat] :
                        ( ( member_nat @ X3 @ A8 )
                       => ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_5729_finite__remove__induct,axiom,
    ! [B2: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ B2 )
     => ( ( P @ bot_bot_set_set_nat )
       => ( ! [A8: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A8 )
             => ( ( A8 != bot_bot_set_set_nat )
               => ( ( ord_le6893508408891458716et_nat @ A8 @ B2 )
                 => ( ! [X3: set_nat] :
                        ( ( member_set_nat @ X3 @ A8 )
                       => ( P @ ( minus_2163939370556025621et_nat @ A8 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_5730_finite__remove__induct,axiom,
    ! [B2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [A8: set_complex] :
              ( ( finite3207457112153483333omplex @ A8 )
             => ( ( A8 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A8 @ B2 )
                 => ( ! [X3: complex] :
                        ( ( member_complex @ X3 @ A8 )
                       => ( P @ ( minus_811609699411566653omplex @ A8 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_5731_finite__remove__induct,axiom,
    ! [B2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ B2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [A8: set_real] :
              ( ( finite_finite_real @ A8 )
             => ( ( A8 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A8 @ B2 )
                 => ( ! [X3: real] :
                        ( ( member_real @ X3 @ A8 )
                       => ( P @ ( minus_minus_set_real @ A8 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_5732_finite__remove__induct,axiom,
    ! [B2: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ B2 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [A8: set_o] :
              ( ( finite_finite_o @ A8 )
             => ( ( A8 != bot_bot_set_o )
               => ( ( ord_less_eq_set_o @ A8 @ B2 )
                 => ( ! [X3: $o] :
                        ( ( member_o @ X3 @ A8 )
                       => ( P @ ( minus_minus_set_o @ A8 @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_5733_finite__remove__induct,axiom,
    ! [B2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ B2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [A8: set_int] :
              ( ( finite_finite_int @ A8 )
             => ( ( A8 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A8 @ B2 )
                 => ( ! [X3: int] :
                        ( ( member_int @ X3 @ A8 )
                       => ( P @ ( minus_minus_set_int @ A8 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_5734_finite__remove__induct,axiom,
    ! [B2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ B2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [A8: set_nat] :
              ( ( finite_finite_nat @ A8 )
             => ( ( A8 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A8 @ B2 )
                 => ( ! [X3: nat] :
                        ( ( member_nat @ X3 @ A8 )
                       => ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_5735_set__encode__insert,axiom,
    ! [A2: set_nat,N: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ N @ A2 )
       => ( ( nat_set_encode @ ( insert_nat @ N @ A2 ) )
          = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).

% set_encode_insert
thf(fact_5736_infinite__remove,axiom,
    ! [S3: set_complex,A: complex] :
      ( ~ ( finite3207457112153483333omplex @ S3 )
     => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ).

% infinite_remove
thf(fact_5737_infinite__remove,axiom,
    ! [S3: set_real,A: real] :
      ( ~ ( finite_finite_real @ S3 )
     => ~ ( finite_finite_real @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).

% infinite_remove
thf(fact_5738_infinite__remove,axiom,
    ! [S3: set_o,A: $o] :
      ( ~ ( finite_finite_o @ S3 )
     => ~ ( finite_finite_o @ ( minus_minus_set_o @ S3 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).

% infinite_remove
thf(fact_5739_infinite__remove,axiom,
    ! [S3: set_int,A: int] :
      ( ~ ( finite_finite_int @ S3 )
     => ~ ( finite_finite_int @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).

% infinite_remove
thf(fact_5740_infinite__remove,axiom,
    ! [S3: set_nat,A: nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).

% infinite_remove
thf(fact_5741_set__encode__empty,axiom,
    ( ( nat_set_encode @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% set_encode_empty
thf(fact_5742_finite__maxlen,axiom,
    ! [M7: set_list_VEBT_VEBT] :
      ( ( finite3004134309566078307T_VEBT @ M7 )
     => ? [N2: nat] :
        ! [X3: list_VEBT_VEBT] :
          ( ( member2936631157270082147T_VEBT @ X3 @ M7 )
         => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X3 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_5743_finite__maxlen,axiom,
    ! [M7: set_list_o] :
      ( ( finite_finite_list_o @ M7 )
     => ? [N2: nat] :
        ! [X3: list_o] :
          ( ( member_list_o @ X3 @ M7 )
         => ( ord_less_nat @ ( size_size_list_o @ X3 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_5744_finite__maxlen,axiom,
    ! [M7: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M7 )
     => ? [N2: nat] :
        ! [X3: list_nat] :
          ( ( member_list_nat @ X3 @ M7 )
         => ( ord_less_nat @ ( size_size_list_nat @ X3 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_5745_finite__maxlen,axiom,
    ! [M7: set_list_int] :
      ( ( finite3922522038869484883st_int @ M7 )
     => ? [N2: nat] :
        ! [X3: list_int] :
          ( ( member_list_int @ X3 @ M7 )
         => ( ord_less_nat @ ( size_size_list_int @ X3 ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_5746_finite__divisors__int,axiom,
    ! [I3: int] :
      ( ( I3 != zero_zero_int )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [D2: int] : ( dvd_dvd_int @ D2 @ I3 ) ) ) ) ).

% finite_divisors_int
thf(fact_5747_set__encode__inf,axiom,
    ! [A2: set_nat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( nat_set_encode @ A2 )
        = zero_zero_nat ) ) ).

% set_encode_inf
thf(fact_5748_finite_OemptyI,axiom,
    finite3207457112153483333omplex @ bot_bot_set_complex ).

% finite.emptyI
thf(fact_5749_finite_OemptyI,axiom,
    finite_finite_real @ bot_bot_set_real ).

% finite.emptyI
thf(fact_5750_finite_OemptyI,axiom,
    finite_finite_o @ bot_bot_set_o ).

% finite.emptyI
thf(fact_5751_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_5752_finite_OemptyI,axiom,
    finite_finite_int @ bot_bot_set_int ).

% finite.emptyI
thf(fact_5753_infinite__imp__nonempty,axiom,
    ! [S3: set_complex] :
      ( ~ ( finite3207457112153483333omplex @ S3 )
     => ( S3 != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
thf(fact_5754_infinite__imp__nonempty,axiom,
    ! [S3: set_real] :
      ( ~ ( finite_finite_real @ S3 )
     => ( S3 != bot_bot_set_real ) ) ).

% infinite_imp_nonempty
thf(fact_5755_infinite__imp__nonempty,axiom,
    ! [S3: set_o] :
      ( ~ ( finite_finite_o @ S3 )
     => ( S3 != bot_bot_set_o ) ) ).

% infinite_imp_nonempty
thf(fact_5756_infinite__imp__nonempty,axiom,
    ! [S3: set_nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ( S3 != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_5757_infinite__imp__nonempty,axiom,
    ! [S3: set_int] :
      ( ~ ( finite_finite_int @ S3 )
     => ( S3 != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
thf(fact_5758_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_5759_finite__has__maximal,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ? [X5: real] :
            ( ( member_real @ X5 @ A2 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A2 )
               => ( ( ord_less_eq_real @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_5760_finite__has__maximal,axiom,
    ! [A2: set_o] :
      ( ( finite_finite_o @ A2 )
     => ( ( A2 != bot_bot_set_o )
       => ? [X5: $o] :
            ( ( member_o @ X5 @ A2 )
            & ! [Xa: $o] :
                ( ( member_o @ Xa @ A2 )
               => ( ( ord_less_eq_o @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_5761_finite__has__maximal,axiom,
    ! [A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( A2 != bot_bot_set_set_nat )
       => ? [X5: set_nat] :
            ( ( member_set_nat @ X5 @ A2 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A2 )
               => ( ( ord_less_eq_set_nat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_5762_finite__has__maximal,axiom,
    ! [A2: set_rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( A2 != bot_bot_set_rat )
       => ? [X5: rat] :
            ( ( member_rat @ X5 @ A2 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A2 )
               => ( ( ord_less_eq_rat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_5763_finite__has__maximal,axiom,
    ! [A2: set_num] :
      ( ( finite_finite_num @ A2 )
     => ( ( A2 != bot_bot_set_num )
       => ? [X5: num] :
            ( ( member_num @ X5 @ A2 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A2 )
               => ( ( ord_less_eq_num @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_5764_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ? [X5: nat] :
            ( ( member_nat @ X5 @ A2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_5765_finite__has__maximal,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ? [X5: int] :
            ( ( member_int @ X5 @ A2 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A2 )
               => ( ( ord_less_eq_int @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_5766_finite__has__minimal,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ? [X5: real] :
            ( ( member_real @ X5 @ A2 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A2 )
               => ( ( ord_less_eq_real @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_5767_finite__has__minimal,axiom,
    ! [A2: set_o] :
      ( ( finite_finite_o @ A2 )
     => ( ( A2 != bot_bot_set_o )
       => ? [X5: $o] :
            ( ( member_o @ X5 @ A2 )
            & ! [Xa: $o] :
                ( ( member_o @ Xa @ A2 )
               => ( ( ord_less_eq_o @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_5768_finite__has__minimal,axiom,
    ! [A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( A2 != bot_bot_set_set_nat )
       => ? [X5: set_nat] :
            ( ( member_set_nat @ X5 @ A2 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A2 )
               => ( ( ord_less_eq_set_nat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_5769_finite__has__minimal,axiom,
    ! [A2: set_rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( A2 != bot_bot_set_rat )
       => ? [X5: rat] :
            ( ( member_rat @ X5 @ A2 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A2 )
               => ( ( ord_less_eq_rat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_5770_finite__has__minimal,axiom,
    ! [A2: set_num] :
      ( ( finite_finite_num @ A2 )
     => ( ( A2 != bot_bot_set_num )
       => ? [X5: num] :
            ( ( member_num @ X5 @ A2 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A2 )
               => ( ( ord_less_eq_num @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_5771_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ? [X5: nat] :
            ( ( member_nat @ X5 @ A2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_5772_finite__has__minimal,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ? [X5: int] :
            ( ( member_int @ X5 @ A2 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A2 )
               => ( ( ord_less_eq_int @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_5773_finite_Ocases,axiom,
    ! [A: set_complex] :
      ( ( finite3207457112153483333omplex @ A )
     => ( ( A != bot_bot_set_complex )
       => ~ ! [A8: set_complex] :
              ( ? [A5: complex] :
                  ( A
                  = ( insert_complex @ A5 @ A8 ) )
             => ~ ( finite3207457112153483333omplex @ A8 ) ) ) ) ).

% finite.cases
thf(fact_5774_finite_Ocases,axiom,
    ! [A: set_real] :
      ( ( finite_finite_real @ A )
     => ( ( A != bot_bot_set_real )
       => ~ ! [A8: set_real] :
              ( ? [A5: real] :
                  ( A
                  = ( insert_real @ A5 @ A8 ) )
             => ~ ( finite_finite_real @ A8 ) ) ) ) ).

% finite.cases
thf(fact_5775_finite_Ocases,axiom,
    ! [A: set_o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ~ ! [A8: set_o] :
              ( ? [A5: $o] :
                  ( A
                  = ( insert_o @ A5 @ A8 ) )
             => ~ ( finite_finite_o @ A8 ) ) ) ) ).

% finite.cases
thf(fact_5776_finite_Ocases,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ~ ! [A8: set_nat] :
              ( ? [A5: nat] :
                  ( A
                  = ( insert_nat @ A5 @ A8 ) )
             => ~ ( finite_finite_nat @ A8 ) ) ) ) ).

% finite.cases
thf(fact_5777_finite_Ocases,axiom,
    ! [A: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ~ ! [A8: set_int] :
              ( ? [A5: int] :
                  ( A
                  = ( insert_int @ A5 @ A8 ) )
             => ~ ( finite_finite_int @ A8 ) ) ) ) ).

% finite.cases
thf(fact_5778_finite_Osimps,axiom,
    ( finite3207457112153483333omplex
    = ( ^ [A3: set_complex] :
          ( ( A3 = bot_bot_set_complex )
          | ? [A6: set_complex,B3: complex] :
              ( ( A3
                = ( insert_complex @ B3 @ A6 ) )
              & ( finite3207457112153483333omplex @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_5779_finite_Osimps,axiom,
    ( finite_finite_real
    = ( ^ [A3: set_real] :
          ( ( A3 = bot_bot_set_real )
          | ? [A6: set_real,B3: real] :
              ( ( A3
                = ( insert_real @ B3 @ A6 ) )
              & ( finite_finite_real @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_5780_finite_Osimps,axiom,
    ( finite_finite_o
    = ( ^ [A3: set_o] :
          ( ( A3 = bot_bot_set_o )
          | ? [A6: set_o,B3: $o] :
              ( ( A3
                = ( insert_o @ B3 @ A6 ) )
              & ( finite_finite_o @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_5781_finite_Osimps,axiom,
    ( finite_finite_nat
    = ( ^ [A3: set_nat] :
          ( ( A3 = bot_bot_set_nat )
          | ? [A6: set_nat,B3: nat] :
              ( ( A3
                = ( insert_nat @ B3 @ A6 ) )
              & ( finite_finite_nat @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_5782_finite_Osimps,axiom,
    ( finite_finite_int
    = ( ^ [A3: set_int] :
          ( ( A3 = bot_bot_set_int )
          | ? [A6: set_int,B3: int] :
              ( ( A3
                = ( insert_int @ B3 @ A6 ) )
              & ( finite_finite_int @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_5783_finite__induct,axiom,
    ! [F3: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F3 )
     => ( ( P @ bot_bot_set_set_nat )
       => ( ! [X5: set_nat,F4: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ F4 )
             => ( ~ ( member_set_nat @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_set_nat @ X5 @ F4 ) ) ) ) )
         => ( P @ F3 ) ) ) ) ).

% finite_induct
thf(fact_5784_finite__induct,axiom,
    ! [F3: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X5: complex,F4: set_complex] :
              ( ( finite3207457112153483333omplex @ F4 )
             => ( ~ ( member_complex @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_complex @ X5 @ F4 ) ) ) ) )
         => ( P @ F3 ) ) ) ) ).

% finite_induct
thf(fact_5785_finite__induct,axiom,
    ! [F3: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F3 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X5: real,F4: set_real] :
              ( ( finite_finite_real @ F4 )
             => ( ~ ( member_real @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_real @ X5 @ F4 ) ) ) ) )
         => ( P @ F3 ) ) ) ) ).

% finite_induct
thf(fact_5786_finite__induct,axiom,
    ! [F3: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F3 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X5: $o,F4: set_o] :
              ( ( finite_finite_o @ F4 )
             => ( ~ ( member_o @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_o @ X5 @ F4 ) ) ) ) )
         => ( P @ F3 ) ) ) ) ).

% finite_induct
thf(fact_5787_finite__induct,axiom,
    ! [F3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X5: nat,F4: set_nat] :
              ( ( finite_finite_nat @ F4 )
             => ( ~ ( member_nat @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_nat @ X5 @ F4 ) ) ) ) )
         => ( P @ F3 ) ) ) ) ).

% finite_induct
thf(fact_5788_finite__induct,axiom,
    ! [F3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X5: int,F4: set_int] :
              ( ( finite_finite_int @ F4 )
             => ( ~ ( member_int @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_int @ X5 @ F4 ) ) ) ) )
         => ( P @ F3 ) ) ) ) ).

% finite_induct
thf(fact_5789_finite__ne__induct,axiom,
    ! [F3: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F3 )
     => ( ( F3 != bot_bot_set_set_nat )
       => ( ! [X5: set_nat] : ( P @ ( insert_set_nat @ X5 @ bot_bot_set_set_nat ) )
         => ( ! [X5: set_nat,F4: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F4 )
               => ( ( F4 != bot_bot_set_set_nat )
                 => ( ~ ( member_set_nat @ X5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_set_nat @ X5 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_5790_finite__ne__induct,axiom,
    ! [F3: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( F3 != bot_bot_set_complex )
       => ( ! [X5: complex] : ( P @ ( insert_complex @ X5 @ bot_bot_set_complex ) )
         => ( ! [X5: complex,F4: set_complex] :
                ( ( finite3207457112153483333omplex @ F4 )
               => ( ( F4 != bot_bot_set_complex )
                 => ( ~ ( member_complex @ X5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_complex @ X5 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_5791_finite__ne__induct,axiom,
    ! [F3: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F3 )
     => ( ( F3 != bot_bot_set_real )
       => ( ! [X5: real] : ( P @ ( insert_real @ X5 @ bot_bot_set_real ) )
         => ( ! [X5: real,F4: set_real] :
                ( ( finite_finite_real @ F4 )
               => ( ( F4 != bot_bot_set_real )
                 => ( ~ ( member_real @ X5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_real @ X5 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_5792_finite__ne__induct,axiom,
    ! [F3: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F3 )
     => ( ( F3 != bot_bot_set_o )
       => ( ! [X5: $o] : ( P @ ( insert_o @ X5 @ bot_bot_set_o ) )
         => ( ! [X5: $o,F4: set_o] :
                ( ( finite_finite_o @ F4 )
               => ( ( F4 != bot_bot_set_o )
                 => ( ~ ( member_o @ X5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_o @ X5 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_5793_finite__ne__induct,axiom,
    ! [F3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F3 )
     => ( ( F3 != bot_bot_set_nat )
       => ( ! [X5: nat] : ( P @ ( insert_nat @ X5 @ bot_bot_set_nat ) )
         => ( ! [X5: nat,F4: set_nat] :
                ( ( finite_finite_nat @ F4 )
               => ( ( F4 != bot_bot_set_nat )
                 => ( ~ ( member_nat @ X5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_nat @ X5 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_5794_finite__ne__induct,axiom,
    ! [F3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F3 )
     => ( ( F3 != bot_bot_set_int )
       => ( ! [X5: int] : ( P @ ( insert_int @ X5 @ bot_bot_set_int ) )
         => ( ! [X5: int,F4: set_int] :
                ( ( finite_finite_int @ F4 )
               => ( ( F4 != bot_bot_set_int )
                 => ( ~ ( member_int @ X5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_int @ X5 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_5795_infinite__finite__induct,axiom,
    ! [P: set_set_nat > $o,A2: set_set_nat] :
      ( ! [A8: set_set_nat] :
          ( ~ ( finite1152437895449049373et_nat @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bot_set_set_nat )
       => ( ! [X5: set_nat,F4: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ F4 )
             => ( ~ ( member_set_nat @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_set_nat @ X5 @ F4 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_5796_infinite__finite__induct,axiom,
    ! [P: set_complex > $o,A2: set_complex] :
      ( ! [A8: set_complex] :
          ( ~ ( finite3207457112153483333omplex @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X5: complex,F4: set_complex] :
              ( ( finite3207457112153483333omplex @ F4 )
             => ( ~ ( member_complex @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_complex @ X5 @ F4 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_5797_infinite__finite__induct,axiom,
    ! [P: set_real > $o,A2: set_real] :
      ( ! [A8: set_real] :
          ( ~ ( finite_finite_real @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X5: real,F4: set_real] :
              ( ( finite_finite_real @ F4 )
             => ( ~ ( member_real @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_real @ X5 @ F4 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_5798_infinite__finite__induct,axiom,
    ! [P: set_o > $o,A2: set_o] :
      ( ! [A8: set_o] :
          ( ~ ( finite_finite_o @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X5: $o,F4: set_o] :
              ( ( finite_finite_o @ F4 )
             => ( ~ ( member_o @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_o @ X5 @ F4 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_5799_infinite__finite__induct,axiom,
    ! [P: set_nat > $o,A2: set_nat] :
      ( ! [A8: set_nat] :
          ( ~ ( finite_finite_nat @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X5: nat,F4: set_nat] :
              ( ( finite_finite_nat @ F4 )
             => ( ~ ( member_nat @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_nat @ X5 @ F4 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_5800_infinite__finite__induct,axiom,
    ! [P: set_int > $o,A2: set_int] :
      ( ! [A8: set_int] :
          ( ~ ( finite_finite_int @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X5: int,F4: set_int] :
              ( ( finite_finite_int @ F4 )
             => ( ~ ( member_int @ X5 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_int @ X5 @ F4 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_5801_finite__subset__induct_H,axiom,
    ! [F3: set_set_nat,A2: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F3 )
     => ( ( ord_le6893508408891458716et_nat @ F3 @ A2 )
       => ( ( P @ bot_bot_set_set_nat )
         => ( ! [A5: set_nat,F4: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F4 )
               => ( ( member_set_nat @ A5 @ A2 )
                 => ( ( ord_le6893508408891458716et_nat @ F4 @ A2 )
                   => ( ~ ( member_set_nat @ A5 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_set_nat @ A5 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_5802_finite__subset__induct_H,axiom,
    ! [F3: set_complex,A2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( ord_le211207098394363844omplex @ F3 @ A2 )
       => ( ( P @ bot_bot_set_complex )
         => ( ! [A5: complex,F4: set_complex] :
                ( ( finite3207457112153483333omplex @ F4 )
               => ( ( member_complex @ A5 @ A2 )
                 => ( ( ord_le211207098394363844omplex @ F4 @ A2 )
                   => ( ~ ( member_complex @ A5 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_complex @ A5 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_5803_finite__subset__induct_H,axiom,
    ! [F3: set_real,A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F3 )
     => ( ( ord_less_eq_set_real @ F3 @ A2 )
       => ( ( P @ bot_bot_set_real )
         => ( ! [A5: real,F4: set_real] :
                ( ( finite_finite_real @ F4 )
               => ( ( member_real @ A5 @ A2 )
                 => ( ( ord_less_eq_set_real @ F4 @ A2 )
                   => ( ~ ( member_real @ A5 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_real @ A5 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_5804_finite__subset__induct_H,axiom,
    ! [F3: set_o,A2: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F3 )
     => ( ( ord_less_eq_set_o @ F3 @ A2 )
       => ( ( P @ bot_bot_set_o )
         => ( ! [A5: $o,F4: set_o] :
                ( ( finite_finite_o @ F4 )
               => ( ( member_o @ A5 @ A2 )
                 => ( ( ord_less_eq_set_o @ F4 @ A2 )
                   => ( ~ ( member_o @ A5 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_o @ A5 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_5805_finite__subset__induct_H,axiom,
    ! [F3: set_int,A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F3 )
     => ( ( ord_less_eq_set_int @ F3 @ A2 )
       => ( ( P @ bot_bot_set_int )
         => ( ! [A5: int,F4: set_int] :
                ( ( finite_finite_int @ F4 )
               => ( ( member_int @ A5 @ A2 )
                 => ( ( ord_less_eq_set_int @ F4 @ A2 )
                   => ( ~ ( member_int @ A5 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_int @ A5 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_5806_finite__subset__induct_H,axiom,
    ! [F3: set_nat,A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F3 )
     => ( ( ord_less_eq_set_nat @ F3 @ A2 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A5: nat,F4: set_nat] :
                ( ( finite_finite_nat @ F4 )
               => ( ( member_nat @ A5 @ A2 )
                 => ( ( ord_less_eq_set_nat @ F4 @ A2 )
                   => ( ~ ( member_nat @ A5 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_nat @ A5 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_5807_finite__subset__induct,axiom,
    ! [F3: set_set_nat,A2: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F3 )
     => ( ( ord_le6893508408891458716et_nat @ F3 @ A2 )
       => ( ( P @ bot_bot_set_set_nat )
         => ( ! [A5: set_nat,F4: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F4 )
               => ( ( member_set_nat @ A5 @ A2 )
                 => ( ~ ( member_set_nat @ A5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_set_nat @ A5 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_5808_finite__subset__induct,axiom,
    ! [F3: set_complex,A2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( ord_le211207098394363844omplex @ F3 @ A2 )
       => ( ( P @ bot_bot_set_complex )
         => ( ! [A5: complex,F4: set_complex] :
                ( ( finite3207457112153483333omplex @ F4 )
               => ( ( member_complex @ A5 @ A2 )
                 => ( ~ ( member_complex @ A5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_complex @ A5 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_5809_finite__subset__induct,axiom,
    ! [F3: set_real,A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F3 )
     => ( ( ord_less_eq_set_real @ F3 @ A2 )
       => ( ( P @ bot_bot_set_real )
         => ( ! [A5: real,F4: set_real] :
                ( ( finite_finite_real @ F4 )
               => ( ( member_real @ A5 @ A2 )
                 => ( ~ ( member_real @ A5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_real @ A5 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_5810_finite__subset__induct,axiom,
    ! [F3: set_o,A2: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F3 )
     => ( ( ord_less_eq_set_o @ F3 @ A2 )
       => ( ( P @ bot_bot_set_o )
         => ( ! [A5: $o,F4: set_o] :
                ( ( finite_finite_o @ F4 )
               => ( ( member_o @ A5 @ A2 )
                 => ( ~ ( member_o @ A5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_o @ A5 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_5811_finite__subset__induct,axiom,
    ! [F3: set_int,A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F3 )
     => ( ( ord_less_eq_set_int @ F3 @ A2 )
       => ( ( P @ bot_bot_set_int )
         => ( ! [A5: int,F4: set_int] :
                ( ( finite_finite_int @ F4 )
               => ( ( member_int @ A5 @ A2 )
                 => ( ~ ( member_int @ A5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_int @ A5 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_5812_finite__subset__induct,axiom,
    ! [F3: set_nat,A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F3 )
     => ( ( ord_less_eq_set_nat @ F3 @ A2 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A5: nat,F4: set_nat] :
                ( ( finite_finite_nat @ F4 )
               => ( ( member_nat @ A5 @ A2 )
                 => ( ~ ( member_nat @ A5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_nat @ A5 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_5813_finite__empty__induct,axiom,
    ! [A2: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: set_nat,A8: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A8 )
             => ( ( member_set_nat @ A5 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_2163939370556025621et_nat @ A8 @ ( insert_set_nat @ A5 @ bot_bot_set_set_nat ) ) ) ) ) )
         => ( P @ bot_bot_set_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_5814_finite__empty__induct,axiom,
    ! [A2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: complex,A8: set_complex] :
              ( ( finite3207457112153483333omplex @ A8 )
             => ( ( member_complex @ A5 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_811609699411566653omplex @ A8 @ ( insert_complex @ A5 @ bot_bot_set_complex ) ) ) ) ) )
         => ( P @ bot_bot_set_complex ) ) ) ) ).

% finite_empty_induct
thf(fact_5815_finite__empty__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: real,A8: set_real] :
              ( ( finite_finite_real @ A8 )
             => ( ( member_real @ A5 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_minus_set_real @ A8 @ ( insert_real @ A5 @ bot_bot_set_real ) ) ) ) ) )
         => ( P @ bot_bot_set_real ) ) ) ) ).

% finite_empty_induct
thf(fact_5816_finite__empty__induct,axiom,
    ! [A2: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: $o,A8: set_o] :
              ( ( finite_finite_o @ A8 )
             => ( ( member_o @ A5 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_minus_set_o @ A8 @ ( insert_o @ A5 @ bot_bot_set_o ) ) ) ) ) )
         => ( P @ bot_bot_set_o ) ) ) ) ).

% finite_empty_induct
thf(fact_5817_finite__empty__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: int,A8: set_int] :
              ( ( finite_finite_int @ A8 )
             => ( ( member_int @ A5 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_minus_set_int @ A8 @ ( insert_int @ A5 @ bot_bot_set_int ) ) ) ) ) )
         => ( P @ bot_bot_set_int ) ) ) ) ).

% finite_empty_induct
thf(fact_5818_finite__empty__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: nat,A8: set_nat] :
              ( ( finite_finite_nat @ A8 )
             => ( ( member_nat @ A5 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ) ) ) )
         => ( P @ bot_bot_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_5819_infinite__coinduct,axiom,
    ! [X9: set_complex > $o,A2: set_complex] :
      ( ( X9 @ A2 )
     => ( ! [A8: set_complex] :
            ( ( X9 @ A8 )
           => ? [X3: complex] :
                ( ( member_complex @ X3 @ A8 )
                & ( ( X9 @ ( minus_811609699411566653omplex @ A8 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) )
                  | ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A8 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) )
       => ~ ( finite3207457112153483333omplex @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_5820_infinite__coinduct,axiom,
    ! [X9: set_real > $o,A2: set_real] :
      ( ( X9 @ A2 )
     => ( ! [A8: set_real] :
            ( ( X9 @ A8 )
           => ? [X3: real] :
                ( ( member_real @ X3 @ A8 )
                & ( ( X9 @ ( minus_minus_set_real @ A8 @ ( insert_real @ X3 @ bot_bot_set_real ) ) )
                  | ~ ( finite_finite_real @ ( minus_minus_set_real @ A8 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) )
       => ~ ( finite_finite_real @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_5821_infinite__coinduct,axiom,
    ! [X9: set_o > $o,A2: set_o] :
      ( ( X9 @ A2 )
     => ( ! [A8: set_o] :
            ( ( X9 @ A8 )
           => ? [X3: $o] :
                ( ( member_o @ X3 @ A8 )
                & ( ( X9 @ ( minus_minus_set_o @ A8 @ ( insert_o @ X3 @ bot_bot_set_o ) ) )
                  | ~ ( finite_finite_o @ ( minus_minus_set_o @ A8 @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) )
       => ~ ( finite_finite_o @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_5822_infinite__coinduct,axiom,
    ! [X9: set_int > $o,A2: set_int] :
      ( ( X9 @ A2 )
     => ( ! [A8: set_int] :
            ( ( X9 @ A8 )
           => ? [X3: int] :
                ( ( member_int @ X3 @ A8 )
                & ( ( X9 @ ( minus_minus_set_int @ A8 @ ( insert_int @ X3 @ bot_bot_set_int ) ) )
                  | ~ ( finite_finite_int @ ( minus_minus_set_int @ A8 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) )
       => ~ ( finite_finite_int @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_5823_infinite__coinduct,axiom,
    ! [X9: set_nat > $o,A2: set_nat] :
      ( ( X9 @ A2 )
     => ( ! [A8: set_nat] :
            ( ( X9 @ A8 )
           => ? [X3: nat] :
                ( ( member_nat @ X3 @ A8 )
                & ( ( X9 @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
                  | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) )
       => ~ ( finite_finite_nat @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_5824_arcosh__1,axiom,
    ( ( arcosh_real @ one_one_real )
    = zero_zero_real ) ).

% arcosh_1
thf(fact_5825_finite__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z3: complex] :
              ( ( power_power_complex @ Z3 @ N )
              = C ) ) ) ) ).

% finite_nth_roots
thf(fact_5826_finite__linorder__min__induct,axiom,
    ! [A2: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [B5: $o,A8: set_o] :
              ( ( finite_finite_o @ A8 )
             => ( ! [X3: $o] :
                    ( ( member_o @ X3 @ A8 )
                   => ( ord_less_o @ B5 @ X3 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_o @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_5827_finite__linorder__min__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B5: real,A8: set_real] :
              ( ( finite_finite_real @ A8 )
             => ( ! [X3: real] :
                    ( ( member_real @ X3 @ A8 )
                   => ( ord_less_real @ B5 @ X3 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_real @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_5828_finite__linorder__min__induct,axiom,
    ! [A2: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A2 )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B5: rat,A8: set_rat] :
              ( ( finite_finite_rat @ A8 )
             => ( ! [X3: rat] :
                    ( ( member_rat @ X3 @ A8 )
                   => ( ord_less_rat @ B5 @ X3 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_rat @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_5829_finite__linorder__min__induct,axiom,
    ! [A2: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A2 )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B5: num,A8: set_num] :
              ( ( finite_finite_num @ A8 )
             => ( ! [X3: num] :
                    ( ( member_num @ X3 @ A8 )
                   => ( ord_less_num @ B5 @ X3 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_num @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_5830_finite__linorder__min__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B5: nat,A8: set_nat] :
              ( ( finite_finite_nat @ A8 )
             => ( ! [X3: nat] :
                    ( ( member_nat @ X3 @ A8 )
                   => ( ord_less_nat @ B5 @ X3 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_nat @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_5831_finite__linorder__min__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B5: int,A8: set_int] :
              ( ( finite_finite_int @ A8 )
             => ( ! [X3: int] :
                    ( ( member_int @ X3 @ A8 )
                   => ( ord_less_int @ B5 @ X3 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_int @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_5832_finite__linorder__min__induct,axiom,
    ! [A2: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [B5: extended_enat,A8: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ A8 )
             => ( ! [X3: extended_enat] :
                    ( ( member_Extended_enat @ X3 @ A8 )
                   => ( ord_le72135733267957522d_enat @ B5 @ X3 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_Extended_enat @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_5833_finite__linorder__max__induct,axiom,
    ! [A2: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [B5: $o,A8: set_o] :
              ( ( finite_finite_o @ A8 )
             => ( ! [X3: $o] :
                    ( ( member_o @ X3 @ A8 )
                   => ( ord_less_o @ X3 @ B5 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_o @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_5834_finite__linorder__max__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B5: real,A8: set_real] :
              ( ( finite_finite_real @ A8 )
             => ( ! [X3: real] :
                    ( ( member_real @ X3 @ A8 )
                   => ( ord_less_real @ X3 @ B5 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_real @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_5835_finite__linorder__max__induct,axiom,
    ! [A2: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A2 )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B5: rat,A8: set_rat] :
              ( ( finite_finite_rat @ A8 )
             => ( ! [X3: rat] :
                    ( ( member_rat @ X3 @ A8 )
                   => ( ord_less_rat @ X3 @ B5 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_rat @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_5836_finite__linorder__max__induct,axiom,
    ! [A2: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A2 )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B5: num,A8: set_num] :
              ( ( finite_finite_num @ A8 )
             => ( ! [X3: num] :
                    ( ( member_num @ X3 @ A8 )
                   => ( ord_less_num @ X3 @ B5 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_num @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_5837_finite__linorder__max__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B5: nat,A8: set_nat] :
              ( ( finite_finite_nat @ A8 )
             => ( ! [X3: nat] :
                    ( ( member_nat @ X3 @ A8 )
                   => ( ord_less_nat @ X3 @ B5 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_nat @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_5838_finite__linorder__max__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B5: int,A8: set_int] :
              ( ( finite_finite_int @ A8 )
             => ( ! [X3: int] :
                    ( ( member_int @ X3 @ A8 )
                   => ( ord_less_int @ X3 @ B5 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_int @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_5839_finite__linorder__max__induct,axiom,
    ! [A2: set_Extended_enat,P: set_Extended_enat > $o] :
      ( ( finite4001608067531595151d_enat @ A2 )
     => ( ( P @ bot_bo7653980558646680370d_enat )
       => ( ! [B5: extended_enat,A8: set_Extended_enat] :
              ( ( finite4001608067531595151d_enat @ A8 )
             => ( ! [X3: extended_enat] :
                    ( ( member_Extended_enat @ X3 @ A8 )
                   => ( ord_le72135733267957522d_enat @ X3 @ B5 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_Extended_enat @ B5 @ A8 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_5840_finite__ranking__induct,axiom,
    ! [S3: set_complex,P: set_complex > $o,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X5: complex,S4: set_complex] :
              ( ( finite3207457112153483333omplex @ S4 )
             => ( ! [Y5: complex] :
                    ( ( member_complex @ Y5 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_complex @ X5 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5841_finite__ranking__induct,axiom,
    ! [S3: set_real,P: set_real > $o,F: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X5: real,S4: set_real] :
              ( ( finite_finite_real @ S4 )
             => ( ! [Y5: real] :
                    ( ( member_real @ Y5 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_real @ X5 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5842_finite__ranking__induct,axiom,
    ! [S3: set_o,P: set_o > $o,F: $o > rat] :
      ( ( finite_finite_o @ S3 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X5: $o,S4: set_o] :
              ( ( finite_finite_o @ S4 )
             => ( ! [Y5: $o] :
                    ( ( member_o @ Y5 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_o @ X5 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5843_finite__ranking__induct,axiom,
    ! [S3: set_nat,P: set_nat > $o,F: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X5: nat,S4: set_nat] :
              ( ( finite_finite_nat @ S4 )
             => ( ! [Y5: nat] :
                    ( ( member_nat @ Y5 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_nat @ X5 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5844_finite__ranking__induct,axiom,
    ! [S3: set_int,P: set_int > $o,F: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X5: int,S4: set_int] :
              ( ( finite_finite_int @ S4 )
             => ( ! [Y5: int] :
                    ( ( member_int @ Y5 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_int @ X5 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5845_finite__ranking__induct,axiom,
    ! [S3: set_complex,P: set_complex > $o,F: complex > num] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X5: complex,S4: set_complex] :
              ( ( finite3207457112153483333omplex @ S4 )
             => ( ! [Y5: complex] :
                    ( ( member_complex @ Y5 @ S4 )
                   => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_complex @ X5 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5846_finite__ranking__induct,axiom,
    ! [S3: set_real,P: set_real > $o,F: real > num] :
      ( ( finite_finite_real @ S3 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X5: real,S4: set_real] :
              ( ( finite_finite_real @ S4 )
             => ( ! [Y5: real] :
                    ( ( member_real @ Y5 @ S4 )
                   => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_real @ X5 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5847_finite__ranking__induct,axiom,
    ! [S3: set_o,P: set_o > $o,F: $o > num] :
      ( ( finite_finite_o @ S3 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X5: $o,S4: set_o] :
              ( ( finite_finite_o @ S4 )
             => ( ! [Y5: $o] :
                    ( ( member_o @ Y5 @ S4 )
                   => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_o @ X5 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5848_finite__ranking__induct,axiom,
    ! [S3: set_nat,P: set_nat > $o,F: nat > num] :
      ( ( finite_finite_nat @ S3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X5: nat,S4: set_nat] :
              ( ( finite_finite_nat @ S4 )
             => ( ! [Y5: nat] :
                    ( ( member_nat @ Y5 @ S4 )
                   => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_nat @ X5 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5849_finite__ranking__induct,axiom,
    ! [S3: set_int,P: set_int > $o,F: int > num] :
      ( ( finite_finite_int @ S3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X5: int,S4: set_int] :
              ( ( finite_finite_int @ S4 )
             => ( ! [Y5: int] :
                    ( ( member_int @ Y5 @ S4 )
                   => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_int @ X5 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_5850_arsinh__0,axiom,
    ( ( arsinh_real @ zero_zero_real )
    = zero_zero_real ) ).

% arsinh_0
thf(fact_5851_artanh__0,axiom,
    ( ( artanh_real @ zero_zero_real )
    = zero_zero_real ) ).

% artanh_0
thf(fact_5852_ex__min__if__finite,axiom,
    ! [S3: set_o] :
      ( ( finite_finite_o @ S3 )
     => ( ( S3 != bot_bot_set_o )
       => ? [X5: $o] :
            ( ( member_o @ X5 @ S3 )
            & ~ ? [Xa: $o] :
                  ( ( member_o @ Xa @ S3 )
                  & ( ord_less_o @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_5853_ex__min__if__finite,axiom,
    ! [S3: set_real] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ? [X5: real] :
            ( ( member_real @ X5 @ S3 )
            & ~ ? [Xa: real] :
                  ( ( member_real @ Xa @ S3 )
                  & ( ord_less_real @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_5854_ex__min__if__finite,axiom,
    ! [S3: set_rat] :
      ( ( finite_finite_rat @ S3 )
     => ( ( S3 != bot_bot_set_rat )
       => ? [X5: rat] :
            ( ( member_rat @ X5 @ S3 )
            & ~ ? [Xa: rat] :
                  ( ( member_rat @ Xa @ S3 )
                  & ( ord_less_rat @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_5855_ex__min__if__finite,axiom,
    ! [S3: set_num] :
      ( ( finite_finite_num @ S3 )
     => ( ( S3 != bot_bot_set_num )
       => ? [X5: num] :
            ( ( member_num @ X5 @ S3 )
            & ~ ? [Xa: num] :
                  ( ( member_num @ Xa @ S3 )
                  & ( ord_less_num @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_5856_ex__min__if__finite,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ? [X5: nat] :
            ( ( member_nat @ X5 @ S3 )
            & ~ ? [Xa: nat] :
                  ( ( member_nat @ Xa @ S3 )
                  & ( ord_less_nat @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_5857_ex__min__if__finite,axiom,
    ! [S3: set_int] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ? [X5: int] :
            ( ( member_int @ X5 @ S3 )
            & ~ ? [Xa: int] :
                  ( ( member_int @ Xa @ S3 )
                  & ( ord_less_int @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_5858_ex__min__if__finite,axiom,
    ! [S3: set_Extended_enat] :
      ( ( finite4001608067531595151d_enat @ S3 )
     => ( ( S3 != bot_bo7653980558646680370d_enat )
       => ? [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ S3 )
            & ~ ? [Xa: extended_enat] :
                  ( ( member_Extended_enat @ Xa @ S3 )
                  & ( ord_le72135733267957522d_enat @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_5859_infinite__growing,axiom,
    ! [X9: set_o] :
      ( ( X9 != bot_bot_set_o )
     => ( ! [X5: $o] :
            ( ( member_o @ X5 @ X9 )
           => ? [Xa: $o] :
                ( ( member_o @ Xa @ X9 )
                & ( ord_less_o @ X5 @ Xa ) ) )
       => ~ ( finite_finite_o @ X9 ) ) ) ).

% infinite_growing
thf(fact_5860_infinite__growing,axiom,
    ! [X9: set_real] :
      ( ( X9 != bot_bot_set_real )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ X9 )
           => ? [Xa: real] :
                ( ( member_real @ Xa @ X9 )
                & ( ord_less_real @ X5 @ Xa ) ) )
       => ~ ( finite_finite_real @ X9 ) ) ) ).

% infinite_growing
thf(fact_5861_infinite__growing,axiom,
    ! [X9: set_rat] :
      ( ( X9 != bot_bot_set_rat )
     => ( ! [X5: rat] :
            ( ( member_rat @ X5 @ X9 )
           => ? [Xa: rat] :
                ( ( member_rat @ Xa @ X9 )
                & ( ord_less_rat @ X5 @ Xa ) ) )
       => ~ ( finite_finite_rat @ X9 ) ) ) ).

% infinite_growing
thf(fact_5862_infinite__growing,axiom,
    ! [X9: set_num] :
      ( ( X9 != bot_bot_set_num )
     => ( ! [X5: num] :
            ( ( member_num @ X5 @ X9 )
           => ? [Xa: num] :
                ( ( member_num @ Xa @ X9 )
                & ( ord_less_num @ X5 @ Xa ) ) )
       => ~ ( finite_finite_num @ X9 ) ) ) ).

% infinite_growing
thf(fact_5863_infinite__growing,axiom,
    ! [X9: set_nat] :
      ( ( X9 != bot_bot_set_nat )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ X9 )
           => ? [Xa: nat] :
                ( ( member_nat @ Xa @ X9 )
                & ( ord_less_nat @ X5 @ Xa ) ) )
       => ~ ( finite_finite_nat @ X9 ) ) ) ).

% infinite_growing
thf(fact_5864_infinite__growing,axiom,
    ! [X9: set_int] :
      ( ( X9 != bot_bot_set_int )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ X9 )
           => ? [Xa: int] :
                ( ( member_int @ Xa @ X9 )
                & ( ord_less_int @ X5 @ Xa ) ) )
       => ~ ( finite_finite_int @ X9 ) ) ) ).

% infinite_growing
thf(fact_5865_infinite__growing,axiom,
    ! [X9: set_Extended_enat] :
      ( ( X9 != bot_bo7653980558646680370d_enat )
     => ( ! [X5: extended_enat] :
            ( ( member_Extended_enat @ X5 @ X9 )
           => ? [Xa: extended_enat] :
                ( ( member_Extended_enat @ Xa @ X9 )
                & ( ord_le72135733267957522d_enat @ X5 @ Xa ) ) )
       => ~ ( finite4001608067531595151d_enat @ X9 ) ) ) ).

% infinite_growing
thf(fact_5866_artanh__def,axiom,
    ( artanh_real
    = ( ^ [X6: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X6 ) @ ( minus_minus_real @ one_one_real @ X6 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% artanh_def
thf(fact_5867_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: real > complex,Y: real > complex] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I: real] :
              ( ( member_real @ I @ I6 )
              & ( ( X @ I )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I: real] :
                ( ( member_real @ I @ I6 )
                & ( ( Y @ I )
                 != one_one_complex ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I: real] :
                ( ( member_real @ I @ I6 )
                & ( ( times_times_complex @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5868_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: nat > complex,Y: nat > complex] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I: nat] :
              ( ( member_nat @ I @ I6 )
              & ( ( X @ I )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I6 )
                & ( ( Y @ I )
                 != one_one_complex ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I6 )
                & ( ( times_times_complex @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5869_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: int > complex,Y: int > complex] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I: int] :
              ( ( member_int @ I @ I6 )
              & ( ( X @ I )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I6 )
                & ( ( Y @ I )
                 != one_one_complex ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I6 )
                & ( ( times_times_complex @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5870_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: complex > complex,Y: complex > complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I: complex] :
              ( ( member_complex @ I @ I6 )
              & ( ( X @ I )
               != one_one_complex ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I: complex] :
                ( ( member_complex @ I @ I6 )
                & ( ( Y @ I )
                 != one_one_complex ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I: complex] :
                ( ( member_complex @ I @ I6 )
                & ( ( times_times_complex @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5871_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: real > real,Y: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I: real] :
              ( ( member_real @ I @ I6 )
              & ( ( X @ I )
               != one_one_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I: real] :
                ( ( member_real @ I @ I6 )
                & ( ( Y @ I )
                 != one_one_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I: real] :
                ( ( member_real @ I @ I6 )
                & ( ( times_times_real @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5872_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: nat > real,Y: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I: nat] :
              ( ( member_nat @ I @ I6 )
              & ( ( X @ I )
               != one_one_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I6 )
                & ( ( Y @ I )
                 != one_one_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I6 )
                & ( ( times_times_real @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5873_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: int > real,Y: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I: int] :
              ( ( member_int @ I @ I6 )
              & ( ( X @ I )
               != one_one_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I6 )
                & ( ( Y @ I )
                 != one_one_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I6 )
                & ( ( times_times_real @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5874_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: complex > real,Y: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I: complex] :
              ( ( member_complex @ I @ I6 )
              & ( ( X @ I )
               != one_one_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I: complex] :
                ( ( member_complex @ I @ I6 )
                & ( ( Y @ I )
                 != one_one_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I: complex] :
                ( ( member_complex @ I @ I6 )
                & ( ( times_times_real @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5875_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: real > rat,Y: real > rat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I: real] :
              ( ( member_real @ I @ I6 )
              & ( ( X @ I )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I: real] :
                ( ( member_real @ I @ I6 )
                & ( ( Y @ I )
                 != one_one_rat ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I: real] :
                ( ( member_real @ I @ I6 )
                & ( ( times_times_rat @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5876_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: nat > rat,Y: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I: nat] :
              ( ( member_nat @ I @ I6 )
              & ( ( X @ I )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I6 )
                & ( ( Y @ I )
                 != one_one_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I6 )
                & ( ( times_times_rat @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5877_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: real > complex,Y: real > complex] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I: real] :
              ( ( member_real @ I @ I6 )
              & ( ( X @ I )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I: real] :
                ( ( member_real @ I @ I6 )
                & ( ( Y @ I )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I: real] :
                ( ( member_real @ I @ I6 )
                & ( ( plus_plus_complex @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5878_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: nat > complex,Y: nat > complex] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I: nat] :
              ( ( member_nat @ I @ I6 )
              & ( ( X @ I )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I6 )
                & ( ( Y @ I )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I6 )
                & ( ( plus_plus_complex @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5879_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: int > complex,Y: int > complex] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I: int] :
              ( ( member_int @ I @ I6 )
              & ( ( X @ I )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I6 )
                & ( ( Y @ I )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I6 )
                & ( ( plus_plus_complex @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5880_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: complex > complex,Y: complex > complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I: complex] :
              ( ( member_complex @ I @ I6 )
              & ( ( X @ I )
               != zero_zero_complex ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I: complex] :
                ( ( member_complex @ I @ I6 )
                & ( ( Y @ I )
                 != zero_zero_complex ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I: complex] :
                ( ( member_complex @ I @ I6 )
                & ( ( plus_plus_complex @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5881_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: real > real,Y: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I: real] :
              ( ( member_real @ I @ I6 )
              & ( ( X @ I )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I: real] :
                ( ( member_real @ I @ I6 )
                & ( ( Y @ I )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I: real] :
                ( ( member_real @ I @ I6 )
                & ( ( plus_plus_real @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5882_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: nat > real,Y: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I: nat] :
              ( ( member_nat @ I @ I6 )
              & ( ( X @ I )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I6 )
                & ( ( Y @ I )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I6 )
                & ( ( plus_plus_real @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5883_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: int > real,Y: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I: int] :
              ( ( member_int @ I @ I6 )
              & ( ( X @ I )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I6 )
                & ( ( Y @ I )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I6 )
                & ( ( plus_plus_real @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5884_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: complex > real,Y: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I: complex] :
              ( ( member_complex @ I @ I6 )
              & ( ( X @ I )
               != zero_zero_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I: complex] :
                ( ( member_complex @ I @ I6 )
                & ( ( Y @ I )
                 != zero_zero_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I: complex] :
                ( ( member_complex @ I @ I6 )
                & ( ( plus_plus_real @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5885_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: real > rat,Y: real > rat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I: real] :
              ( ( member_real @ I @ I6 )
              & ( ( X @ I )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I: real] :
                ( ( member_real @ I @ I6 )
                & ( ( Y @ I )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I: real] :
                ( ( member_real @ I @ I6 )
                & ( ( plus_plus_rat @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5886_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: nat > rat,Y: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I: nat] :
              ( ( member_nat @ I @ I6 )
              & ( ( X @ I )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I6 )
                & ( ( Y @ I )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I6 )
                & ( ( plus_plus_rat @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5887_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R2: int] :
      ( ( adjust_div @ ( product_Pair_int_int @ Q2 @ R2 ) )
      = ( plus_plus_int @ Q2 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).

% Divides.adjust_div_eq
thf(fact_5888_signed__take__bit__rec,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N3: nat,A3: code_integer] : ( if_Code_integer @ ( N3 = 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 @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_5889_signed__take__bit__rec,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N3: nat,A3: int] : ( if_int @ ( N3 = 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 @ N3 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_5890_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X )
        = Y )
     => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( vEBT_Leaf @ $false @ $false ) )
             => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( vEBT_Leaf @ $false @ $false ) )
               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va: nat] :
                  ( ( X
                    = ( suc @ ( suc @ Va ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                       => ( Y
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                       => ( Y
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.pelims
thf(fact_5891_option_Osize__gen_I2_J,axiom,
    ! [X: nat > nat,X2: nat] :
      ( ( size_option_nat @ X @ ( some_nat @ X2 ) )
      = ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_5892_option_Osize__gen_I2_J,axiom,
    ! [X: product_prod_nat_nat > nat,X2: product_prod_nat_nat] :
      ( ( size_o8335143837870341156at_nat @ X @ ( some_P7363390416028606310at_nat @ X2 ) )
      = ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_5893_option_Osize__gen_I2_J,axiom,
    ! [X: num > nat,X2: num] :
      ( ( size_option_num @ X @ ( some_num @ X2 ) )
      = ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_5894_verit__minus__simplify_I4_J,axiom,
    ! [B: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
      = B ) ).

% verit_minus_simplify(4)
thf(fact_5895_verit__minus__simplify_I4_J,axiom,
    ! [B: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
      = B ) ).

% verit_minus_simplify(4)
thf(fact_5896_verit__minus__simplify_I4_J,axiom,
    ! [B: complex] :
      ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ B ) )
      = B ) ).

% verit_minus_simplify(4)
thf(fact_5897_verit__minus__simplify_I4_J,axiom,
    ! [B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ B ) )
      = B ) ).

% verit_minus_simplify(4)
thf(fact_5898_verit__minus__simplify_I4_J,axiom,
    ! [B: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ B ) )
      = B ) ).

% verit_minus_simplify(4)
thf(fact_5899_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_5900_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_5901_neg__equal__iff__equal,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = ( uminus1482373934393186551omplex @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_5902_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = ( uminus1351360451143612070nteger @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_5903_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_5904_add_Oinverse__inverse,axiom,
    ! [A: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_5905_add_Oinverse__inverse,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_5906_add_Oinverse__inverse,axiom,
    ! [A: complex] :
      ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_5907_add_Oinverse__inverse,axiom,
    ! [A: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_5908_add_Oinverse__inverse,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_5909_Compl__anti__mono,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B2 ) @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).

% Compl_anti_mono
thf(fact_5910_Compl__subset__Compl__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( uminus5710092332889474511et_nat @ B2 ) )
      = ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).

% Compl_subset_Compl_iff
thf(fact_5911_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_5912_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_5913_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_5914_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_5915_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = A )
      = ( A = zero_zero_real ) ) ).

% neg_equal_zero
thf(fact_5916_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = A )
      = ( A = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_5917_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
thf(fact_5918_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = A )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_zero
thf(fact_5919_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A
        = ( uminus_uminus_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% equal_neg_zero
thf(fact_5920_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A
        = ( uminus_uminus_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_5921_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
thf(fact_5922_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A
        = ( uminus_uminus_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% equal_neg_zero
thf(fact_5923_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_5924_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_5925_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
thf(fact_5926_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
thf(fact_5927_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_5928_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_5929_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_5930_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex
        = ( uminus1482373934393186551omplex @ A ) )
      = ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_5931_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( uminus1351360451143612070nteger @ A ) )
      = ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_5932_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_5933_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_real @ zero_zero_real )
    = zero_zero_real ) ).

% add.inverse_neutral
thf(fact_5934_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_5935_add_Oinverse__neutral,axiom,
    ( ( uminus1482373934393186551omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% add.inverse_neutral
thf(fact_5936_add_Oinverse__neutral,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% add.inverse_neutral
thf(fact_5937_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% add.inverse_neutral
thf(fact_5938_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_5939_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_5940_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_5941_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_5942_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_5943_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_5944_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_5945_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_5946_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_5947_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_5948_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_5949_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_5950_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_5951_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_5952_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_5953_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_5954_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_5955_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_5956_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_5957_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_5958_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_5959_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_5960_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_5961_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_5962_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_5963_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_5964_add__minus__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_5965_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_5966_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_5967_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_5968_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_5969_minus__add__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_5970_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_5971_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_5972_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_5973_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_5974_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_5975_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_5976_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_5977_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_5978_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_5979_minus__diff__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
      = ( minus_minus_complex @ B @ A ) ) ).

% minus_diff_eq
thf(fact_5980_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( minus_8373710615458151222nteger @ B @ A ) ) ).

% minus_diff_eq
thf(fact_5981_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_5982_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_5983_div__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( divide6298287555418463151nteger @ A @ B ) ) ).

% div_minus_minus
thf(fact_5984_ln__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
          = ( ord_less_real @ X @ Y ) ) ) ) ).

% ln_less_cancel_iff
thf(fact_5985_ln__inj__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ( ln_ln_real @ X )
            = ( ln_ln_real @ Y ) )
          = ( X = Y ) ) ) ) ).

% ln_inj_iff
thf(fact_5986_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_5987_mod__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% mod_minus_minus
thf(fact_5988_Compl__disjoint,axiom,
    ! [A2: set_real] :
      ( ( inf_inf_set_real @ A2 @ ( uminus612125837232591019t_real @ A2 ) )
      = bot_bot_set_real ) ).

% Compl_disjoint
thf(fact_5989_Compl__disjoint,axiom,
    ! [A2: set_o] :
      ( ( inf_inf_set_o @ A2 @ ( uminus_uminus_set_o @ A2 ) )
      = bot_bot_set_o ) ).

% Compl_disjoint
thf(fact_5990_Compl__disjoint,axiom,
    ! [A2: set_nat] :
      ( ( inf_inf_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
      = bot_bot_set_nat ) ).

% Compl_disjoint
thf(fact_5991_Compl__disjoint,axiom,
    ! [A2: set_int] :
      ( ( inf_inf_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
      = bot_bot_set_int ) ).

% Compl_disjoint
thf(fact_5992_Compl__disjoint2,axiom,
    ! [A2: set_real] :
      ( ( inf_inf_set_real @ ( uminus612125837232591019t_real @ A2 ) @ A2 )
      = bot_bot_set_real ) ).

% Compl_disjoint2
thf(fact_5993_Compl__disjoint2,axiom,
    ! [A2: set_o] :
      ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ A2 ) @ A2 )
      = bot_bot_set_o ) ).

% Compl_disjoint2
thf(fact_5994_Compl__disjoint2,axiom,
    ! [A2: set_nat] :
      ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ A2 )
      = bot_bot_set_nat ) ).

% Compl_disjoint2
thf(fact_5995_Compl__disjoint2,axiom,
    ! [A2: set_int] :
      ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ A2 )
      = bot_bot_set_int ) ).

% Compl_disjoint2
thf(fact_5996_real__add__minus__iff,axiom,
    ! [X: real,A: real] :
      ( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
        = zero_zero_real )
      = ( X = A ) ) ).

% real_add_minus_iff
thf(fact_5997_Diff__Compl,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ B2 ) )
      = ( inf_inf_set_nat @ A2 @ B2 ) ) ).

% Diff_Compl
thf(fact_5998_Compl__Diff__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
      = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ B2 ) ) ).

% Compl_Diff_eq
thf(fact_5999_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_6000_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_6001_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_6002_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_6003_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_6004_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_eq_neg_nonpos
thf(fact_6005_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_6006_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_6007_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_6008_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_6009_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_6010_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_6011_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_6012_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_6013_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_6014_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_6015_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_6016_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_6017_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_neg_neg
thf(fact_6018_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_6019_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_6020_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_6021_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_pos
thf(fact_6022_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_6023_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_6024_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_6025_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_6026_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_6027_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_6028_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_6029_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_6030_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_6031_ab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_left_minus
thf(fact_6032_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_6033_ab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_left_minus
thf(fact_6034_ab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_left_minus
thf(fact_6035_ab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_6036_add_Oright__inverse,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
      = zero_zero_real ) ).

% add.right_inverse
thf(fact_6037_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_6038_add_Oright__inverse,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
      = zero_zero_complex ) ).

% add.right_inverse
thf(fact_6039_add_Oright__inverse,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = zero_z3403309356797280102nteger ) ).

% add.right_inverse
thf(fact_6040_add_Oright__inverse,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_6041_diff__0,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ zero_zero_real @ A )
      = ( uminus_uminus_real @ A ) ) ).

% diff_0
thf(fact_6042_diff__0,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ zero_zero_int @ A )
      = ( uminus_uminus_int @ A ) ) ).

% diff_0
thf(fact_6043_diff__0,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ A )
      = ( uminus1482373934393186551omplex @ A ) ) ).

% diff_0
thf(fact_6044_diff__0,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% diff_0
thf(fact_6045_diff__0,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ A )
      = ( uminus_uminus_rat @ A ) ) ).

% diff_0
thf(fact_6046_verit__minus__simplify_I3_J,axiom,
    ! [B: real] :
      ( ( minus_minus_real @ zero_zero_real @ B )
      = ( uminus_uminus_real @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_6047_verit__minus__simplify_I3_J,axiom,
    ! [B: int] :
      ( ( minus_minus_int @ zero_zero_int @ B )
      = ( uminus_uminus_int @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_6048_verit__minus__simplify_I3_J,axiom,
    ! [B: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ B )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_6049_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_6050_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ B )
      = ( uminus_uminus_rat @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_6051_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_6052_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_6053_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_6054_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_6055_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_6056_mult__minus1__right,axiom,
    ! [Z2: real] :
      ( ( times_times_real @ Z2 @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ Z2 ) ) ).

% mult_minus1_right
thf(fact_6057_mult__minus1__right,axiom,
    ! [Z2: int] :
      ( ( times_times_int @ Z2 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ Z2 ) ) ).

% mult_minus1_right
thf(fact_6058_mult__minus1__right,axiom,
    ! [Z2: complex] :
      ( ( times_times_complex @ Z2 @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ Z2 ) ) ).

% mult_minus1_right
thf(fact_6059_mult__minus1__right,axiom,
    ! [Z2: code_integer] :
      ( ( times_3573771949741848930nteger @ Z2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ Z2 ) ) ).

% mult_minus1_right
thf(fact_6060_mult__minus1__right,axiom,
    ! [Z2: rat] :
      ( ( times_times_rat @ Z2 @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ Z2 ) ) ).

% mult_minus1_right
thf(fact_6061_mult__minus1,axiom,
    ! [Z2: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z2 )
      = ( uminus_uminus_real @ Z2 ) ) ).

% mult_minus1
thf(fact_6062_mult__minus1,axiom,
    ! [Z2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z2 )
      = ( uminus_uminus_int @ Z2 ) ) ).

% mult_minus1
thf(fact_6063_mult__minus1,axiom,
    ! [Z2: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z2 )
      = ( uminus1482373934393186551omplex @ Z2 ) ) ).

% mult_minus1
thf(fact_6064_mult__minus1,axiom,
    ! [Z2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z2 )
      = ( uminus1351360451143612070nteger @ Z2 ) ) ).

% mult_minus1
thf(fact_6065_mult__minus1,axiom,
    ! [Z2: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z2 )
      = ( uminus_uminus_rat @ Z2 ) ) ).

% mult_minus1
thf(fact_6066_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_6067_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_6068_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_6069_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_6070_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_6071_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_6072_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_6073_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_6074_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_6075_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_6076_divide__minus1,axiom,
    ! [X: real] :
      ( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ X ) ) ).

% divide_minus1
thf(fact_6077_divide__minus1,axiom,
    ! [X: complex] :
      ( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ X ) ) ).

% divide_minus1
thf(fact_6078_divide__minus1,axiom,
    ! [X: rat] :
      ( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ X ) ) ).

% divide_minus1
thf(fact_6079_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_6080_div__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% div_minus1_right
thf(fact_6081_inf__compl__bot__left1,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X ) @ ( inf_inf_set_real @ X @ Y ) )
      = bot_bot_set_real ) ).

% inf_compl_bot_left1
thf(fact_6082_inf__compl__bot__left1,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ ( inf_inf_set_o @ X @ Y ) )
      = bot_bot_set_o ) ).

% inf_compl_bot_left1
thf(fact_6083_inf__compl__bot__left1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( inf_inf_set_nat @ X @ Y ) )
      = bot_bot_set_nat ) ).

% inf_compl_bot_left1
thf(fact_6084_inf__compl__bot__left1,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( inf_inf_set_int @ X @ Y ) )
      = bot_bot_set_int ) ).

% inf_compl_bot_left1
thf(fact_6085_inf__compl__bot__left2,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( inf_inf_set_real @ X @ ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X ) @ Y ) )
      = bot_bot_set_real ) ).

% inf_compl_bot_left2
thf(fact_6086_inf__compl__bot__left2,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( inf_inf_set_o @ X @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ Y ) )
      = bot_bot_set_o ) ).

% inf_compl_bot_left2
thf(fact_6087_inf__compl__bot__left2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) )
      = bot_bot_set_nat ) ).

% inf_compl_bot_left2
thf(fact_6088_inf__compl__bot__left2,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( inf_inf_set_int @ X @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y ) )
      = bot_bot_set_int ) ).

% inf_compl_bot_left2
thf(fact_6089_inf__compl__bot__right,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( inf_inf_set_real @ X @ ( inf_inf_set_real @ Y @ ( uminus612125837232591019t_real @ X ) ) )
      = bot_bot_set_real ) ).

% inf_compl_bot_right
thf(fact_6090_inf__compl__bot__right,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( inf_inf_set_o @ X @ ( inf_inf_set_o @ Y @ ( uminus_uminus_set_o @ X ) ) )
      = bot_bot_set_o ) ).

% inf_compl_bot_right
thf(fact_6091_inf__compl__bot__right,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) ) )
      = bot_bot_set_nat ) ).

% inf_compl_bot_right
thf(fact_6092_inf__compl__bot__right,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( inf_inf_set_int @ X @ ( inf_inf_set_int @ Y @ ( uminus1532241313380277803et_int @ X ) ) )
      = bot_bot_set_int ) ).

% inf_compl_bot_right
thf(fact_6093_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_real] :
      ( ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X ) @ X )
      = bot_bot_set_real ) ).

% boolean_algebra.conj_cancel_left
thf(fact_6094_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ X )
      = bot_bot_set_o ) ).

% boolean_algebra.conj_cancel_left
thf(fact_6095_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ X )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_cancel_left
thf(fact_6096_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_int] :
      ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ X )
      = bot_bot_set_int ) ).

% boolean_algebra.conj_cancel_left
thf(fact_6097_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_real] :
      ( ( inf_inf_set_real @ X @ ( uminus612125837232591019t_real @ X ) )
      = bot_bot_set_real ) ).

% boolean_algebra.conj_cancel_right
thf(fact_6098_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ X @ ( uminus_uminus_set_o @ X ) )
      = bot_bot_set_o ) ).

% boolean_algebra.conj_cancel_right
thf(fact_6099_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( uminus5710092332889474511et_nat @ X ) )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_cancel_right
thf(fact_6100_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_int] :
      ( ( inf_inf_set_int @ X @ ( uminus1532241313380277803et_int @ X ) )
      = bot_bot_set_int ) ).

% boolean_algebra.conj_cancel_right
thf(fact_6101_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_6102_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_6103_subset__Compl__singleton,axiom,
    ! [A2: set_complex,B: complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ ( uminus8566677241136511917omplex @ ( insert_complex @ B @ bot_bot_set_complex ) ) )
      = ( ~ ( member_complex @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_6104_subset__Compl__singleton,axiom,
    ! [A2: set_set_nat,B: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) )
      = ( ~ ( member_set_nat @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_6105_subset__Compl__singleton,axiom,
    ! [A2: set_real,B: real] :
      ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ ( insert_real @ B @ bot_bot_set_real ) ) )
      = ( ~ ( member_real @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_6106_subset__Compl__singleton,axiom,
    ! [A2: set_o,B: $o] :
      ( ( ord_less_eq_set_o @ A2 @ ( uminus_uminus_set_o @ ( insert_o @ B @ bot_bot_set_o ) ) )
      = ( ~ ( member_o @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_6107_subset__Compl__singleton,axiom,
    ! [A2: set_int,B: int] :
      ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
      = ( ~ ( member_int @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_6108_subset__Compl__singleton,axiom,
    ! [A2: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
      = ( ~ ( member_nat @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_6109_ln__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
          = ( ord_less_eq_real @ X @ Y ) ) ) ) ).

% ln_le_cancel_iff
thf(fact_6110_ln__less__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
        = ( ord_less_real @ X @ one_one_real ) ) ) ).

% ln_less_zero_iff
thf(fact_6111_ln__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
        = ( ord_less_real @ one_one_real @ X ) ) ) ).

% ln_gt_zero_iff
thf(fact_6112_ln__eq__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ln_ln_real @ X )
          = zero_zero_real )
        = ( X = one_one_real ) ) ) ).

% ln_eq_zero_iff
thf(fact_6113_ln__one,axiom,
    ( ( ln_ln_real @ one_one_real )
    = zero_zero_real ) ).

% ln_one
thf(fact_6114_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] :
      ( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% signed_take_bit_of_minus_1
thf(fact_6115_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] :
      ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% signed_take_bit_of_minus_1
thf(fact_6116_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_6117_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_6118_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_6119_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_6120_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_6121_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_6122_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_6123_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_6124_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_6125_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_6126_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_6127_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_6128_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_6129_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_6130_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_6131_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_6132_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_6133_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_6134_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_6135_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_6136_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_6137_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_6138_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_6139_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_6140_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_6141_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_real @ one_one_real )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_6142_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_int @ one_one_int )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_6143_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1482373934393186551omplex @ one_one_complex )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_6144_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_6145_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_rat @ one_one_rat )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_6146_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
      = one_one_real ) ).

% minus_one_mult_self
thf(fact_6147_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
      = one_one_int ) ).

% minus_one_mult_self
thf(fact_6148_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
      = one_one_complex ) ).

% minus_one_mult_self
thf(fact_6149_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
      = one_one_Code_integer ) ).

% minus_one_mult_self
thf(fact_6150_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
      = one_one_rat ) ).

% minus_one_mult_self
thf(fact_6151_left__minus__one__mult__self,axiom,
    ! [N: nat,A: real] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_6152_left__minus__one__mult__self,axiom,
    ! [N: nat,A: int] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_6153_left__minus__one__mult__self,axiom,
    ! [N: nat,A: complex] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_6154_left__minus__one__mult__self,axiom,
    ! [N: nat,A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_6155_left__minus__one__mult__self,axiom,
    ! [N: nat,A: rat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_6156_mod__minus1__right,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_6157_mod__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = zero_z3403309356797280102nteger ) ).

% mod_minus1_right
thf(fact_6158_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
      & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_6159_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_6160_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
      & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_6161_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_6162_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
          = ( numeral_numeral_real @ V ) ) )
      & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_6163_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
          = ( numera6620942414471956472nteger @ V ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_6164_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
          = ( numeral_numeral_rat @ V ) ) )
      & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_6165_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
          = ( numeral_numeral_int @ V ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_6166_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
      & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
          = ( numeral_numeral_real @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_6167_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
          = ( numera6620942414471956472nteger @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_6168_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
      & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
          = ( numeral_numeral_rat @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_6169_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
          = ( numeral_numeral_int @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_6170_ln__le__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
        = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).

% ln_le_zero_iff
thf(fact_6171_ln__ge__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
        = ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% ln_ge_zero_iff
thf(fact_6172_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_6173_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_6174_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_6175_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_6176_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_6177_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_6178_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_6179_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_6180_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_6181_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_6182_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_6183_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_6184_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_6185_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_6186_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_6187_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_6188_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_6189_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_6190_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_6191_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_6192_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_6193_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_6194_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_6195_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_6196_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_6197_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_6198_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_6199_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_6200_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_6201_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_6202_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_6203_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_6204_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_6205_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_6206_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_6207_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_6208_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_6209_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_6210_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_6211_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_6212_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_6213_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_6214_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_6215_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_6216_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_6217_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( ord_less_eq_num @ N @ M ) ) ).

% neg_numeral_le_iff
thf(fact_6218_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( ord_less_eq_num @ N @ M ) ) ).

% neg_numeral_le_iff
thf(fact_6219_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( ord_less_eq_num @ N @ M ) ) ).

% neg_numeral_le_iff
thf(fact_6220_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( ord_less_eq_num @ N @ M ) ) ).

% neg_numeral_le_iff
thf(fact_6221_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( ord_less_num @ N @ M ) ) ).

% neg_numeral_less_iff
thf(fact_6222_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( ord_less_num @ N @ M ) ) ).

% neg_numeral_less_iff
thf(fact_6223_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( ord_less_num @ N @ M ) ) ).

% neg_numeral_less_iff
thf(fact_6224_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( ord_less_num @ N @ M ) ) ).

% neg_numeral_less_iff
thf(fact_6225_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_6226_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_6227_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_6228_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_6229_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_6230_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_6231_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_6232_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_6233_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_6234_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_6235_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_6236_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_6237_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_6238_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_6239_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_6240_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_6241_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_6242_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_6243_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_6244_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_6245_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_6246_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_6247_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_6248_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_6249_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_6250_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_6251_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_6252_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_6253_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_6254_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_6255_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_6256_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_6257_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_6258_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_6259_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_6260_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_6261_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_6262_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_6263_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_6264_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_6265_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_6266_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_6267_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_6268_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_6269_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_6270_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_6271_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_6272_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_6273_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_6274_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_6275_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_6276_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_6277_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_6278_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
        = ( power_power_real @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_6279_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
        = ( power_power_int @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_6280_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: complex] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
        = ( power_power_complex @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_6281_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
        = ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_6282_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
        = ( power_power_rat @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_6283_power__minus__odd,axiom,
    ! [N: nat,A: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
        = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_6284_power__minus__odd,axiom,
    ! [N: nat,A: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
        = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_6285_power__minus__odd,axiom,
    ! [N: nat,A: complex] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
        = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_6286_power__minus__odd,axiom,
    ! [N: nat,A: code_integer] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
        = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_6287_power__minus__odd,axiom,
    ! [N: nat,A: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
        = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_6288_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_6289_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_6290_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_6291_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_6292_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_6293_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_6294_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_6295_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_6296_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_6297_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_6298_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_minus_bit0
thf(fact_6299_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_6300_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_6301_dbl__simps_I4_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_6302_dbl__simps_I4_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_6303_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_6304_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_real ) ).

% power_minus1_even
thf(fact_6305_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_int ) ).

% power_minus1_even
thf(fact_6306_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_complex ) ).

% power_minus1_even
thf(fact_6307_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_Code_integer ) ).

% power_minus1_even
thf(fact_6308_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_rat ) ).

% power_minus1_even
thf(fact_6309_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
        = one_one_real ) ) ).

% neg_one_even_power
thf(fact_6310_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
        = one_one_int ) ) ).

% neg_one_even_power
thf(fact_6311_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
        = one_one_complex ) ) ).

% neg_one_even_power
thf(fact_6312_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
        = one_one_Code_integer ) ) ).

% neg_one_even_power
thf(fact_6313_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
        = one_one_rat ) ) ).

% neg_one_even_power
thf(fact_6314_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
        = ( uminus_uminus_real @ one_one_real ) ) ) ).

% neg_one_odd_power
thf(fact_6315_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% neg_one_odd_power
thf(fact_6316_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
        = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).

% neg_one_odd_power
thf(fact_6317_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).

% neg_one_odd_power
thf(fact_6318_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
        = ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% neg_one_odd_power
thf(fact_6319_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_6320_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_6321_verit__negate__coefficient_I3_J,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
     => ( ( uminus_uminus_real @ A )
        = ( uminus_uminus_real @ B ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_6322_verit__negate__coefficient_I3_J,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( ( uminus_uminus_int @ A )
        = ( uminus_uminus_int @ B ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_6323_verit__negate__coefficient_I3_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
     => ( ( uminus1351360451143612070nteger @ A )
        = ( uminus1351360451143612070nteger @ B ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_6324_verit__negate__coefficient_I3_J,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
     => ( ( uminus_uminus_rat @ A )
        = ( uminus_uminus_rat @ B ) ) ) ).

% verit_negate_coefficient(3)
thf(fact_6325_minus__equation__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = B )
      = ( ( uminus_uminus_real @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_6326_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( uminus_uminus_int @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_6327_minus__equation__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = B )
      = ( ( uminus1482373934393186551omplex @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_6328_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B )
      = ( ( uminus1351360451143612070nteger @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_6329_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( uminus_uminus_rat @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_6330_equation__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( uminus_uminus_real @ B ) )
      = ( B
        = ( uminus_uminus_real @ A ) ) ) ).

% equation_minus_iff
thf(fact_6331_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% equation_minus_iff
thf(fact_6332_equation__minus__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ B ) )
      = ( B
        = ( uminus1482373934393186551omplex @ A ) ) ) ).

% equation_minus_iff
thf(fact_6333_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B ) )
      = ( B
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% equation_minus_iff
thf(fact_6334_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% equation_minus_iff
thf(fact_6335_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_6336_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_6337_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_6338_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_6339_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_6340_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_6341_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_6342_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_6343_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_6344_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_6345_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_6346_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_6347_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_6348_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_6349_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_6350_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_6351_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_6352_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_6353_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_6354_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_6355_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_6356_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_6357_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_6358_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_6359_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_real @ M )
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_6360_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_int @ M )
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_6361_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numera6690914467698888265omplex @ M )
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_6362_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numera6620942414471956472nteger @ M )
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_6363_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_rat @ M )
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_6364_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
     != ( numeral_numeral_real @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_6365_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
     != ( numeral_numeral_int @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_6366_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
     != ( numera6690914467698888265omplex @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_6367_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
     != ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_6368_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
     != ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_6369_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_6370_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_6371_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_6372_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_6373_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_6374_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_6375_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_6376_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_6377_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_6378_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_6379_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_6380_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_6381_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_6382_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_6383_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_6384_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_6385_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_6386_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_6387_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_6388_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_6389_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_6390_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_6391_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_6392_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_6393_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_6394_one__neq__neg__one,axiom,
    ( one_one_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% one_neq_neg_one
thf(fact_6395_one__neq__neg__one,axiom,
    ( one_one_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% one_neq_neg_one
thf(fact_6396_one__neq__neg__one,axiom,
    ( one_one_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% one_neq_neg_one
thf(fact_6397_one__neq__neg__one,axiom,
    ( one_one_Code_integer
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% one_neq_neg_one
thf(fact_6398_one__neq__neg__one,axiom,
    ( one_one_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% one_neq_neg_one
thf(fact_6399_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_6400_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_6401_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_6402_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_6403_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_6404_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_6405_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_6406_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_6407_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_6408_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_6409_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_6410_minus__divide__right,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_divide_right
thf(fact_6411_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_6412_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_6413_minus__divide__divide,axiom,
    ! [A: complex,B: complex] :
      ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
      = ( divide1717551699836669952omplex @ A @ B ) ) ).

% minus_divide_divide
thf(fact_6414_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_6415_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_6416_minus__divide__left,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).

% minus_divide_left
thf(fact_6417_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_6418_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_6419_div__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% div_minus_right
thf(fact_6420_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_6421_mod__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% mod_minus_right
thf(fact_6422_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_6423_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_6424_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_6425_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_6426_uminus__int__code_I1_J,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% uminus_int_code(1)
thf(fact_6427_signed__take__bit__minus,axiom,
    ! [N: nat,K: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N @ K ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).

% signed_take_bit_minus
thf(fact_6428_Collect__imp__eq,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ( collect_real
        @ ^ [X6: real] :
            ( ( P @ X6 )
           => ( Q @ X6 ) ) )
      = ( sup_sup_set_real @ ( uminus612125837232591019t_real @ ( collect_real @ P ) ) @ ( collect_real @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_6429_Collect__imp__eq,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X6: list_nat] :
            ( ( P @ X6 )
           => ( Q @ X6 ) ) )
      = ( sup_sup_set_list_nat @ ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P ) ) @ ( collect_list_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_6430_Collect__imp__eq,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X6: set_nat] :
            ( ( P @ X6 )
           => ( Q @ X6 ) ) )
      = ( sup_sup_set_set_nat @ ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_6431_Collect__imp__eq,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( collect_int
        @ ^ [X6: int] :
            ( ( P @ X6 )
           => ( Q @ X6 ) ) )
      = ( sup_sup_set_int @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) @ ( collect_int @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_6432_Collect__imp__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X6: nat] :
            ( ( P @ X6 )
           => ( Q @ X6 ) ) )
      = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) @ ( collect_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_6433_ln__less__self,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).

% ln_less_self
thf(fact_6434_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_6435_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_6436_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_6437_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_6438_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_6439_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_6440_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_6441_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_6442_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_6443_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_6444_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_6445_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_z3403309356797280102nteger
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_6446_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_6447_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_6448_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_6449_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_6450_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_6451_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_6452_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_6453_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_6454_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_6455_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_6456_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_6457_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_6458_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_6459_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_6460_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_6461_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_6462_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_6463_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_6464_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_6465_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_6466_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_6467_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_6468_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_6469_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_6470_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_6471_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_6472_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_6473_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_6474_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_6475_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_6476_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( ( uminus1351360451143612070nteger @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_6477_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_6478_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_6479_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_6480_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_6481_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_6482_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_6483_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_6484_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_6485_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_6486_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_6487_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_6488_zero__neq__neg__one,axiom,
    ( zero_zero_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% zero_neq_neg_one
thf(fact_6489_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_6490_zero__neq__neg__one,axiom,
    ( zero_zero_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% zero_neq_neg_one
thf(fact_6491_zero__neq__neg__one,axiom,
    ( zero_z3403309356797280102nteger
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% zero_neq_neg_one
thf(fact_6492_zero__neq__neg__one,axiom,
    ( zero_zero_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% zero_neq_neg_one
thf(fact_6493_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_6494_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_6495_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_6496_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_6497_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_6498_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_6499_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_6500_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_6501_numeral__times__minus__swap,axiom,
    ! [W: num,X: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X ) )
      = ( times_times_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_6502_numeral__times__minus__swap,axiom,
    ! [W: num,X: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X ) )
      = ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_6503_numeral__times__minus__swap,axiom,
    ! [W: num,X: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X ) )
      = ( times_times_complex @ X @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_6504_numeral__times__minus__swap,axiom,
    ! [W: num,X: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X ) )
      = ( times_3573771949741848930nteger @ X @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_6505_numeral__times__minus__swap,axiom,
    ! [W: num,X: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X ) )
      = ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_6506_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_6507_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_6508_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_6509_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_6510_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_6511_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_6512_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ N )
     != ( uminus_uminus_real @ one_one_real ) ) ).

% numeral_neq_neg_one
thf(fact_6513_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ N )
     != ( uminus_uminus_int @ one_one_int ) ) ).

% numeral_neq_neg_one
thf(fact_6514_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ N )
     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% numeral_neq_neg_one
thf(fact_6515_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger @ N )
     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% numeral_neq_neg_one
thf(fact_6516_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ N )
     != ( uminus_uminus_rat @ one_one_rat ) ) ).

% numeral_neq_neg_one
thf(fact_6517_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_6518_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_6519_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_6520_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_Code_integer
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_6521_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_6522_square__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( times_times_real @ X @ X )
        = one_one_real )
      = ( ( X = one_one_real )
        | ( X
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% square_eq_1_iff
thf(fact_6523_square__eq__1__iff,axiom,
    ! [X: int] :
      ( ( ( times_times_int @ X @ X )
        = one_one_int )
      = ( ( X = one_one_int )
        | ( X
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% square_eq_1_iff
thf(fact_6524_square__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( ( times_times_complex @ X @ X )
        = one_one_complex )
      = ( ( X = one_one_complex )
        | ( X
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% square_eq_1_iff
thf(fact_6525_square__eq__1__iff,axiom,
    ! [X: code_integer] :
      ( ( ( times_3573771949741848930nteger @ X @ X )
        = one_one_Code_integer )
      = ( ( X = one_one_Code_integer )
        | ( X
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% square_eq_1_iff
thf(fact_6526_square__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( ( times_times_rat @ X @ X )
        = one_one_rat )
      = ( ( X = one_one_rat )
        | ( X
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% square_eq_1_iff
thf(fact_6527_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A3: real,B3: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_6528_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A3: int,B3: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B3 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_6529_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A3: complex,B3: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_6530_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A3: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_6531_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A3: rat,B3: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_6532_diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A3: real,B3: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_6533_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A3: int,B3: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B3 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_6534_diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A3: complex,B3: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_6535_diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A3: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_6536_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A3: rat,B3: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_6537_group__cancel_Osub2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2
        = ( plus_plus_real @ K @ B ) )
     => ( ( minus_minus_real @ A @ B2 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_6538_group__cancel_Osub2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2
        = ( plus_plus_int @ K @ B ) )
     => ( ( minus_minus_int @ A @ B2 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_6539_group__cancel_Osub2,axiom,
    ! [B2: complex,K: complex,B: complex,A: complex] :
      ( ( B2
        = ( plus_plus_complex @ K @ B ) )
     => ( ( minus_minus_complex @ A @ B2 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_6540_group__cancel_Osub2,axiom,
    ! [B2: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B2
        = ( plus_p5714425477246183910nteger @ K @ B ) )
     => ( ( minus_8373710615458151222nteger @ A @ B2 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_6541_group__cancel_Osub2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2
        = ( plus_plus_rat @ K @ B ) )
     => ( ( minus_minus_rat @ A @ B2 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_6542_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_6543_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_6544_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_6545_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_6546_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_6547_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_6548_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_6549_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_6550_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_6551_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_6552_inf__cancel__left1,axiom,
    ! [X: set_real,A: set_real,B: set_real] :
      ( ( inf_inf_set_real @ ( inf_inf_set_real @ X @ A ) @ ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X ) @ B ) )
      = bot_bot_set_real ) ).

% inf_cancel_left1
thf(fact_6553_inf__cancel__left1,axiom,
    ! [X: set_o,A: set_o,B: set_o] :
      ( ( inf_inf_set_o @ ( inf_inf_set_o @ X @ A ) @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ B ) )
      = bot_bot_set_o ) ).

% inf_cancel_left1
thf(fact_6554_inf__cancel__left1,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ A ) @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ B ) )
      = bot_bot_set_nat ) ).

% inf_cancel_left1
thf(fact_6555_inf__cancel__left1,axiom,
    ! [X: set_int,A: set_int,B: set_int] :
      ( ( inf_inf_set_int @ ( inf_inf_set_int @ X @ A ) @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ B ) )
      = bot_bot_set_int ) ).

% inf_cancel_left1
thf(fact_6556_inf__cancel__left2,axiom,
    ! [X: set_real,A: set_real,B: set_real] :
      ( ( inf_inf_set_real @ ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X ) @ A ) @ ( inf_inf_set_real @ X @ B ) )
      = bot_bot_set_real ) ).

% inf_cancel_left2
thf(fact_6557_inf__cancel__left2,axiom,
    ! [X: set_o,A: set_o,B: set_o] :
      ( ( inf_inf_set_o @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X ) @ A ) @ ( inf_inf_set_o @ X @ B ) )
      = bot_bot_set_o ) ).

% inf_cancel_left2
thf(fact_6558_inf__cancel__left2,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ A ) @ ( inf_inf_set_nat @ X @ B ) )
      = bot_bot_set_nat ) ).

% inf_cancel_left2
thf(fact_6559_inf__cancel__left2,axiom,
    ! [X: set_int,A: set_int,B: set_int] :
      ( ( inf_inf_set_int @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X ) @ A ) @ ( inf_inf_set_int @ X @ B ) )
      = bot_bot_set_int ) ).

% inf_cancel_left2
thf(fact_6560_subset__Compl__self__eq,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ A2 ) )
      = ( A2 = bot_bot_set_real ) ) ).

% subset_Compl_self_eq
thf(fact_6561_subset__Compl__self__eq,axiom,
    ! [A2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ ( uminus_uminus_set_o @ A2 ) )
      = ( A2 = bot_bot_set_o ) ) ).

% subset_Compl_self_eq
thf(fact_6562_subset__Compl__self__eq,axiom,
    ! [A2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
      = ( A2 = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
thf(fact_6563_subset__Compl__self__eq,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
      = ( A2 = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
thf(fact_6564_real__minus__mult__self__le,axiom,
    ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).

% real_minus_mult_self_le
thf(fact_6565_Compl__Un,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( sup_sup_set_nat @ A2 @ B2 ) )
      = ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( uminus5710092332889474511et_nat @ B2 ) ) ) ).

% Compl_Un
thf(fact_6566_Compl__Int,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( inf_inf_set_nat @ A2 @ B2 ) )
      = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( uminus5710092332889474511et_nat @ B2 ) ) ) ).

% Compl_Int
thf(fact_6567_Diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] : ( inf_inf_set_nat @ A6 @ ( uminus5710092332889474511et_nat @ B7 ) ) ) ) ).

% Diff_eq
thf(fact_6568_zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( ( times_times_int @ M @ N )
        = one_one_int )
      = ( ( ( M = one_one_int )
          & ( N = one_one_int ) )
        | ( ( M
            = ( uminus_uminus_int @ one_one_int ) )
          & ( N
            = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zmult_eq_1_iff
thf(fact_6569_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N: int] :
      ( ( ( times_times_int @ M @ N )
        = one_one_int )
     => ( ( M = one_one_int )
        | ( M
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
thf(fact_6570_minus__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( minus_minus_int @ zero_zero_int @ L )
      = ( uminus_uminus_int @ L ) ) ).

% minus_int_code(2)
thf(fact_6571_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_6572_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_6573_minus__real__def,axiom,
    ( minus_minus_real
    = ( ^ [X6: real,Y6: real] : ( plus_plus_real @ X6 @ ( uminus_uminus_real @ Y6 ) ) ) ) ).

% minus_real_def
thf(fact_6574_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).

% ln_one_minus_pos_upper_bound
thf(fact_6575_ln__bound,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).

% ln_bound
thf(fact_6576_ln__gt__zero__imp__gt__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_real @ one_one_real @ X ) ) ) ).

% ln_gt_zero_imp_gt_one
thf(fact_6577_ln__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).

% ln_less_zero
thf(fact_6578_ln__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).

% ln_gt_zero
thf(fact_6579_ln__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).

% ln_ge_zero
thf(fact_6580_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).

% neg_numeral_le_zero
thf(fact_6581_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_le_zero
thf(fact_6582_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).

% neg_numeral_le_zero
thf(fact_6583_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).

% neg_numeral_le_zero
thf(fact_6584_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_6585_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_6586_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_6587_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_6588_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_6589_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_6590_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_6591_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_6592_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).

% neg_numeral_less_zero
thf(fact_6593_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).

% neg_numeral_less_zero
thf(fact_6594_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_less_zero
thf(fact_6595_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).

% neg_numeral_less_zero
thf(fact_6596_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_6597_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(3)
thf(fact_6598_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_6599_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_6600_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_6601_le__minus__one__simps_I1_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% le_minus_one_simps(1)
thf(fact_6602_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_6603_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_6604_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_6605_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_6606_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(3)
thf(fact_6607_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_6608_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_6609_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_6610_less__minus__one__simps_I1_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% less_minus_one_simps(1)
thf(fact_6611_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_6612_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_6613_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_6614_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_6615_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_6616_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_6617_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_le_neg_one
thf(fact_6618_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_6619_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_6620_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_6621_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_6622_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_6623_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_6624_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_6625_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_le_numeral
thf(fact_6626_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_6627_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_6628_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_6629_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_le_one
thf(fact_6630_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_6631_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_6632_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_6633_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_6634_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_6635_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_6636_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_6637_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_6638_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_6639_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_6640_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_6641_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_6642_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_less_neg_one
thf(fact_6643_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_6644_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_6645_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_6646_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_less_numeral
thf(fact_6647_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_6648_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_6649_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_6650_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_less_one
thf(fact_6651_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_6652_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_6653_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_6654_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_6655_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_6656_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_6657_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_6658_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_6659_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_6660_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_6661_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_6662_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_6663_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_6664_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_6665_mult__1s__ring__1_I1_J,axiom,
    ! [B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
      = ( uminus_uminus_int @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_6666_mult__1s__ring__1_I1_J,axiom,
    ! [B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_6667_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_6668_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_6669_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_6670_mult__1s__ring__1_I2_J,axiom,
    ! [B: int] :
      ( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
      = ( uminus_uminus_int @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_6671_mult__1s__ring__1_I2_J,axiom,
    ! [B: complex] :
      ( ( times_times_complex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_6672_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_6673_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_6674_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_6675_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_6676_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_6677_uminus__numeral__One,axiom,
    ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% uminus_numeral_One
thf(fact_6678_uminus__numeral__One,axiom,
    ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% uminus_numeral_One
thf(fact_6679_uminus__numeral__One,axiom,
    ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% uminus_numeral_One
thf(fact_6680_uminus__numeral__One,axiom,
    ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% uminus_numeral_One
thf(fact_6681_uminus__numeral__One,axiom,
    ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% uminus_numeral_One
thf(fact_6682_power__minus,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).

% power_minus
thf(fact_6683_power__minus,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).

% power_minus
thf(fact_6684_power__minus,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).

% power_minus
thf(fact_6685_power__minus,axiom,
    ! [A: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% power_minus
thf(fact_6686_power__minus,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).

% power_minus
thf(fact_6687_inf__shunt,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( ( inf_inf_set_real @ X @ Y )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ X @ ( uminus612125837232591019t_real @ Y ) ) ) ).

% inf_shunt
thf(fact_6688_inf__shunt,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( ( inf_inf_set_o @ X @ Y )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ X @ ( uminus_uminus_set_o @ Y ) ) ) ).

% inf_shunt
thf(fact_6689_inf__shunt,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ( inf_inf_set_int @ X @ Y )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ X @ ( uminus1532241313380277803et_int @ Y ) ) ) ).

% inf_shunt
thf(fact_6690_inf__shunt,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( inf_inf_set_nat @ X @ Y )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).

% inf_shunt
thf(fact_6691_power__minus__Bit0,axiom,
    ! [X: real,K: num] :
      ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_6692_power__minus__Bit0,axiom,
    ! [X: int,K: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_6693_power__minus__Bit0,axiom,
    ! [X: complex,K: num] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_6694_power__minus__Bit0,axiom,
    ! [X: code_integer,K: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_6695_power__minus__Bit0,axiom,
    ! [X: rat,K: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_6696_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( ( inf_inf_set_real @ A2 @ B2 )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ B2 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_6697_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_o,B2: set_o] :
      ( ( ( inf_inf_set_o @ A2 @ B2 )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ A2 @ ( uminus_uminus_set_o @ B2 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_6698_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ( inf_inf_set_int @ A2 @ B2 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ B2 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_6699_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ( inf_inf_set_nat @ A2 @ B2 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ B2 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_6700_Compl__insert,axiom,
    ! [X: real,A2: set_real] :
      ( ( uminus612125837232591019t_real @ ( insert_real @ X @ A2 ) )
      = ( minus_minus_set_real @ ( uminus612125837232591019t_real @ A2 ) @ ( insert_real @ X @ bot_bot_set_real ) ) ) ).

% Compl_insert
thf(fact_6701_Compl__insert,axiom,
    ! [X: $o,A2: set_o] :
      ( ( uminus_uminus_set_o @ ( insert_o @ X @ A2 ) )
      = ( minus_minus_set_o @ ( uminus_uminus_set_o @ A2 ) @ ( insert_o @ X @ bot_bot_set_o ) ) ) ).

% Compl_insert
thf(fact_6702_Compl__insert,axiom,
    ! [X: int,A2: set_int] :
      ( ( uminus1532241313380277803et_int @ ( insert_int @ X @ A2 ) )
      = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( insert_int @ X @ bot_bot_set_int ) ) ) ).

% Compl_insert
thf(fact_6703_Compl__insert,axiom,
    ! [X: nat,A2: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( insert_nat @ X @ A2 ) )
      = ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% Compl_insert
thf(fact_6704_real__add__less__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
      = ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).

% real_add_less_0_iff
thf(fact_6705_real__0__less__add__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
      = ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).

% real_0_less_add_iff
thf(fact_6706_real__0__le__add__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).

% real_0_le_add_iff
thf(fact_6707_real__add__le__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
      = ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).

% real_add_le_0_iff
thf(fact_6708_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_6709_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_6710_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% ln_ge_zero_imp_ge_one
thf(fact_6711_ln__add__one__self__le__self,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).

% ln_add_one_self_le_self
thf(fact_6712_ln__mult,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ln_ln_real @ ( times_times_real @ X @ Y ) )
          = ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).

% ln_mult
thf(fact_6713_ln__eq__minus__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ln_ln_real @ X )
          = ( minus_minus_real @ X @ one_one_real ) )
       => ( X = one_one_real ) ) ) ).

% ln_eq_minus_one
thf(fact_6714_ln__div,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ln_ln_real @ ( divide_divide_real @ X @ Y ) )
          = ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).

% ln_div
thf(fact_6715_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_6716_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_6717_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_6718_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_6719_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_6720_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_6721_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_6722_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_6723_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_6724_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_6725_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_6726_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_6727_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_6728_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_6729_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_6730_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_6731_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_6732_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_6733_minus__divide__add__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z2 ) ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_6734_minus__divide__add__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z2 ) ) @ Y )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_6735_minus__divide__add__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z2 ) ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_6736_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
          = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_6737_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_6738_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_6739_minus__divide__diff__eq__iff,axiom,
    ! [Z2: real,X: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z2 ) ) @ Y )
        = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_6740_minus__divide__diff__eq__iff,axiom,
    ! [Z2: complex,X: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z2 ) ) @ Y )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_6741_minus__divide__diff__eq__iff,axiom,
    ! [Z2: rat,X: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z2 ) ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_6742_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
          = ( uminus_uminus_real @ B ) ) )
      & ( ( Z2 != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
          = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_6743_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
          = ( uminus1482373934393186551omplex @ B ) ) )
      & ( ( Z2 != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_6744_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
          = ( uminus_uminus_rat @ B ) ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
          = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_6745_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
          = ( uminus_uminus_real @ B ) ) )
      & ( ( Z2 != zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
          = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_6746_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
          = ( uminus1482373934393186551omplex @ B ) ) )
      & ( ( Z2 != zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_6747_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
          = ( uminus_uminus_rat @ B ) ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_6748_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_6749_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_6750_power2__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_real @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_6751_power2__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_int @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_6752_power2__eq__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus1482373934393186551omplex @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_6753_power2__eq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus1351360451143612070nteger @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_6754_power2__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_rat @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_6755_verit__less__mono__div__int2,axiom,
    ! [A2: int,B2: int,N: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
       => ( ord_less_eq_int @ ( divide_divide_int @ B2 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).

% verit_less_mono_div_int2
thf(fact_6756_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_6757_ln__le__minus__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).

% ln_le_minus_one
thf(fact_6758_ln__diff__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y ) @ Y ) ) ) ) ).

% ln_diff_le
thf(fact_6759_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_6760_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_6761_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_6762_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_6763_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_6764_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_6765_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_6766_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_6767_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_6768_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_6769_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_6770_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_6771_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_6772_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_6773_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_6774_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_6775_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_6776_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_6777_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_6778_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_6779_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_6780_uminus__power__if,axiom,
    ! [N: nat,A: real] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
          = ( power_power_real @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
          = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_6781_uminus__power__if,axiom,
    ! [N: nat,A: int] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
          = ( power_power_int @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
          = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_6782_uminus__power__if,axiom,
    ! [N: nat,A: complex] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
          = ( power_power_complex @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
          = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_6783_uminus__power__if,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
          = ( power_8256067586552552935nteger @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
          = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_6784_uminus__power__if,axiom,
    ! [N: nat,A: rat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
          = ( power_power_rat @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
          = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_6785_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_6786_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_6787_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_6788_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_6789_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_6790_realpow__square__minus__le,axiom,
    ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% realpow_square_minus_le
thf(fact_6791_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% ln_one_minus_pos_lower_bound
thf(fact_6792_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_6793_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_6794_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_6795_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_6796_zminus1__lemma,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R2 ) )
     => ( ( B != zero_zero_int )
       => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R2 = zero_zero_int ) @ ( uminus_uminus_int @ Q2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q2 ) @ one_one_int ) ) @ ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R2 ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_6797_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_6798_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_6799_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_6800_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_6801_square__le__1,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).

% square_le_1
thf(fact_6802_square__le__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
     => ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% square_le_1
thf(fact_6803_square__le__1,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
     => ( ( ord_less_eq_rat @ X @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).

% square_le_1
thf(fact_6804_square__le__1,axiom,
    ! [X: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
     => ( ( ord_less_eq_int @ X @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% square_le_1
thf(fact_6805_minus__power__mult__self,axiom,
    ! [A: real,N: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
      = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_6806_minus__power__mult__self,axiom,
    ! [A: int,N: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
      = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_6807_minus__power__mult__self,axiom,
    ! [A: complex,N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) )
      = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_6808_minus__power__mult__self,axiom,
    ! [A: code_integer,N: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
      = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_6809_minus__power__mult__self,axiom,
    ! [A: rat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
      = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_6810_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
          = one_one_real ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% minus_one_power_iff
thf(fact_6811_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
          = one_one_int ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% minus_one_power_iff
thf(fact_6812_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
          = one_one_complex ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% minus_one_power_iff
thf(fact_6813_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
          = one_one_Code_integer ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% minus_one_power_iff
thf(fact_6814_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
          = one_one_rat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% minus_one_power_iff
thf(fact_6815_minus__1__div__exp__eq__int,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% minus_1_div_exp_eq_int
thf(fact_6816_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_6817_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ).

% signed_take_bit_int_greater_eq_minus_exp
thf(fact_6818_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).

% signed_take_bit_int_less_eq_self_iff
thf(fact_6819_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
      = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% signed_take_bit_int_greater_self_iff
thf(fact_6820_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus_uminus_real @ one_one_real ) ) ).

% power_minus1_odd
thf(fact_6821_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% power_minus1_odd
thf(fact_6822_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power_minus1_odd
thf(fact_6823_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% power_minus1_odd
thf(fact_6824_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus_uminus_rat @ one_one_rat ) ) ).

% power_minus1_odd
thf(fact_6825_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_6826_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_ri631733984087533419it_int @ N @ K )
        = K )
      = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% signed_take_bit_int_eq_self_iff
thf(fact_6827_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_ri631733984087533419it_int @ N @ K )
          = K ) ) ) ).

% signed_take_bit_int_eq_self
thf(fact_6828_option_Osize__gen_I1_J,axiom,
    ! [X: product_prod_nat_nat > nat] :
      ( ( size_o8335143837870341156at_nat @ X @ none_P5556105721700978146at_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_6829_option_Osize__gen_I1_J,axiom,
    ! [X: nat > nat] :
      ( ( size_option_nat @ X @ none_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_6830_option_Osize__gen_I1_J,axiom,
    ! [X: num > nat] :
      ( ( size_option_num @ X @ none_num )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_6831_ln__one__plus__pos__lower__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( minus_minus_real @ X @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) ) ) ) ).

% ln_one_plus_pos_lower_bound
thf(fact_6832_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).

% signed_take_bit_int_greater_eq
thf(fact_6833_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_6834_and__int_Oelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
        = Y )
     => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
        & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.elims
thf(fact_6835_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_6836_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ zero_zero_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_6837_tanh__ln__real,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( tanh_real @ ( ln_ln_real @ X ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% tanh_ln_real
thf(fact_6838_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_minus_bit1
thf(fact_6839_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound
thf(fact_6840_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( ( bit1 @ X32 )
        = ( bit1 @ Y32 ) )
      = ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
thf(fact_6841_semiring__norm_I90_J,axiom,
    ! [M: num,N: num] :
      ( ( ( bit1 @ M )
        = ( bit1 @ N ) )
      = ( M = N ) ) ).

% semiring_norm(90)
thf(fact_6842_abs__idempotent,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_idempotent
thf(fact_6843_abs__idempotent,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_idempotent
thf(fact_6844_abs__idempotent,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_idempotent
thf(fact_6845_abs__idempotent,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_idempotent
thf(fact_6846_and_Oidem,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ A )
      = A ) ).

% and.idem
thf(fact_6847_and_Oidem,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ A )
      = A ) ).

% and.idem
thf(fact_6848_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_6849_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_6850_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_6851_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_6852_Compl__iff,axiom,
    ! [C: complex,A2: set_complex] :
      ( ( member_complex @ C @ ( uminus8566677241136511917omplex @ A2 ) )
      = ( ~ ( member_complex @ C @ A2 ) ) ) ).

% Compl_iff
thf(fact_6853_Compl__iff,axiom,
    ! [C: real,A2: set_real] :
      ( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
      = ( ~ ( member_real @ C @ A2 ) ) ) ).

% Compl_iff
thf(fact_6854_Compl__iff,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
      = ( ~ ( member_set_nat @ C @ A2 ) ) ) ).

% Compl_iff
thf(fact_6855_Compl__iff,axiom,
    ! [C: nat,A2: set_nat] :
      ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
      = ( ~ ( member_nat @ C @ A2 ) ) ) ).

% Compl_iff
thf(fact_6856_Compl__iff,axiom,
    ! [C: int,A2: set_int] :
      ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
      = ( ~ ( member_int @ C @ A2 ) ) ) ).

% Compl_iff
thf(fact_6857_ComplI,axiom,
    ! [C: complex,A2: set_complex] :
      ( ~ ( member_complex @ C @ A2 )
     => ( member_complex @ C @ ( uminus8566677241136511917omplex @ A2 ) ) ) ).

% ComplI
thf(fact_6858_ComplI,axiom,
    ! [C: real,A2: set_real] :
      ( ~ ( member_real @ C @ A2 )
     => ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) ) ) ).

% ComplI
thf(fact_6859_ComplI,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( ~ ( member_set_nat @ C @ A2 )
     => ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) ) ) ).

% ComplI
thf(fact_6860_ComplI,axiom,
    ! [C: nat,A2: set_nat] :
      ( ~ ( member_nat @ C @ A2 )
     => ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).

% ComplI
thf(fact_6861_ComplI,axiom,
    ! [C: int,A2: set_int] :
      ( ~ ( member_int @ C @ A2 )
     => ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).

% ComplI
thf(fact_6862_abs__0,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_0
thf(fact_6863_abs__0,axiom,
    ( ( abs_abs_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% abs_0
thf(fact_6864_abs__0,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_0
thf(fact_6865_abs__0,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_0
thf(fact_6866_abs__0,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_0
thf(fact_6867_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( abs_abs_Code_integer @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
thf(fact_6868_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( abs_abs_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% abs_0_eq
thf(fact_6869_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( abs_abs_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% abs_0_eq
thf(fact_6870_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_6871_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
thf(fact_6872_abs__eq__0,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0
thf(fact_6873_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0
thf(fact_6874_abs__eq__0,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_6875_abs__zero,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_zero
thf(fact_6876_abs__zero,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_zero
thf(fact_6877_abs__zero,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_zero
thf(fact_6878_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_6879_semiring__norm_I89_J,axiom,
    ! [M: num,N: num] :
      ( ( bit1 @ M )
     != ( bit0 @ N ) ) ).

% semiring_norm(89)
thf(fact_6880_semiring__norm_I88_J,axiom,
    ! [M: num,N: num] :
      ( ( bit0 @ M )
     != ( bit1 @ N ) ) ).

% semiring_norm(88)
thf(fact_6881_semiring__norm_I86_J,axiom,
    ! [M: num] :
      ( ( bit1 @ M )
     != one ) ).

% semiring_norm(86)
thf(fact_6882_semiring__norm_I84_J,axiom,
    ! [N: num] :
      ( one
     != ( bit1 @ N ) ) ).

% semiring_norm(84)
thf(fact_6883_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% abs_numeral
thf(fact_6884_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ N ) ) ).

% abs_numeral
thf(fact_6885_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% abs_numeral
thf(fact_6886_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ N ) ) ).

% abs_numeral
thf(fact_6887_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_6888_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_6889_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_6890_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_6891_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_6892_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_6893_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_6894_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_6895_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_6896_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_6897_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_6898_abs__minus__cancel,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus_cancel
thf(fact_6899_abs__minus__cancel,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus_cancel
thf(fact_6900_abs__minus__cancel,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus_cancel
thf(fact_6901_abs__minus__cancel,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus_cancel
thf(fact_6902_and__zero__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% and_zero_eq
thf(fact_6903_and__zero__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% and_zero_eq
thf(fact_6904_zero__and__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% zero_and_eq
thf(fact_6905_zero__and__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_and_eq
thf(fact_6906_bit_Oconj__zero__left,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ X )
      = zero_zero_int ) ).

% bit.conj_zero_left
thf(fact_6907_bit_Oconj__zero__right,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ X @ zero_zero_int )
      = zero_zero_int ) ).

% bit.conj_zero_right
thf(fact_6908_tanh__0,axiom,
    ( ( tanh_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% tanh_0
thf(fact_6909_tanh__0,axiom,
    ( ( tanh_real @ zero_zero_real )
    = zero_zero_real ) ).

% tanh_0
thf(fact_6910_tanh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( ( tanh_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% tanh_real_zero_iff
thf(fact_6911_semiring__norm_I73_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(73)
thf(fact_6912_semiring__norm_I80_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(80)
thf(fact_6913_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_6914_abs__of__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_6915_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_6916_abs__of__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_6917_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_6918_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_6919_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_6920_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_6921_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_6922_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_6923_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_6924_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_6925_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_6926_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_6927_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_6928_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_6929_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ N ) ) ).

% abs_neg_numeral
thf(fact_6930_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ N ) ) ).

% abs_neg_numeral
thf(fact_6931_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% abs_neg_numeral
thf(fact_6932_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ N ) ) ).

% abs_neg_numeral
thf(fact_6933_abs__neg__one,axiom,
    ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
    = one_one_real ) ).

% abs_neg_one
thf(fact_6934_abs__neg__one,axiom,
    ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
    = one_one_int ) ).

% abs_neg_one
thf(fact_6935_abs__neg__one,axiom,
    ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = one_one_Code_integer ) ).

% abs_neg_one
thf(fact_6936_abs__neg__one,axiom,
    ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = one_one_rat ) ).

% abs_neg_one
thf(fact_6937_abs__power__minus,axiom,
    ! [A: real,N: nat] :
      ( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
      = ( abs_abs_real @ ( power_power_real @ A @ N ) ) ) ).

% abs_power_minus
thf(fact_6938_abs__power__minus,axiom,
    ! [A: int,N: nat] :
      ( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
      = ( abs_abs_int @ ( power_power_int @ A @ N ) ) ) ).

% abs_power_minus
thf(fact_6939_abs__power__minus,axiom,
    ! [A: code_integer,N: nat] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
      = ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% abs_power_minus
thf(fact_6940_abs__power__minus,axiom,
    ! [A: rat,N: nat] :
      ( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
      = ( abs_abs_rat @ ( power_power_rat @ A @ N ) ) ) ).

% abs_power_minus
thf(fact_6941_and_Oleft__neutral,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ A )
      = A ) ).

% and.left_neutral
thf(fact_6942_and_Oleft__neutral,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) @ A )
      = A ) ).

% and.left_neutral
thf(fact_6943_and_Oright__neutral,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = A ) ).

% and.right_neutral
thf(fact_6944_and_Oright__neutral,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = A ) ).

% and.right_neutral
thf(fact_6945_bit_Oconj__one__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se3949692690581998587nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = X ) ).

% bit.conj_one_right
thf(fact_6946_bit_Oconj__one__right,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ X @ ( uminus_uminus_int @ one_one_int ) )
      = X ) ).

% bit.conj_one_right
thf(fact_6947_semiring__norm_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).

% semiring_norm(7)
thf(fact_6948_semiring__norm_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).

% semiring_norm(9)
thf(fact_6949_semiring__norm_I14_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).

% semiring_norm(14)
thf(fact_6950_semiring__norm_I15_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).

% semiring_norm(15)
thf(fact_6951_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_6952_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_6953_semiring__norm_I72_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(72)
thf(fact_6954_semiring__norm_I81_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(81)
thf(fact_6955_semiring__norm_I70_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).

% semiring_norm(70)
thf(fact_6956_semiring__norm_I77_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).

% semiring_norm(77)
thf(fact_6957_tanh__real__neg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( tanh_real @ X ) @ zero_zero_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% tanh_real_neg_iff
thf(fact_6958_tanh__real__pos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% tanh_real_pos_iff
thf(fact_6959_tanh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% tanh_real_nonpos_iff
thf(fact_6960_tanh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% tanh_real_nonneg_iff
thf(fact_6961_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_6962_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_6963_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_6964_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_6965_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_6966_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_nonpos
thf(fact_6967_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_6968_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_6969_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_6970_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_6971_and__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
      = one_one_int ) ).

% and_numerals(8)
thf(fact_6972_and__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
      = one_one_nat ) ).

% and_numerals(8)
thf(fact_6973_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_6974_semiring__norm_I10_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).

% semiring_norm(10)
thf(fact_6975_semiring__norm_I8_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ one )
      = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).

% semiring_norm(8)
thf(fact_6976_semiring__norm_I5_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ one )
      = ( bit1 @ M ) ) ).

% semiring_norm(5)
thf(fact_6977_semiring__norm_I4_J,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bit1 @ N ) )
      = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).

% semiring_norm(4)
thf(fact_6978_semiring__norm_I3_J,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bit0 @ N ) )
      = ( bit1 @ N ) ) ).

% semiring_norm(3)
thf(fact_6979_semiring__norm_I16_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).

% semiring_norm(16)
thf(fact_6980_semiring__norm_I74_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(74)
thf(fact_6981_semiring__norm_I79_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(79)
thf(fact_6982_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
      = ( ( A != zero_z3403309356797280102nteger )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_6983_zero__less__power__abs__iff,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
      = ( ( A != zero_zero_real )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_6984_zero__less__power__abs__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
      = ( ( A != zero_zero_rat )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_6985_zero__less__power__abs__iff,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
      = ( ( A != zero_zero_int )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_6986_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_numerals(5)
thf(fact_6987_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
      = zero_zero_nat ) ).

% and_numerals(5)
thf(fact_6988_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_6989_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_6990_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_6991_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_6992_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_6993_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_6994_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_6995_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_6996_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_6997_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_6998_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(3)
thf(fact_6999_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(3)
thf(fact_7000_and__minus__numerals_I2_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = one_one_int ) ).

% and_minus_numerals(2)
thf(fact_7001_and__minus__numerals_I6_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
      = one_one_int ) ).

% and_minus_numerals(6)
thf(fact_7002_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(6)
thf(fact_7003_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(6)
thf(fact_7004_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(4)
thf(fact_7005_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(4)
thf(fact_7006_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_7007_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_7008_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_7009_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_7010_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
      = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).

% div_Suc_eq_div_add3
thf(fact_7011_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_7012_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
      = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).

% mod_Suc_eq_mod_add3
thf(fact_7013_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_7014_and__minus__numerals_I5_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_minus_numerals(5)
thf(fact_7015_and__minus__numerals_I1_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = zero_zero_int ) ).

% and_minus_numerals(1)
thf(fact_7016_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_7017_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_7018_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_7019_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_bit1
thf(fact_7020_ComplD,axiom,
    ! [C: complex,A2: set_complex] :
      ( ( member_complex @ C @ ( uminus8566677241136511917omplex @ A2 ) )
     => ~ ( member_complex @ C @ A2 ) ) ).

% ComplD
thf(fact_7021_ComplD,axiom,
    ! [C: real,A2: set_real] :
      ( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
     => ~ ( member_real @ C @ A2 ) ) ).

% ComplD
thf(fact_7022_ComplD,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
     => ~ ( member_set_nat @ C @ A2 ) ) ).

% ComplD
thf(fact_7023_ComplD,axiom,
    ! [C: nat,A2: set_nat] :
      ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
     => ~ ( member_nat @ C @ A2 ) ) ).

% ComplD
thf(fact_7024_ComplD,axiom,
    ! [C: int,A2: set_int] :
      ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
     => ~ ( member_int @ C @ A2 ) ) ).

% ComplD
thf(fact_7025_Compl__eq,axiom,
    ( uminus8566677241136511917omplex
    = ( ^ [A6: set_complex] :
          ( collect_complex
          @ ^ [X6: complex] :
              ~ ( member_complex @ X6 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_7026_Compl__eq,axiom,
    ( uminus612125837232591019t_real
    = ( ^ [A6: set_real] :
          ( collect_real
          @ ^ [X6: real] :
              ~ ( member_real @ X6 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_7027_Compl__eq,axiom,
    ( uminus3195874150345416415st_nat
    = ( ^ [A6: set_list_nat] :
          ( collect_list_nat
          @ ^ [X6: list_nat] :
              ~ ( member_list_nat @ X6 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_7028_Compl__eq,axiom,
    ( uminus613421341184616069et_nat
    = ( ^ [A6: set_set_nat] :
          ( collect_set_nat
          @ ^ [X6: set_nat] :
              ~ ( member_set_nat @ X6 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_7029_Compl__eq,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A6: set_nat] :
          ( collect_nat
          @ ^ [X6: nat] :
              ~ ( member_nat @ X6 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_7030_Compl__eq,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A6: set_int] :
          ( collect_int
          @ ^ [X6: int] :
              ~ ( member_int @ X6 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_7031_Collect__neg__eq,axiom,
    ! [P: real > $o] :
      ( ( collect_real
        @ ^ [X6: real] :
            ~ ( P @ X6 ) )
      = ( uminus612125837232591019t_real @ ( collect_real @ P ) ) ) ).

% Collect_neg_eq
thf(fact_7032_Collect__neg__eq,axiom,
    ! [P: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X6: list_nat] :
            ~ ( P @ X6 ) )
      = ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_7033_Collect__neg__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X6: set_nat] :
            ~ ( P @ X6 ) )
      = ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_7034_Collect__neg__eq,axiom,
    ! [P: nat > $o] :
      ( ( collect_nat
        @ ^ [X6: nat] :
            ~ ( P @ X6 ) )
      = ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_7035_Collect__neg__eq,axiom,
    ! [P: int > $o] :
      ( ( collect_int
        @ ^ [X6: int] :
            ~ ( P @ X6 ) )
      = ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ).

% Collect_neg_eq
thf(fact_7036_uminus__set__def,axiom,
    ( uminus8566677241136511917omplex
    = ( ^ [A6: set_complex] :
          ( collect_complex
          @ ( uminus1680532995456772888plex_o
            @ ^ [X6: complex] : ( member_complex @ X6 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_7037_uminus__set__def,axiom,
    ( uminus612125837232591019t_real
    = ( ^ [A6: set_real] :
          ( collect_real
          @ ( uminus_uminus_real_o
            @ ^ [X6: real] : ( member_real @ X6 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_7038_uminus__set__def,axiom,
    ( uminus3195874150345416415st_nat
    = ( ^ [A6: set_list_nat] :
          ( collect_list_nat
          @ ( uminus5770388063884162150_nat_o
            @ ^ [X6: list_nat] : ( member_list_nat @ X6 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_7039_uminus__set__def,axiom,
    ( uminus613421341184616069et_nat
    = ( ^ [A6: set_set_nat] :
          ( collect_set_nat
          @ ( uminus6401447641752708672_nat_o
            @ ^ [X6: set_nat] : ( member_set_nat @ X6 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_7040_uminus__set__def,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A6: set_nat] :
          ( collect_nat
          @ ( uminus_uminus_nat_o
            @ ^ [X6: nat] : ( member_nat @ X6 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_7041_uminus__set__def,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A6: set_int] :
          ( collect_int
          @ ( uminus_uminus_int_o
            @ ^ [X6: int] : ( member_int @ X6 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_7042_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_7043_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_7044_and_Ocommute,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [A3: int,B3: int] : ( bit_se725231765392027082nd_int @ B3 @ A3 ) ) ) ).

% and.commute
thf(fact_7045_and_Ocommute,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [A3: nat,B3: nat] : ( bit_se727722235901077358nd_nat @ B3 @ A3 ) ) ) ).

% and.commute
thf(fact_7046_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_7047_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_7048_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_7049_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_7050_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_7051_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_7052_abs__ge__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).

% abs_ge_self
thf(fact_7053_abs__ge__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_self
thf(fact_7054_abs__ge__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).

% abs_ge_self
thf(fact_7055_abs__ge__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).

% abs_ge_self
thf(fact_7056_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
thf(fact_7057_abs__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( ( abs_abs_complex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% abs_eq_0_iff
thf(fact_7058_abs__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0_iff
thf(fact_7059_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
thf(fact_7060_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
thf(fact_7061_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_7062_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_7063_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_7064_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_7065_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_7066_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_7067_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_7068_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_7069_power__abs,axiom,
    ! [A: code_integer,N: nat] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
      = ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).

% power_abs
thf(fact_7070_power__abs,axiom,
    ! [A: rat,N: nat] :
      ( ( abs_abs_rat @ ( power_power_rat @ A @ N ) )
      = ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).

% power_abs
thf(fact_7071_power__abs,axiom,
    ! [A: real,N: nat] :
      ( ( abs_abs_real @ ( power_power_real @ A @ N ) )
      = ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).

% power_abs
thf(fact_7072_power__abs,axiom,
    ! [A: int,N: nat] :
      ( ( abs_abs_int @ ( power_power_int @ A @ N ) )
      = ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).

% power_abs
thf(fact_7073_verit__eq__simplify_I14_J,axiom,
    ! [X2: num,X32: num] :
      ( ( bit0 @ X2 )
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(14)
thf(fact_7074_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] :
      ( one
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(12)
thf(fact_7075_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_7076_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_7077_abs__ge__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_zero
thf(fact_7078_abs__ge__zero,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).

% abs_ge_zero
thf(fact_7079_abs__ge__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).

% abs_ge_zero
thf(fact_7080_abs__ge__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).

% abs_ge_zero
thf(fact_7081_abs__not__less__zero,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).

% abs_not_less_zero
thf(fact_7082_abs__not__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).

% abs_not_less_zero
thf(fact_7083_abs__not__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_7084_abs__not__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_7085_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_7086_abs__of__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_7087_abs__of__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_7088_abs__of__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_7089_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_7090_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_7091_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_7092_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_7093_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_7094_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_7095_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_7096_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_7097_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_7098_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_7099_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_7100_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_7101_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_7102_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_7103_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_7104_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_7105_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_7106_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_7107_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_7108_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_7109_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_7110_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_7111_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_7112_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_7113_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_7114_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_7115_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_7116_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_7117_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_7118_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_7119_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_7120_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_7121_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_7122_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_7123_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_7124_abs__ge__minus__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_minus_self
thf(fact_7125_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_7126_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_7127_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_7128_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_7129_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_7130_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_7131_AND__lower,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X @ Y ) ) ) ).

% AND_lower
thf(fact_7132_AND__upper1,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ X ) ) ).

% AND_upper1
thf(fact_7133_AND__upper2,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Y ) ) ).

% AND_upper2
thf(fact_7134_AND__upper1_H,axiom,
    ! [Y: int,Z2: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z2 )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z2 ) ) ) ).

% AND_upper1'
thf(fact_7135_AND__upper2_H,axiom,
    ! [Y: int,Z2: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z2 )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z2 ) ) ) ).

% AND_upper2'
thf(fact_7136_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_7137_xor__num_Ocases,axiom,
    ! [X: product_prod_num_num] :
      ( ( X
       != ( product_Pair_num_num @ one @ one ) )
     => ( ! [N2: num] :
            ( X
           != ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) )
       => ( ! [N2: num] :
              ( X
             != ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) )
         => ( ! [M3: num] :
                ( X
               != ( product_Pair_num_num @ ( bit0 @ M3 ) @ one ) )
           => ( ! [M3: num,N2: num] :
                  ( X
                 != ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit0 @ N2 ) ) )
             => ( ! [M3: num,N2: num] :
                    ( X
                   != ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit1 @ N2 ) ) )
               => ( ! [M3: num] :
                      ( X
                     != ( product_Pair_num_num @ ( bit1 @ M3 ) @ one ) )
                 => ( ! [M3: num,N2: num] :
                        ( X
                       != ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit0 @ N2 ) ) )
                   => ~ ! [M3: num,N2: num] :
                          ( X
                         != ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.cases
thf(fact_7138_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X23: num] :
            ( Y
           != ( bit0 @ X23 ) )
       => ~ ! [X33: num] :
              ( Y
             != ( bit1 @ X33 ) ) ) ) ).

% num.exhaust
thf(fact_7139_dense__eq0__I,axiom,
    ! [X: real] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E ) )
     => ( X = zero_zero_real ) ) ).

% dense_eq0_I
thf(fact_7140_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E )
         => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E ) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
thf(fact_7141_abs__mult__pos,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y ) @ X )
        = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_7142_abs__mult__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( times_times_real @ ( abs_abs_real @ Y ) @ X )
        = ( abs_abs_real @ ( times_times_real @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_7143_abs__mult__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( times_times_rat @ ( abs_abs_rat @ Y ) @ X )
        = ( abs_abs_rat @ ( times_times_rat @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_7144_abs__mult__pos,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
        = ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_7145_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_7146_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_7147_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_7148_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_7149_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_7150_abs__minus__le__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).

% abs_minus_le_zero
thf(fact_7151_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_7152_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_7153_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_7154_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_7155_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_7156_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_7157_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_7158_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_7159_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_7160_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_7161_abs__div__pos,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( divide_divide_real @ ( abs_abs_real @ X ) @ Y )
        = ( abs_abs_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% abs_div_pos
thf(fact_7162_abs__div__pos,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y )
        = ( abs_abs_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% abs_div_pos
thf(fact_7163_zero__le__power__abs,axiom,
    ! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_7164_zero__le__power__abs,axiom,
    ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_7165_zero__le__power__abs,axiom,
    ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_7166_zero__le__power__abs,axiom,
    ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_7167_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_7168_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_7169_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_7170_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_7171_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_7172_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_7173_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_neg
thf(fact_7174_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_7175_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_7176_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_7177_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_7178_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_7179_abs__diff__le__iff,axiom,
    ! [X: code_integer,A: code_integer,R2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
      = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
        & ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_7180_abs__diff__le__iff,axiom,
    ! [X: real,A: real,R2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
      = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R2 ) @ X )
        & ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_7181_abs__diff__le__iff,axiom,
    ! [X: rat,A: rat,R2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
      = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
        & ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_7182_abs__diff__le__iff,axiom,
    ! [X: int,A: int,R2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
      = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X )
        & ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_7183_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_7184_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_7185_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_7186_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_7187_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_7188_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_7189_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_7190_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_7191_abs__diff__less__iff,axiom,
    ! [X: code_integer,A: code_integer,R2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R2 )
      = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X )
        & ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_7192_abs__diff__less__iff,axiom,
    ! [X: real,A: real,R2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R2 )
      = ( ( ord_less_real @ ( minus_minus_real @ A @ R2 ) @ X )
        & ( ord_less_real @ X @ ( plus_plus_real @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_7193_abs__diff__less__iff,axiom,
    ! [X: rat,A: rat,R2: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R2 )
      = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X )
        & ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_7194_abs__diff__less__iff,axiom,
    ! [X: int,A: int,R2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R2 )
      = ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X )
        & ( ord_less_int @ X @ ( plus_plus_int @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_7195_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_7196_AND__upper1_H_H,axiom,
    ! [Y: int,Z2: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z2 )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z2 ) ) ) ).

% AND_upper1''
thf(fact_7197_AND__upper2_H_H,axiom,
    ! [Y: int,Z2: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z2 )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z2 ) ) ) ).

% AND_upper2''
thf(fact_7198_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numera1916890842035813515d_enat @ ( bit1 @ N ) )
      = ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) @ one_on7984719198319812577d_enat ) ) ).

% numeral_Bit1
thf(fact_7199_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).

% numeral_Bit1
thf(fact_7200_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).

% numeral_Bit1
thf(fact_7201_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).

% numeral_Bit1
thf(fact_7202_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).

% numeral_Bit1
thf(fact_7203_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).

% numeral_Bit1
thf(fact_7204_eval__nat__numeral_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N ) )
      = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).

% eval_nat_numeral(3)
thf(fact_7205_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_7206_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_7207_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_7208_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_7209_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_7210_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_7211_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_7212_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_7213_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_7214_power__minus__Bit1,axiom,
    ! [X: real,K: num] :
      ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_7215_power__minus__Bit1,axiom,
    ! [X: int,K: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_7216_power__minus__Bit1,axiom,
    ! [X: complex,K: num] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_7217_power__minus__Bit1,axiom,
    ! [X: code_integer,K: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_7218_power__minus__Bit1,axiom,
    ! [X: rat,K: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_7219_lemma__interval__lt,axiom,
    ! [A: real,X: real,B: real] :
      ( ( ord_less_real @ A @ X )
     => ( ( ord_less_real @ X @ B )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [Y5: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D3 )
               => ( ( ord_less_real @ A @ Y5 )
                  & ( ord_less_real @ Y5 @ B ) ) ) ) ) ) ).

% lemma_interval_lt
thf(fact_7220_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numera1916890842035813515d_enat @ ( bit1 @ N ) )
      = ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) @ one_on7984719198319812577d_enat ) ) ).

% numeral_code(3)
thf(fact_7221_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).

% numeral_code(3)
thf(fact_7222_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).

% numeral_code(3)
thf(fact_7223_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).

% numeral_code(3)
thf(fact_7224_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).

% numeral_code(3)
thf(fact_7225_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).

% numeral_code(3)
thf(fact_7226_power__numeral__odd,axiom,
    ! [Z2: complex,W: num] :
      ( ( power_power_complex @ Z2 @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_complex @ ( times_times_complex @ Z2 @ ( power_power_complex @ Z2 @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_complex @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_7227_power__numeral__odd,axiom,
    ! [Z2: real,W: num] :
      ( ( power_power_real @ Z2 @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_real @ ( times_times_real @ Z2 @ ( power_power_real @ Z2 @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_real @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_7228_power__numeral__odd,axiom,
    ! [Z2: rat,W: num] :
      ( ( power_power_rat @ Z2 @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_rat @ ( times_times_rat @ Z2 @ ( power_power_rat @ Z2 @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_rat @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_7229_power__numeral__odd,axiom,
    ! [Z2: nat,W: num] :
      ( ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_nat @ ( times_times_nat @ Z2 @ ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_7230_power__numeral__odd,axiom,
    ! [Z2: int,W: num] :
      ( ( power_power_int @ Z2 @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_int @ ( times_times_int @ Z2 @ ( power_power_int @ Z2 @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_int @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_7231_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_7232_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_7233_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_7234_abs__add__one__gt__zero,axiom,
    ! [X: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_7235_abs__add__one__gt__zero,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_7236_abs__add__one__gt__zero,axiom,
    ! [X: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_7237_abs__add__one__gt__zero,axiom,
    ! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_7238_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_7239_numeral__Bit1__div__2,axiom,
    ! [N: num] :
      ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( numeral_numeral_nat @ N ) ) ).

% numeral_Bit1_div_2
thf(fact_7240_numeral__Bit1__div__2,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( numeral_numeral_int @ N ) ) ).

% numeral_Bit1_div_2
thf(fact_7241_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_7242_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_7243_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_7244_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != zero_zero_nat ) ).

% cong_exp_iff_simps(3)
thf(fact_7245_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != zero_zero_int ) ).

% cong_exp_iff_simps(3)
thf(fact_7246_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(3)
thf(fact_7247_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_7248_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_7249_power3__eq__cube,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_7250_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_7251_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_7252_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_7253_Suc3__eq__add__3,axiom,
    ! [N: nat] :
      ( ( suc @ ( suc @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).

% Suc3_eq_add_3
thf(fact_7254_lemma__interval,axiom,
    ! [A: real,X: real,B: real] :
      ( ( ord_less_real @ A @ X )
     => ( ( ord_less_real @ X @ B )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [Y5: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D3 )
               => ( ( ord_less_eq_real @ A @ Y5 )
                  & ( ord_less_eq_real @ Y5 @ B ) ) ) ) ) ) ).

% lemma_interval
thf(fact_7255_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_7256_and__one__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ A @ one_one_Code_integer )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_7257_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_7258_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_7259_one__and__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ A )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_7260_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_7261_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_7262_abs__le__square__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ ( abs_abs_Code_integer @ Y ) )
      = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_7263_abs__le__square__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) )
      = ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_7264_abs__le__square__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y ) )
      = ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_7265_abs__le__square__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) )
      = ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_7266_abs__square__eq__1,axiom,
    ! [X: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_Code_integer )
      = ( ( abs_abs_Code_integer @ X )
        = one_one_Code_integer ) ) ).

% abs_square_eq_1
thf(fact_7267_abs__square__eq__1,axiom,
    ! [X: rat] :
      ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_rat )
      = ( ( abs_abs_rat @ X )
        = one_one_rat ) ) ).

% abs_square_eq_1
thf(fact_7268_abs__square__eq__1,axiom,
    ! [X: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
      = ( ( abs_abs_real @ X )
        = one_one_real ) ) ).

% abs_square_eq_1
thf(fact_7269_abs__square__eq__1,axiom,
    ! [X: int] :
      ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_int )
      = ( ( abs_abs_int @ X )
        = one_one_int ) ) ).

% abs_square_eq_1
thf(fact_7270_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_7271_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_7272_power__even__abs,axiom,
    ! [N: nat,A: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N )
        = ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% power_even_abs
thf(fact_7273_power__even__abs,axiom,
    ! [N: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ N )
        = ( power_power_rat @ A @ N ) ) ) ).

% power_even_abs
thf(fact_7274_power__even__abs,axiom,
    ! [N: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( abs_abs_real @ A ) @ N )
        = ( power_power_real @ A @ N ) ) ) ).

% power_even_abs
thf(fact_7275_power__even__abs,axiom,
    ! [N: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( abs_abs_int @ A ) @ N )
        = ( power_power_int @ A @ N ) ) ) ).

% power_even_abs
thf(fact_7276_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
thf(fact_7277_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
thf(fact_7278_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
thf(fact_7279_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
thf(fact_7280_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
thf(fact_7281_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
thf(fact_7282_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).

% Suc_div_eq_add3_div
thf(fact_7283_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).

% Suc_mod_eq_add3_mod
thf(fact_7284_abs__sqrt__wlog,axiom,
    ! [P: code_integer > code_integer > $o,X: 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 @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_7285_abs__sqrt__wlog,axiom,
    ! [P: real > real > $o,X: real] :
      ( ! [X5: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X5 )
         => ( P @ X5 @ ( power_power_real @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_7286_abs__sqrt__wlog,axiom,
    ! [P: rat > rat > $o,X: 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 @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_7287_abs__sqrt__wlog,axiom,
    ! [P: int > int > $o,X: int] :
      ( ! [X5: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X5 )
         => ( P @ X5 @ ( power_power_int @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_7288_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_7289_power2__le__iff__abs__le,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_7290_power2__le__iff__abs__le,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_7291_power2__le__iff__abs__le,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_7292_abs__square__le__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).

% abs_square_le_1
thf(fact_7293_abs__square__le__1,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).

% abs_square_le_1
thf(fact_7294_abs__square__le__1,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).

% abs_square_le_1
thf(fact_7295_abs__square__le__1,axiom,
    ! [X: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).

% abs_square_le_1
thf(fact_7296_abs__square__less__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).

% abs_square_less_1
thf(fact_7297_abs__square__less__1,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).

% abs_square_less_1
thf(fact_7298_abs__square__less__1,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).

% abs_square_less_1
thf(fact_7299_abs__square__less__1,axiom,
    ! [X: int] :
      ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).

% abs_square_less_1
thf(fact_7300_power__mono__even,axiom,
    ! [N: nat,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).

% power_mono_even
thf(fact_7301_power__mono__even,axiom,
    ! [N: nat,A: real,B: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).

% power_mono_even
thf(fact_7302_power__mono__even,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).

% power_mono_even
thf(fact_7303_power__mono__even,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).

% power_mono_even
thf(fact_7304_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_7305_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_7306_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_7307_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
     => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% odd_mod_4_div_2
thf(fact_7308_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_7309_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_7310_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_7311_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_7312_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_7313_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_7314_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_num @ M @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_7315_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_num @ M @ N )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_7316_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_num @ M @ N )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_7317_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_num @ M @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_7318_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_num @ M @ N )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_7319_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_num @ M @ N )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_7320_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_7321_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_7322_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ one_one_complex )
    = one_one_complex ) ).

% dbl_dec_simps(3)
thf(fact_7323_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ one_one_real )
    = one_one_real ) ).

% dbl_dec_simps(3)
thf(fact_7324_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
    = one_one_rat ) ).

% dbl_dec_simps(3)
thf(fact_7325_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ one_one_int )
    = one_one_int ) ).

% dbl_dec_simps(3)
thf(fact_7326_zabs__less__one__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
      = ( Z2 = zero_zero_int ) ) ).

% zabs_less_one_iff
thf(fact_7327_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one )
    = zero_zero_nat ) ).

% pred_numeral_simps(1)
thf(fact_7328_Suc__eq__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ( suc @ N )
        = ( numeral_numeral_nat @ K ) )
      = ( N
        = ( pred_numeral @ K ) ) ) ).

% Suc_eq_numeral
thf(fact_7329_eq__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ( numeral_numeral_nat @ K )
        = ( suc @ N ) )
      = ( ( pred_numeral @ K )
        = N ) ) ).

% eq_numeral_Suc
thf(fact_7330_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_7331_and__nat__numerals_I3_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% and_nat_numerals(3)
thf(fact_7332_less__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).

% less_numeral_Suc
thf(fact_7333_less__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).

% less_Suc_numeral
thf(fact_7334_pred__numeral__simps_I3_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit1 @ K ) )
      = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).

% pred_numeral_simps(3)
thf(fact_7335_le__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).

% le_Suc_numeral
thf(fact_7336_le__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).

% le_numeral_Suc
thf(fact_7337_diff__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( minus_minus_nat @ N @ ( pred_numeral @ K ) ) ) ).

% diff_Suc_numeral
thf(fact_7338_diff__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( minus_minus_nat @ ( pred_numeral @ K ) @ N ) ) ).

% diff_numeral_Suc
thf(fact_7339_max__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K ) ) ) ) ).

% max_Suc_numeral
thf(fact_7340_max__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N ) ) ) ).

% max_numeral_Suc
thf(fact_7341_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_dec_simps(2)
thf(fact_7342_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_dec_simps(2)
thf(fact_7343_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_dec_simps(2)
thf(fact_7344_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_dec_simps(2)
thf(fact_7345_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_dec_simps(2)
thf(fact_7346_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_7347_and__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% and_nat_numerals(4)
thf(fact_7348_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_7349_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_7350_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
      = ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_7351_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_7352_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_7353_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique3479559517661332726nteger @ M @ one )
      = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).

% divmod_algorithm_code(2)
thf(fact_7354_and__Suc__0__eq,axiom,
    ! [N: nat] :
      ( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% and_Suc_0_eq
thf(fact_7355_Suc__0__and__eq,axiom,
    ! [N: nat] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Suc_0_and_eq
thf(fact_7356_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_7357_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_7358_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_7359_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_7360_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_7361_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_7362_one__div__minus__numeral,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).

% one_div_minus_numeral
thf(fact_7363_minus__one__div__numeral,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).

% minus_one_div_numeral
thf(fact_7364_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_7365_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_7366_abs__zmult__eq__1,axiom,
    ! [M: int,N: int] :
      ( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
        = one_one_int )
     => ( ( abs_abs_int @ M )
        = one_one_int ) ) ).

% abs_zmult_eq_1
thf(fact_7367_numeral__eq__Suc,axiom,
    ( numeral_numeral_nat
    = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).

% numeral_eq_Suc
thf(fact_7368_zabs__def,axiom,
    ( abs_abs_int
    = ( ^ [I: int] : ( if_int @ ( ord_less_int @ I @ zero_zero_int ) @ ( uminus_uminus_int @ I ) @ I ) ) ) ).

% zabs_def
thf(fact_7369_dvd__imp__le__int,axiom,
    ! [I3: int,D: int] :
      ( ( I3 != zero_zero_int )
     => ( ( dvd_dvd_int @ D @ I3 )
       => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I3 ) ) ) ) ).

% dvd_imp_le_int
thf(fact_7370_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_7371_pred__numeral__def,axiom,
    ( pred_numeral
    = ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).

% pred_numeral_def
thf(fact_7372_zdvd__mult__cancel1,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
        = ( ( abs_abs_int @ N )
          = one_one_int ) ) ) ).

% zdvd_mult_cancel1
thf(fact_7373_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_7374_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_7375_divmod__int__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M4: num,N3: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M4 ) @ ( numeral_numeral_int @ N3 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M4 ) @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).

% divmod_int_def
thf(fact_7376_divmod__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M4: num,N3: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M4 ) @ ( numeral_numeral_int @ N3 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M4 ) @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).

% divmod_def
thf(fact_7377_divmod__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M4: num,N3: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M4 ) @ ( numeral_numeral_nat @ N3 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M4 ) @ ( numeral_numeral_nat @ N3 ) ) ) ) ) ).

% divmod_def
thf(fact_7378_divmod__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M4: num,N3: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M4 ) @ ( numera6620942414471956472nteger @ N3 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M4 ) @ ( numera6620942414471956472nteger @ N3 ) ) ) ) ) ).

% divmod_def
thf(fact_7379_divmod_H__nat__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M4: num,N3: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M4 ) @ ( numeral_numeral_nat @ N3 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M4 ) @ ( numeral_numeral_nat @ N3 ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_7380_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ( ord_less_eq_nat @ M @ I2 )
            & ( ord_less_nat @ I2 @ N ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( ord_less_eq_int @ ( F @ M ) @ K )
         => ( ( ord_less_eq_int @ K @ ( F @ N ) )
           => ? [I2: nat] :
                ( ( ord_less_eq_nat @ M @ I2 )
                & ( ord_less_eq_nat @ I2 @ N )
                & ( ( F @ I2 )
                  = K ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_7381_dbl__dec__def,axiom,
    ( neg_nu6511756317524482435omplex
    = ( ^ [X6: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X6 @ X6 ) @ one_one_complex ) ) ) ).

% dbl_dec_def
thf(fact_7382_dbl__dec__def,axiom,
    ( neg_nu6075765906172075777c_real
    = ( ^ [X6: real] : ( minus_minus_real @ ( plus_plus_real @ X6 @ X6 ) @ one_one_real ) ) ) ).

% dbl_dec_def
thf(fact_7383_dbl__dec__def,axiom,
    ( neg_nu3179335615603231917ec_rat
    = ( ^ [X6: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X6 @ X6 ) @ one_one_rat ) ) ) ).

% dbl_dec_def
thf(fact_7384_dbl__dec__def,axiom,
    ( neg_nu3811975205180677377ec_int
    = ( ^ [X6: int] : ( minus_minus_int @ ( plus_plus_int @ X6 @ X6 ) @ one_one_int ) ) ) ).

% dbl_dec_def
thf(fact_7385_decr__lemma,axiom,
    ! [D: int,X: int,Z2: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D ) ) @ Z2 ) ) ).

% decr_lemma
thf(fact_7386_incr__lemma,axiom,
    ! [D: int,Z2: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ Z2 @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z2 ) ) @ one_one_int ) @ D ) ) ) ) ).

% incr_lemma
thf(fact_7387_nat__ivt__aux,axiom,
    ! [N: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N ) )
         => ? [I2: nat] :
              ( ( ord_less_eq_nat @ I2 @ N )
              & ( ( F @ I2 )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_7388_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M4: nat,N3: nat] :
          ( if_nat
          @ ( ( M4 = zero_zero_nat )
            | ( N3 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_7389_and__nat__rec,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M4: nat,N3: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 )
              & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_7390_nat0__intermed__int__val,axiom,
    ! [N: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N ) )
         => ? [I2: nat] :
              ( ( ord_less_eq_nat @ I2 @ N )
              & ( ( F @ I2 )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_7391_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [K2: int,L3: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L3 ) )
           => ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ L3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K2 @ L3 ) ) )
       => ( P @ A0 @ A1 ) ) ) ).

% and_int.pinduct
thf(fact_7392_divmod__divmod__step,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M4: num,N3: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M4 @ N3 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M4 ) ) @ ( unique5026877609467782581ep_nat @ N3 @ ( unique5055182867167087721od_nat @ M4 @ ( bit0 @ N3 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_7393_divmod__divmod__step,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M4: num,N3: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M4 @ N3 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M4 ) ) @ ( unique5024387138958732305ep_int @ N3 @ ( unique5052692396658037445od_int @ M4 @ ( bit0 @ N3 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_7394_divmod__divmod__step,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M4: num,N3: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M4 @ N3 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M4 ) ) @ ( unique4921790084139445826nteger @ N3 @ ( unique3479559517661332726nteger @ M4 @ ( bit0 @ N3 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_7395_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
       => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                  & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( uminus_uminus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
              & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( plus_plus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
                    @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).

% and_int.pelims
thf(fact_7396_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [I2: int,J3: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J3 ) )
           => ( ( ( ord_less_eq_int @ I2 @ J3 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) )
             => ( P @ I2 @ J3 ) ) )
       => ( P @ A0 @ A1 ) ) ) ).

% upto.pinduct
thf(fact_7397_arctan__double,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X ) )
        = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% arctan_double
thf(fact_7398_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_7399_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_7400_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_7401_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_7402_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_7403_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_7404_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_7405_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_7406_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_7407_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) @ one_one_int ) )
        @ ( unique5052692396658037445od_int @ M @ N ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_7408_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) @ one_one_nat ) )
        @ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_7409_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( produc6916734918728496179nteger
        @ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) @ one_one_Code_integer ) )
        @ ( unique3479559517661332726nteger @ M @ N ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_7410_arctan__eq__zero__iff,axiom,
    ! [X: real] :
      ( ( ( arctan @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% arctan_eq_zero_iff
thf(fact_7411_arctan__zero__zero,axiom,
    ( ( arctan @ zero_zero_real )
    = zero_zero_real ) ).

% arctan_zero_zero
thf(fact_7412_of__int__eq__0__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_1_of_int_rat @ Z2 )
        = zero_zero_rat )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_7413_of__int__eq__0__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_1_of_int_int @ Z2 )
        = zero_zero_int )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_7414_of__int__eq__0__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_1_of_int_real @ Z2 )
        = zero_zero_real )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_7415_of__int__eq__0__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_17405671764205052669omplex @ Z2 )
        = zero_zero_complex )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_7416_of__int__0__eq__iff,axiom,
    ! [Z2: int] :
      ( ( zero_zero_rat
        = ( ring_1_of_int_rat @ Z2 ) )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_7417_of__int__0__eq__iff,axiom,
    ! [Z2: int] :
      ( ( zero_zero_int
        = ( ring_1_of_int_int @ Z2 ) )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_7418_of__int__0__eq__iff,axiom,
    ! [Z2: int] :
      ( ( zero_zero_real
        = ( ring_1_of_int_real @ Z2 ) )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_7419_of__int__0__eq__iff,axiom,
    ! [Z2: int] :
      ( ( zero_zero_complex
        = ( ring_17405671764205052669omplex @ Z2 ) )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_7420_of__int__0,axiom,
    ( ( ring_1_of_int_rat @ zero_zero_int )
    = zero_zero_rat ) ).

% of_int_0
thf(fact_7421_of__int__0,axiom,
    ( ( ring_1_of_int_int @ zero_zero_int )
    = zero_zero_int ) ).

% of_int_0
thf(fact_7422_of__int__0,axiom,
    ( ( ring_1_of_int_real @ zero_zero_int )
    = zero_zero_real ) ).

% of_int_0
thf(fact_7423_of__int__0,axiom,
    ( ( ring_17405671764205052669omplex @ zero_zero_int )
    = zero_zero_complex ) ).

% of_int_0
thf(fact_7424_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
      = ( numera6690914467698888265omplex @ K ) ) ).

% of_int_numeral
thf(fact_7425_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_real @ K ) ) ).

% of_int_numeral
thf(fact_7426_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% of_int_numeral
thf(fact_7427_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_rat @ K ) ) ).

% of_int_numeral
thf(fact_7428_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ( ring_17405671764205052669omplex @ Z2 )
        = ( numera6690914467698888265omplex @ N ) )
      = ( Z2
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_7429_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ( ring_1_of_int_real @ Z2 )
        = ( numeral_numeral_real @ N ) )
      = ( Z2
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_7430_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ( ring_1_of_int_int @ Z2 )
        = ( numeral_numeral_int @ N ) )
      = ( Z2
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_7431_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ( ring_1_of_int_rat @ Z2 )
        = ( numeral_numeral_rat @ N ) )
      = ( Z2
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_7432_of__int__less__iff,axiom,
    ! [W: int,Z2: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_int @ W @ Z2 ) ) ).

% of_int_less_iff
thf(fact_7433_of__int__less__iff,axiom,
    ! [W: int,Z2: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_int @ W @ Z2 ) ) ).

% of_int_less_iff
thf(fact_7434_of__int__less__iff,axiom,
    ! [W: int,Z2: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_int @ W @ Z2 ) ) ).

% of_int_less_iff
thf(fact_7435_of__int__mult,axiom,
    ! [W: int,Z2: int] :
      ( ( ring_17405671764205052669omplex @ ( times_times_int @ W @ Z2 ) )
      = ( times_times_complex @ ( ring_17405671764205052669omplex @ W ) @ ( ring_17405671764205052669omplex @ Z2 ) ) ) ).

% of_int_mult
thf(fact_7436_of__int__mult,axiom,
    ! [W: int,Z2: int] :
      ( ( ring_1_of_int_real @ ( times_times_int @ W @ Z2 ) )
      = ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).

% of_int_mult
thf(fact_7437_of__int__mult,axiom,
    ! [W: int,Z2: int] :
      ( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z2 ) )
      = ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_int_mult
thf(fact_7438_of__int__mult,axiom,
    ! [W: int,Z2: int] :
      ( ( ring_1_of_int_int @ ( times_times_int @ W @ Z2 ) )
      = ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_mult
thf(fact_7439_of__int__add,axiom,
    ! [W: int,Z2: int] :
      ( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z2 ) )
      = ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_int_add
thf(fact_7440_of__int__add,axiom,
    ! [W: int,Z2: int] :
      ( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z2 ) )
      = ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_add
thf(fact_7441_of__int__add,axiom,
    ! [W: int,Z2: int] :
      ( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z2 ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).

% of_int_add
thf(fact_7442_of__int__add,axiom,
    ! [W: int,Z2: int] :
      ( ( ring_17405671764205052669omplex @ ( plus_plus_int @ W @ Z2 ) )
      = ( plus_plus_complex @ ( ring_17405671764205052669omplex @ W ) @ ( ring_17405671764205052669omplex @ Z2 ) ) ) ).

% of_int_add
thf(fact_7443_arctan__less__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( arctan @ X ) @ zero_zero_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% arctan_less_zero_iff
thf(fact_7444_zero__less__arctan__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( arctan @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% zero_less_arctan_iff
thf(fact_7445_of__int__power,axiom,
    ! [Z2: int,N: nat] :
      ( ( ring_1_of_int_real @ ( power_power_int @ Z2 @ N ) )
      = ( power_power_real @ ( ring_1_of_int_real @ Z2 ) @ N ) ) ).

% of_int_power
thf(fact_7446_of__int__power,axiom,
    ! [Z2: int,N: nat] :
      ( ( ring_1_of_int_int @ ( power_power_int @ Z2 @ N ) )
      = ( power_power_int @ ( ring_1_of_int_int @ Z2 ) @ N ) ) ).

% of_int_power
thf(fact_7447_of__int__power,axiom,
    ! [Z2: int,N: nat] :
      ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z2 @ N ) )
      = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z2 ) @ N ) ) ).

% of_int_power
thf(fact_7448_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
        = ( ring_1_of_int_real @ X ) )
      = ( ( power_power_int @ B @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_7449_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
        = ( ring_1_of_int_int @ X ) )
      = ( ( power_power_int @ B @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_7450_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W )
        = ( ring_17405671764205052669omplex @ X ) )
      = ( ( power_power_int @ B @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_7451_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_real @ X )
        = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( X
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_7452_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_int @ X )
        = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( X
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_7453_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ( ring_17405671764205052669omplex @ X )
        = ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W ) )
      = ( X
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_7454_zero__le__arctan__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% zero_le_arctan_iff
thf(fact_7455_arctan__le__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( arctan @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% arctan_le_zero_iff
thf(fact_7456_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
    = one_one_complex ) ).

% dbl_inc_simps(2)
thf(fact_7457_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
    = one_one_real ) ).

% dbl_inc_simps(2)
thf(fact_7458_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
    = one_one_rat ) ).

% dbl_inc_simps(2)
thf(fact_7459_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
    = one_one_int ) ).

% dbl_inc_simps(2)
thf(fact_7460_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_7461_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_7462_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_inc_simps(4)
thf(fact_7463_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_inc_simps(4)
thf(fact_7464_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_7465_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
      = ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_7466_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_7467_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_7468_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) )
      = ( numeral_numeral_rat @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_7469_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_7470_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_7471_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_7472_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_7473_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_7474_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_7475_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_7476_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_7477_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_7478_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_7479_of__int__le__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
      = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_7480_of__int__le__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
      = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_7481_of__int__le__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
      = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_7482_of__int__0__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_le_iff
thf(fact_7483_of__int__0__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_le_iff
thf(fact_7484_of__int__0__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_le_iff
thf(fact_7485_of__int__less__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
      = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_7486_of__int__less__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
      = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_7487_of__int__less__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
      = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_7488_of__int__0__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_less_iff
thf(fact_7489_of__int__0__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_less_iff
thf(fact_7490_of__int__0__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_less_iff
thf(fact_7491_of__int__numeral__le__iff,axiom,
    ! [N: num,Z2: int] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).

% of_int_numeral_le_iff
thf(fact_7492_of__int__numeral__le__iff,axiom,
    ! [N: num,Z2: int] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).

% of_int_numeral_le_iff
thf(fact_7493_of__int__numeral__le__iff,axiom,
    ! [N: num,Z2: int] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).

% of_int_numeral_le_iff
thf(fact_7494_of__int__le__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_7495_of__int__le__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_7496_of__int__le__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_7497_of__int__less__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_7498_of__int__less__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_7499_of__int__less__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_7500_of__int__numeral__less__iff,axiom,
    ! [N: num,Z2: int] :
      ( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).

% of_int_numeral_less_iff
thf(fact_7501_of__int__numeral__less__iff,axiom,
    ! [N: num,Z2: int] :
      ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).

% of_int_numeral_less_iff
thf(fact_7502_of__int__numeral__less__iff,axiom,
    ! [N: num,Z2: int] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).

% of_int_numeral_less_iff
thf(fact_7503_of__int__1__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_int @ one_one_int @ Z2 ) ) ).

% of_int_1_less_iff
thf(fact_7504_of__int__1__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_int @ one_one_int @ Z2 ) ) ).

% of_int_1_less_iff
thf(fact_7505_of__int__1__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_int @ one_one_int @ Z2 ) ) ).

% of_int_1_less_iff
thf(fact_7506_of__int__less__1__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
      = ( ord_less_int @ Z2 @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_7507_of__int__less__1__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
      = ( ord_less_int @ Z2 @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_7508_of__int__less__1__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
      = ( ord_less_int @ Z2 @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_7509_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
        = ( ring_17405671764205052669omplex @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_7510_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
        = ( ring_1_of_int_real @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_7511_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = ( ring_1_of_int_int @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_7512_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
        = ( ring_1_of_int_rat @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_7513_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_7514_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_real @ Y )
        = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_7515_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_int @ Y )
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_7516_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_rat @ Y )
        = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_7517_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_7518_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_7519_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_7520_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_7521_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_7522_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_7523_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_7524_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_7525_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_7526_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_7527_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_7528_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_7529_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_7530_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_7531_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_7532_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_7533_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_7534_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_7535_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_7536_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_7537_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_7538_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_7539_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_7540_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_7541_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_real @ Y )
        = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7542_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_int @ Y )
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7543_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7544_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_18347121197199848620nteger @ Y )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7545_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_rat @ Y )
        = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7546_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N )
        = ( ring_1_of_int_real @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7547_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = ( ring_1_of_int_int @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7548_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N )
        = ( ring_17405671764205052669omplex @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7549_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N )
        = ( ring_18347121197199848620nteger @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7550_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N )
        = ( ring_1_of_int_rat @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7551_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
        @ ( unique5052692396658037445od_int @ M @ N ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_7552_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
        @ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_7553_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( produc6916734918728496179nteger
        @ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) )
        @ ( unique3479559517661332726nteger @ M @ N ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_7554_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_7555_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_7556_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_7557_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_7558_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_7559_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_7560_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_7561_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_7562_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_7563_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_7564_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_7565_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_7566_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_7567_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_7568_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_7569_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_7570_mult__of__int__commute,axiom,
    ! [X: int,Y: complex] :
      ( ( times_times_complex @ ( ring_17405671764205052669omplex @ X ) @ Y )
      = ( times_times_complex @ Y @ ( ring_17405671764205052669omplex @ X ) ) ) ).

% mult_of_int_commute
thf(fact_7571_mult__of__int__commute,axiom,
    ! [X: int,Y: real] :
      ( ( times_times_real @ ( ring_1_of_int_real @ X ) @ Y )
      = ( times_times_real @ Y @ ( ring_1_of_int_real @ X ) ) ) ).

% mult_of_int_commute
thf(fact_7572_mult__of__int__commute,axiom,
    ! [X: int,Y: rat] :
      ( ( times_times_rat @ ( ring_1_of_int_rat @ X ) @ Y )
      = ( times_times_rat @ Y @ ( ring_1_of_int_rat @ X ) ) ) ).

% mult_of_int_commute
thf(fact_7573_mult__of__int__commute,axiom,
    ! [X: int,Y: int] :
      ( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y )
      = ( times_times_int @ Y @ ( ring_1_of_int_int @ X ) ) ) ).

% mult_of_int_commute
thf(fact_7574_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_7575_real__of__int__div4,axiom,
    ! [N: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) ) ).

% real_of_int_div4
thf(fact_7576_real__of__int__div,axiom,
    ! [D: int,N: int] :
      ( ( dvd_dvd_int @ D @ N )
     => ( ( ring_1_of_int_real @ ( divide_divide_int @ N @ D ) )
        = ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ D ) ) ) ) ).

% real_of_int_div
thf(fact_7577_of__int__nonneg,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).

% of_int_nonneg
thf(fact_7578_of__int__nonneg,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_int_nonneg
thf(fact_7579_of__int__nonneg,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_nonneg
thf(fact_7580_of__int__leD,axiom,
    ! [N: int,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).

% of_int_leD
thf(fact_7581_of__int__leD,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% of_int_leD
thf(fact_7582_of__int__leD,axiom,
    ! [N: int,X: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).

% of_int_leD
thf(fact_7583_of__int__leD,axiom,
    ! [N: int,X: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).

% of_int_leD
thf(fact_7584_of__int__pos,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).

% of_int_pos
thf(fact_7585_of__int__pos,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_int_pos
thf(fact_7586_of__int__pos,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_pos
thf(fact_7587_of__int__lessD,axiom,
    ! [N: int,X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).

% of_int_lessD
thf(fact_7588_of__int__lessD,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_real @ one_one_real @ X ) ) ) ).

% of_int_lessD
thf(fact_7589_of__int__lessD,axiom,
    ! [N: int,X: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_rat @ one_one_rat @ X ) ) ) ).

% of_int_lessD
thf(fact_7590_of__int__lessD,axiom,
    ! [N: int,X: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_int @ one_one_int @ X ) ) ) ).

% of_int_lessD
thf(fact_7591_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_7592_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_7593_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_7594_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_7595_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_7596_int__le__real__less,axiom,
    ( ord_less_eq_int
    = ( ^ [N3: int,M4: int] : ( ord_less_real @ ( ring_1_of_int_real @ N3 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M4 ) @ one_one_real ) ) ) ) ).

% int_le_real_less
thf(fact_7597_int__less__real__le,axiom,
    ( ord_less_int
    = ( ^ [N3: int,M4: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N3 ) @ one_one_real ) @ ( ring_1_of_int_real @ M4 ) ) ) ) ).

% int_less_real_le
thf(fact_7598_dbl__inc__def,axiom,
    ( neg_nu8557863876264182079omplex
    = ( ^ [X6: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X6 @ X6 ) @ one_one_complex ) ) ) ).

% dbl_inc_def
thf(fact_7599_dbl__inc__def,axiom,
    ( neg_nu8295874005876285629c_real
    = ( ^ [X6: real] : ( plus_plus_real @ ( plus_plus_real @ X6 @ X6 ) @ one_one_real ) ) ) ).

% dbl_inc_def
thf(fact_7600_dbl__inc__def,axiom,
    ( neg_nu5219082963157363817nc_rat
    = ( ^ [X6: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X6 @ X6 ) @ one_one_rat ) ) ) ).

% dbl_inc_def
thf(fact_7601_dbl__inc__def,axiom,
    ( neg_nu5851722552734809277nc_int
    = ( ^ [X6: int] : ( plus_plus_int @ ( plus_plus_int @ X6 @ X6 ) @ one_one_int ) ) ) ).

% dbl_inc_def
thf(fact_7602_real__of__int__div__aux,axiom,
    ! [X: int,D: int] :
      ( ( divide_divide_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ D ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).

% real_of_int_div_aux
thf(fact_7603_real__of__int__div2,axiom,
    ! [N: int,X: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) ) ).

% real_of_int_div2
thf(fact_7604_real__of__int__div3,axiom,
    ! [N: int,X: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) @ one_one_real ) ).

% real_of_int_div3
thf(fact_7605_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_7606_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_7607_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L2: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q4: 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 ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_7608_arctan__add,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y ) )
          = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y ) ) ) ) ) ) ) ).

% arctan_add
thf(fact_7609_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L2: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q4: 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 ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_7610_divmod__step__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L2: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q4: 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 ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_def
thf(fact_7611_divmod__step__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L2: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q4: 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 ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_def
thf(fact_7612_divmod__step__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L2: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_def
thf(fact_7613_floor__exists,axiom,
    ! [X: real] :
    ? [Z4: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z4 ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_7614_floor__exists,axiom,
    ! [X: rat] :
    ? [Z4: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z4 ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_7615_floor__exists1,axiom,
    ! [X: real] :
    ? [X5: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X5 ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X5 @ one_one_int ) ) )
      & ! [Y5: int] :
          ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y5 ) @ X )
            & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
         => ( Y5 = X5 ) ) ) ).

% floor_exists1
thf(fact_7616_floor__exists1,axiom,
    ! [X: rat] :
    ? [X5: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X5 ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X5 @ one_one_int ) ) )
      & ! [Y5: int] :
          ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y5 ) @ X )
            & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
         => ( Y5 = X5 ) ) ) ).

% floor_exists1
thf(fact_7617_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M4: nat,N3: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N3 = zero_zero_nat )
            | ( ord_less_nat @ M4 @ N3 ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M4 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M4 @ N3 ) @ N3 ) ) ) ) ) ).

% divmod_nat_if
thf(fact_7618_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_7619_set__decode__0,axiom,
    ! [X: nat] :
      ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).

% set_decode_0
thf(fact_7620_set__decode__Suc,axiom,
    ! [N: nat,X: nat] :
      ( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
      = ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_decode_Suc
thf(fact_7621_set__decode__zero,axiom,
    ( ( nat_set_decode @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% set_decode_zero
thf(fact_7622_dbl__dec__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) )
      = ( numera6690914467698888265omplex @ ( bitM @ K ) ) ) ).

% dbl_dec_simps(5)
thf(fact_7623_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_7624_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_7625_dbl__dec__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) )
      = ( numeral_numeral_rat @ ( bitM @ K ) ) ) ).

% dbl_dec_simps(5)
thf(fact_7626_pred__numeral__simps_I2_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit0 @ K ) )
      = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).

% pred_numeral_simps(2)
thf(fact_7627_semiring__norm_I26_J,axiom,
    ( ( bitM @ one )
    = one ) ).

% semiring_norm(26)
thf(fact_7628_semiring__norm_I27_J,axiom,
    ! [N: num] :
      ( ( bitM @ ( bit0 @ N ) )
      = ( bit1 @ ( bitM @ N ) ) ) ).

% semiring_norm(27)
thf(fact_7629_semiring__norm_I28_J,axiom,
    ! [N: num] :
      ( ( bitM @ ( bit1 @ N ) )
      = ( bit1 @ ( bit0 @ N ) ) ) ).

% semiring_norm(28)
thf(fact_7630_eval__nat__numeral_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N ) )
      = ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).

% eval_nat_numeral(2)
thf(fact_7631_one__plus__BitM,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bitM @ N ) )
      = ( bit0 @ N ) ) ).

% one_plus_BitM
thf(fact_7632_BitM__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ ( bitM @ N ) @ one )
      = ( bit0 @ N ) ) ).

% BitM_plus_one
thf(fact_7633_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bitM @ N ) )
      = ( minus_minus_complex @ ( numera6690914467698888265omplex @ ( bit0 @ N ) ) @ one_one_complex ) ) ).

% numeral_BitM
thf(fact_7634_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bitM @ N ) )
      = ( minus_minus_real @ ( numeral_numeral_real @ ( bit0 @ N ) ) @ one_one_real ) ) ).

% numeral_BitM
thf(fact_7635_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bitM @ N ) )
      = ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ one_one_int ) ) ).

% numeral_BitM
thf(fact_7636_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bitM @ N ) )
      = ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N ) ) @ one_one_rat ) ) ).

% numeral_BitM
thf(fact_7637_odd__numeral__BitM,axiom,
    ! [W: num] :
      ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bitM @ W ) ) ) ).

% odd_numeral_BitM
thf(fact_7638_odd__numeral__BitM,axiom,
    ! [W: num] :
      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bitM @ W ) ) ) ).

% odd_numeral_BitM
thf(fact_7639_odd__numeral__BitM,axiom,
    ! [W: num] :
      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bitM @ W ) ) ) ).

% odd_numeral_BitM
thf(fact_7640_divmod__nat__def,axiom,
    ( divmod_nat
    = ( ^ [M4: nat,N3: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M4 @ N3 ) @ ( modulo_modulo_nat @ M4 @ N3 ) ) ) ) ).

% divmod_nat_def
thf(fact_7641_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_1: nat] : ( P @ X_1 )
       => ? [N2: nat] :
            ( ~ ( P @ N2 )
            & ( P @ ( suc @ N2 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_7642_ex__of__int__less,axiom,
    ! [X: real] :
    ? [Z4: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z4 ) @ X ) ).

% ex_of_int_less
thf(fact_7643_ex__of__int__less,axiom,
    ! [X: rat] :
    ? [Z4: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z4 ) @ X ) ).

% ex_of_int_less
thf(fact_7644_ex__less__of__int,axiom,
    ! [X: real] :
    ? [Z4: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z4 ) ) ).

% ex_less_of_int
thf(fact_7645_ex__less__of__int,axiom,
    ! [X: rat] :
    ? [Z4: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z4 ) ) ).

% ex_less_of_int
thf(fact_7646_set__decode__plus__power__2,axiom,
    ! [N: nat,Z2: nat] :
      ( ~ ( member_nat @ N @ ( nat_set_decode @ Z2 ) )
     => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z2 ) )
        = ( insert_nat @ N @ ( nat_set_decode @ Z2 ) ) ) ) ).

% set_decode_plus_power_2
thf(fact_7647_set__decode__def,axiom,
    ( nat_set_decode
    = ( ^ [X6: nat] :
          ( collect_nat
          @ ^ [N3: nat] :
              ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ) ).

% set_decode_def
thf(fact_7648_round__unique,axiom,
    ! [X: real,Y: int] :
      ( ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y ) )
     => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( archim8280529875227126926d_real @ X )
          = Y ) ) ) ).

% round_unique
thf(fact_7649_round__unique,axiom,
    ! [X: rat,Y: int] :
      ( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y ) )
     => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
       => ( ( archim7778729529865785530nd_rat @ X )
          = Y ) ) ) ).

% round_unique
thf(fact_7650_round__unique_H,axiom,
    ! [X: real,N: int] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( archim8280529875227126926d_real @ X )
        = N ) ) ).

% round_unique'
thf(fact_7651_round__unique_H,axiom,
    ! [X: rat,N: int] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
     => ( ( archim7778729529865785530nd_rat @ X )
        = N ) ) ).

% round_unique'
thf(fact_7652_of__int__round__abs__le,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ X ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_7653_of__int__round__abs__le,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ X ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_7654_of__int__round__gt,axiom,
    ! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).

% of_int_round_gt
thf(fact_7655_of__int__round__gt,axiom,
    ! [X: rat] : ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).

% of_int_round_gt
thf(fact_7656_of__int__round__ge,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).

% of_int_round_ge
thf(fact_7657_of__int__round__ge,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).

% of_int_round_ge
thf(fact_7658_of__int__round__le,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_7659_of__int__round__le,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_7660_round__0,axiom,
    ( ( archim8280529875227126926d_real @ zero_zero_real )
    = zero_zero_int ) ).

% round_0
thf(fact_7661_round__0,axiom,
    ( ( archim7778729529865785530nd_rat @ zero_zero_rat )
    = zero_zero_int ) ).

% round_0
thf(fact_7662_round__numeral,axiom,
    ! [N: num] :
      ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% round_numeral
thf(fact_7663_round__numeral,axiom,
    ! [N: num] :
      ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% round_numeral
thf(fact_7664_round__neg__numeral,axiom,
    ! [N: num] :
      ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% round_neg_numeral
thf(fact_7665_round__neg__numeral,axiom,
    ! [N: num] :
      ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% round_neg_numeral
thf(fact_7666_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ M @ N )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X6: int] : X6
          @ ( set_or1266510415728281911st_int @ M @ N ) )
        = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% Sum_Icc_int
thf(fact_7667_mask__numeral,axiom,
    ! [N: num] :
      ( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N ) ) ) ) ) ).

% mask_numeral
thf(fact_7668_mask__numeral,axiom,
    ! [N: num] :
      ( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N ) ) ) ) ) ).

% mask_numeral
thf(fact_7669_take__bit__rec,axiom,
    ( bit_se1745604003318907178nteger
    = ( ^ [N3: nat,A3: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_7670_take__bit__rec,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N3: nat,A3: int] : ( if_int @ ( N3 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ 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_7671_take__bit__rec,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N3: nat,A3: nat] : ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_nat @ 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_7672_tanh__real__altdef,axiom,
    ( tanh_real
    = ( ^ [X6: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X6 ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X6 ) ) ) ) ) ) ).

% tanh_real_altdef
thf(fact_7673_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_7674_or_Oright__idem,axiom,
    ! [A: int,B: int] :
      ( ( bit_se1409905431419307370or_int @ ( bit_se1409905431419307370or_int @ A @ B ) @ B )
      = ( bit_se1409905431419307370or_int @ A @ B ) ) ).

% or.right_idem
thf(fact_7675_or_Oright__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( bit_se1412395901928357646or_nat @ ( bit_se1412395901928357646or_nat @ A @ B ) @ B )
      = ( bit_se1412395901928357646or_nat @ A @ B ) ) ).

% or.right_idem
thf(fact_7676_or_Oleft__idem,axiom,
    ! [A: int,B: int] :
      ( ( bit_se1409905431419307370or_int @ A @ ( bit_se1409905431419307370or_int @ A @ B ) )
      = ( bit_se1409905431419307370or_int @ A @ B ) ) ).

% or.left_idem
thf(fact_7677_or_Oleft__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( bit_se1412395901928357646or_nat @ A @ ( bit_se1412395901928357646or_nat @ A @ B ) )
      = ( bit_se1412395901928357646or_nat @ A @ B ) ) ).

% or.left_idem
thf(fact_7678_or_Oidem,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ A @ A )
      = A ) ).

% or.idem
thf(fact_7679_or_Oidem,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ A @ A )
      = A ) ).

% or.idem
thf(fact_7680_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% mask_nat_positive_iff
thf(fact_7681_take__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ N @ zero_zero_int )
      = zero_zero_int ) ).

% take_bit_of_0
thf(fact_7682_take__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% take_bit_of_0
thf(fact_7683_or_Oright__neutral,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ A @ zero_zero_int )
      = A ) ).

% or.right_neutral
thf(fact_7684_or_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ A @ zero_zero_nat )
      = A ) ).

% or.right_neutral
thf(fact_7685_or_Oleft__neutral,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ zero_zero_int @ A )
      = A ) ).

% or.left_neutral
thf(fact_7686_or_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ zero_zero_nat @ A )
      = A ) ).

% or.left_neutral
thf(fact_7687_take__bit__and,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( bit_se725231765392027082nd_int @ A @ B ) )
      = ( bit_se725231765392027082nd_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) ) ).

% take_bit_and
thf(fact_7688_take__bit__and,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se727722235901077358nd_nat @ A @ B ) )
      = ( bit_se727722235901077358nd_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) ) ).

% take_bit_and
thf(fact_7689_take__bit__or,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( bit_se1409905431419307370or_int @ A @ B ) )
      = ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) ) ).

% take_bit_or
thf(fact_7690_take__bit__or,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se1412395901928357646or_nat @ A @ B ) )
      = ( bit_se1412395901928357646or_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) ) ).

% take_bit_or
thf(fact_7691_concat__bit__of__zero__2,axiom,
    ! [N: nat,K: int] :
      ( ( bit_concat_bit @ N @ K @ zero_zero_int )
      = ( bit_se2923211474154528505it_int @ N @ K ) ) ).

% concat_bit_of_zero_2
thf(fact_7692_sum_Oneutral__const,axiom,
    ! [A2: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [Uu3: int] : zero_zero_int
        @ A2 )
      = zero_zero_int ) ).

% sum.neutral_const
thf(fact_7693_sum_Oneutral__const,axiom,
    ! [A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [Uu3: complex] : zero_zero_complex
        @ A2 )
      = zero_zero_complex ) ).

% sum.neutral_const
thf(fact_7694_sum_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [Uu3: nat] : zero_zero_nat
        @ A2 )
      = zero_zero_nat ) ).

% sum.neutral_const
thf(fact_7695_sum_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [Uu3: nat] : zero_zero_real
        @ A2 )
      = zero_zero_real ) ).

% sum.neutral_const
thf(fact_7696_sum_Oempty,axiom,
    ! [G: real > complex] :
      ( ( groups5754745047067104278omplex @ G @ bot_bot_set_real )
      = zero_zero_complex ) ).

% sum.empty
thf(fact_7697_sum_Oempty,axiom,
    ! [G: real > real] :
      ( ( groups8097168146408367636l_real @ G @ bot_bot_set_real )
      = zero_zero_real ) ).

% sum.empty
thf(fact_7698_sum_Oempty,axiom,
    ! [G: real > rat] :
      ( ( groups1300246762558778688al_rat @ G @ bot_bot_set_real )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_7699_sum_Oempty,axiom,
    ! [G: real > nat] :
      ( ( groups1935376822645274424al_nat @ G @ bot_bot_set_real )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_7700_sum_Oempty,axiom,
    ! [G: real > int] :
      ( ( groups1932886352136224148al_int @ G @ bot_bot_set_real )
      = zero_zero_int ) ).

% sum.empty
thf(fact_7701_sum_Oempty,axiom,
    ! [G: $o > complex] :
      ( ( groups5328290441151304332omplex @ G @ bot_bot_set_o )
      = zero_zero_complex ) ).

% sum.empty
thf(fact_7702_sum_Oempty,axiom,
    ! [G: $o > real] :
      ( ( groups8691415230153176458o_real @ G @ bot_bot_set_o )
      = zero_zero_real ) ).

% sum.empty
thf(fact_7703_sum_Oempty,axiom,
    ! [G: $o > rat] :
      ( ( groups7872700643590313910_o_rat @ G @ bot_bot_set_o )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_7704_sum_Oempty,axiom,
    ! [G: $o > nat] :
      ( ( groups8507830703676809646_o_nat @ G @ bot_bot_set_o )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_7705_sum_Oempty,axiom,
    ! [G: $o > int] :
      ( ( groups8505340233167759370_o_int @ G @ bot_bot_set_o )
      = zero_zero_int ) ).

% sum.empty
thf(fact_7706_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > complex] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups2073611262835488442omplex @ G @ A2 )
        = zero_zero_complex ) ) ).

% sum.infinite
thf(fact_7707_sum_Oinfinite,axiom,
    ! [A2: set_int,G: int > complex] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups3049146728041665814omplex @ G @ A2 )
        = zero_zero_complex ) ) ).

% sum.infinite
thf(fact_7708_sum_Oinfinite,axiom,
    ! [A2: set_int,G: int > real] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G @ A2 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_7709_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > real] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G @ A2 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_7710_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > rat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups2906978787729119204at_rat @ G @ A2 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_7711_sum_Oinfinite,axiom,
    ! [A2: set_int,G: int > rat] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat @ G @ A2 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_7712_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > rat] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G @ A2 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_7713_sum_Oinfinite,axiom,
    ! [A2: set_int,G: int > nat] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat @ G @ A2 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_7714_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > nat] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G @ A2 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_7715_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > int] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups3539618377306564664at_int @ G @ A2 )
        = zero_zero_int ) ) ).

% sum.infinite
thf(fact_7716_sum__eq__0__iff,axiom,
    ! [F3: set_int,F: int > nat] :
      ( ( finite_finite_int @ F3 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ F3 )
          = zero_zero_nat )
        = ( ! [X6: int] :
              ( ( member_int @ X6 @ F3 )
             => ( ( F @ X6 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_7717_sum__eq__0__iff,axiom,
    ! [F3: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ F3 )
          = zero_zero_nat )
        = ( ! [X6: complex] :
              ( ( member_complex @ X6 @ F3 )
             => ( ( F @ X6 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_7718_sum__eq__0__iff,axiom,
    ! [F3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ F3 )
     => ( ( ( groups3542108847815614940at_nat @ F @ F3 )
          = zero_zero_nat )
        = ( ! [X6: nat] :
              ( ( member_nat @ X6 @ F3 )
             => ( ( F @ X6 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_7719_take__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
      = zero_zero_int ) ).

% take_bit_0
thf(fact_7720_take__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% take_bit_0
thf(fact_7721_exp__zero,axiom,
    ( ( exp_complex @ zero_zero_complex )
    = one_one_complex ) ).

% exp_zero
thf(fact_7722_exp__zero,axiom,
    ( ( exp_real @ zero_zero_real )
    = one_one_real ) ).

% exp_zero
thf(fact_7723_take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ one_one_int )
      = one_one_int ) ).

% take_bit_Suc_1
thf(fact_7724_take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ one_one_nat )
      = one_one_nat ) ).

% take_bit_Suc_1
thf(fact_7725_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_7726_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_7727_bit_Odisj__one__left,axiom,
    ! [X: code_integer] :
      ( ( bit_se1080825931792720795nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.disj_one_left
thf(fact_7728_bit_Odisj__one__left,axiom,
    ! [X: int] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ one_one_int ) @ X )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.disj_one_left
thf(fact_7729_bit_Odisj__one__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se1080825931792720795nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.disj_one_right
thf(fact_7730_bit_Odisj__one__right,axiom,
    ! [X: int] :
      ( ( bit_se1409905431419307370or_int @ X @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.disj_one_right
thf(fact_7731_mask__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se2002935070580805687sk_nat @ N )
        = zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% mask_eq_0_iff
thf(fact_7732_mask__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se2000444600071755411sk_int @ N )
        = zero_zero_int )
      = ( N = zero_zero_nat ) ) ).

% mask_eq_0_iff
thf(fact_7733_mask__0,axiom,
    ( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% mask_0
thf(fact_7734_mask__0,axiom,
    ( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
    = zero_zero_int ) ).

% mask_0
thf(fact_7735_exp__eq__one__iff,axiom,
    ! [X: real] :
      ( ( ( exp_real @ X )
        = one_one_real )
      = ( X = zero_zero_real ) ) ).

% exp_eq_one_iff
thf(fact_7736_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_7737_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_7738_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > complex] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_7739_sum_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_7740_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_7741_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_7742_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_7743_sum_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_7744_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_7745_sum_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_7746_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_7747_sum_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_7748_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > complex] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_7749_sum_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_7750_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_7751_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_7752_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_7753_sum_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_7754_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_7755_sum_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_7756_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_7757_sum_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_7758_sum_Oinsert,axiom,
    ! [A2: set_o,X: $o,G: $o > real] :
      ( ( finite_finite_o @ A2 )
     => ( ~ ( member_o @ X @ A2 )
       => ( ( groups8691415230153176458o_real @ G @ ( insert_o @ X @ A2 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups8691415230153176458o_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7759_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7760_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7761_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7762_sum_Oinsert,axiom,
    ! [A2: set_o,X: $o,G: $o > rat] :
      ( ( finite_finite_o @ A2 )
     => ( ~ ( member_o @ X @ A2 )
       => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7763_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7764_sum_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X @ A2 )
       => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7765_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7766_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7767_sum_Oinsert,axiom,
    ! [A2: set_o,X: $o,G: $o > nat] :
      ( ( finite_finite_o @ A2 )
     => ( ~ ( member_o @ X @ A2 )
       => ( ( groups8507830703676809646_o_nat @ G @ ( insert_o @ X @ A2 ) )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups8507830703676809646_o_nat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7768_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
        = zero_zero_int )
      = ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_7769_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
        = zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_7770_or__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
      = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).

% or_numerals(8)
thf(fact_7771_or__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_numerals(8)
thf(fact_7772_or__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_int @ ( bit1 @ Y ) ) ) ).

% or_numerals(2)
thf(fact_7773_or__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_numerals(2)
thf(fact_7774_mask__Suc__0,axiom,
    ( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% mask_Suc_0
thf(fact_7775_mask__Suc__0,axiom,
    ( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% mask_Suc_0
thf(fact_7776_one__less__exp__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ ( exp_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% one_less_exp_iff
thf(fact_7777_exp__less__one__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( exp_real @ X ) @ one_one_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% exp_less_one_iff
thf(fact_7778_exp__le__one__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X ) @ one_one_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% exp_le_one_iff
thf(fact_7779_one__le__exp__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% one_le_exp_iff
thf(fact_7780_take__bit__minus__one__eq__mask,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( bit_se2119862282449309892nteger @ N ) ) ).

% take_bit_minus_one_eq_mask
thf(fact_7781_take__bit__minus__one__eq__mask,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( bit_se2000444600071755411sk_int @ N ) ) ).

% take_bit_minus_one_eq_mask
thf(fact_7782_exp__ln__iff,axiom,
    ! [X: real] :
      ( ( ( exp_real @ ( ln_ln_real @ X ) )
        = X )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% exp_ln_iff
thf(fact_7783_exp__ln,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( exp_real @ ( ln_ln_real @ X ) )
        = X ) ) ).

% exp_ln
thf(fact_7784_take__bit__of__Suc__0,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% take_bit_of_Suc_0
thf(fact_7785_sum__abs__ge__zero,axiom,
    ! [F: int > int,A2: set_int] :
      ( ord_less_eq_int @ zero_zero_int
      @ ( groups4538972089207619220nt_int
        @ ^ [I: int] : ( abs_abs_int @ ( F @ I ) )
        @ A2 ) ) ).

% sum_abs_ge_zero
thf(fact_7786_sum__abs__ge__zero,axiom,
    ! [F: nat > real,A2: set_nat] :
      ( ord_less_eq_real @ zero_zero_real
      @ ( groups6591440286371151544t_real
        @ ^ [I: nat] : ( abs_abs_real @ ( F @ I ) )
        @ A2 ) ) ).

% sum_abs_ge_zero
thf(fact_7787_or__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% or_numerals(3)
thf(fact_7788_or__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% or_numerals(3)
thf(fact_7789_or__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_int @ ( bit1 @ Y ) ) ) ).

% or_numerals(1)
thf(fact_7790_or__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_numerals(1)
thf(fact_7791_or__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
      = ( numeral_numeral_int @ ( bit1 @ X ) ) ) ).

% or_numerals(5)
thf(fact_7792_or__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_numerals(5)
thf(fact_7793_take__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ N @ one_one_Code_integer )
      = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% take_bit_of_1
thf(fact_7794_take__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% take_bit_of_1
thf(fact_7795_take__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% take_bit_of_1
thf(fact_7796_or__minus__numerals_I6_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).

% or_minus_numerals(6)
thf(fact_7797_or__minus__numerals_I2_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).

% or_minus_numerals(2)
thf(fact_7798_sum__of__bool__mult__eq,axiom,
    ! [A2: set_real,P: real > $o,F: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups8097168146408367636l_real
          @ ^ [X6: real] : ( times_times_real @ ( zero_n3304061248610475627l_real @ ( P @ X6 ) ) @ ( F @ X6 ) )
          @ A2 )
        = ( groups8097168146408367636l_real @ F @ ( inf_inf_set_real @ A2 @ ( collect_real @ P ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_7799_sum__of__bool__mult__eq,axiom,
    ! [A2: set_int,P: int > $o,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real
          @ ^ [X6: int] : ( times_times_real @ ( zero_n3304061248610475627l_real @ ( P @ X6 ) ) @ ( F @ X6 ) )
          @ A2 )
        = ( groups8778361861064173332t_real @ F @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_7800_sum__of__bool__mult__eq,axiom,
    ! [A2: set_complex,P: complex > $o,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real
          @ ^ [X6: complex] : ( times_times_real @ ( zero_n3304061248610475627l_real @ ( P @ X6 ) ) @ ( F @ X6 ) )
          @ A2 )
        = ( groups5808333547571424918x_real @ F @ ( inf_inf_set_complex @ A2 @ ( collect_complex @ P ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_7801_sum__of__bool__mult__eq,axiom,
    ! [A2: set_real,P: real > $o,F: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1300246762558778688al_rat
          @ ^ [X6: real] : ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ ( P @ X6 ) ) @ ( F @ X6 ) )
          @ A2 )
        = ( groups1300246762558778688al_rat @ F @ ( inf_inf_set_real @ A2 @ ( collect_real @ P ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_7802_sum__of__bool__mult__eq,axiom,
    ! [A2: set_int,P: int > $o,F: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat
          @ ^ [X6: int] : ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ ( P @ X6 ) ) @ ( F @ X6 ) )
          @ A2 )
        = ( groups3906332499630173760nt_rat @ F @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_7803_sum__of__bool__mult__eq,axiom,
    ! [A2: set_complex,P: complex > $o,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat
          @ ^ [X6: complex] : ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ ( P @ X6 ) ) @ ( F @ X6 ) )
          @ A2 )
        = ( groups5058264527183730370ex_rat @ F @ ( inf_inf_set_complex @ A2 @ ( collect_complex @ P ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_7804_sum__of__bool__mult__eq,axiom,
    ! [A2: set_nat,P: nat > $o,F: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [X6: nat] : ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ ( P @ X6 ) ) @ ( F @ X6 ) )
          @ A2 )
        = ( groups2906978787729119204at_rat @ F @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_7805_sum__of__bool__mult__eq,axiom,
    ! [A2: set_real,P: real > $o,F: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1935376822645274424al_nat
          @ ^ [X6: real] : ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( P @ X6 ) ) @ ( F @ X6 ) )
          @ A2 )
        = ( groups1935376822645274424al_nat @ F @ ( inf_inf_set_real @ A2 @ ( collect_real @ P ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_7806_sum__of__bool__mult__eq,axiom,
    ! [A2: set_int,P: int > $o,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X6: int] : ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( P @ X6 ) ) @ ( F @ X6 ) )
          @ A2 )
        = ( groups4541462559716669496nt_nat @ F @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_7807_sum__of__bool__mult__eq,axiom,
    ! [A2: set_complex,P: complex > $o,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat
          @ ^ [X6: complex] : ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( P @ X6 ) ) @ ( F @ X6 ) )
          @ A2 )
        = ( groups5693394587270226106ex_nat @ F @ ( inf_inf_set_complex @ A2 @ ( collect_complex @ P ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_7808_sum__mult__of__bool__eq,axiom,
    ! [A2: set_real,F: real > real,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups8097168146408367636l_real
          @ ^ [X6: real] : ( times_times_real @ ( F @ X6 ) @ ( zero_n3304061248610475627l_real @ ( P @ X6 ) ) )
          @ A2 )
        = ( groups8097168146408367636l_real @ F @ ( inf_inf_set_real @ A2 @ ( collect_real @ P ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_7809_sum__mult__of__bool__eq,axiom,
    ! [A2: set_int,F: int > real,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real
          @ ^ [X6: int] : ( times_times_real @ ( F @ X6 ) @ ( zero_n3304061248610475627l_real @ ( P @ X6 ) ) )
          @ A2 )
        = ( groups8778361861064173332t_real @ F @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_7810_sum__mult__of__bool__eq,axiom,
    ! [A2: set_complex,F: complex > real,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real
          @ ^ [X6: complex] : ( times_times_real @ ( F @ X6 ) @ ( zero_n3304061248610475627l_real @ ( P @ X6 ) ) )
          @ A2 )
        = ( groups5808333547571424918x_real @ F @ ( inf_inf_set_complex @ A2 @ ( collect_complex @ P ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_7811_sum__mult__of__bool__eq,axiom,
    ! [A2: set_real,F: real > rat,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1300246762558778688al_rat
          @ ^ [X6: real] : ( times_times_rat @ ( F @ X6 ) @ ( zero_n2052037380579107095ol_rat @ ( P @ X6 ) ) )
          @ A2 )
        = ( groups1300246762558778688al_rat @ F @ ( inf_inf_set_real @ A2 @ ( collect_real @ P ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_7812_sum__mult__of__bool__eq,axiom,
    ! [A2: set_int,F: int > rat,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat
          @ ^ [X6: int] : ( times_times_rat @ ( F @ X6 ) @ ( zero_n2052037380579107095ol_rat @ ( P @ X6 ) ) )
          @ A2 )
        = ( groups3906332499630173760nt_rat @ F @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_7813_sum__mult__of__bool__eq,axiom,
    ! [A2: set_complex,F: complex > rat,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat
          @ ^ [X6: complex] : ( times_times_rat @ ( F @ X6 ) @ ( zero_n2052037380579107095ol_rat @ ( P @ X6 ) ) )
          @ A2 )
        = ( groups5058264527183730370ex_rat @ F @ ( inf_inf_set_complex @ A2 @ ( collect_complex @ P ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_7814_sum__mult__of__bool__eq,axiom,
    ! [A2: set_nat,F: nat > rat,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [X6: nat] : ( times_times_rat @ ( F @ X6 ) @ ( zero_n2052037380579107095ol_rat @ ( P @ X6 ) ) )
          @ A2 )
        = ( groups2906978787729119204at_rat @ F @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_7815_sum__mult__of__bool__eq,axiom,
    ! [A2: set_real,F: real > nat,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1935376822645274424al_nat
          @ ^ [X6: real] : ( times_times_nat @ ( F @ X6 ) @ ( zero_n2687167440665602831ol_nat @ ( P @ X6 ) ) )
          @ A2 )
        = ( groups1935376822645274424al_nat @ F @ ( inf_inf_set_real @ A2 @ ( collect_real @ P ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_7816_sum__mult__of__bool__eq,axiom,
    ! [A2: set_int,F: int > nat,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X6: int] : ( times_times_nat @ ( F @ X6 ) @ ( zero_n2687167440665602831ol_nat @ ( P @ X6 ) ) )
          @ A2 )
        = ( groups4541462559716669496nt_nat @ F @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_7817_sum__mult__of__bool__eq,axiom,
    ! [A2: set_complex,F: complex > nat,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat
          @ ^ [X6: complex] : ( times_times_nat @ ( F @ X6 ) @ ( zero_n2687167440665602831ol_nat @ ( P @ X6 ) ) )
          @ A2 )
        = ( groups5693394587270226106ex_nat @ F @ ( inf_inf_set_complex @ A2 @ ( collect_complex @ P ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_7818_even__take__bit__eq,axiom,
    ! [N: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1745604003318907178nteger @ N @ A ) )
      = ( ( N = zero_zero_nat )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_take_bit_eq
thf(fact_7819_even__take__bit__eq,axiom,
    ! [N: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N @ A ) )
      = ( ( N = zero_zero_nat )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_take_bit_eq
thf(fact_7820_even__take__bit__eq,axiom,
    ! [N: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N @ A ) )
      = ( ( N = zero_zero_nat )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_take_bit_eq
thf(fact_7821_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_7822_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_7823_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_7824_or__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% or_numerals(4)
thf(fact_7825_or__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% or_numerals(4)
thf(fact_7826_or__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% or_numerals(6)
thf(fact_7827_or__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% or_numerals(6)
thf(fact_7828_or__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% or_numerals(7)
thf(fact_7829_or__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% or_numerals(7)
thf(fact_7830_take__bit__of__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( bit_se1745604003318907178nteger @ M @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ N @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_of_exp
thf(fact_7831_take__bit__of__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_of_exp
thf(fact_7832_take__bit__of__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_of_exp
thf(fact_7833_take__bit__of__2,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_7834_take__bit__of__2,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_7835_take__bit__of__2,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_7836_take__bit__eq__mask,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N3: nat,A3: int] : ( bit_se725231765392027082nd_int @ A3 @ ( bit_se2000444600071755411sk_int @ N3 ) ) ) ) ).

% take_bit_eq_mask
thf(fact_7837_take__bit__eq__mask,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N3: nat,A3: nat] : ( bit_se727722235901077358nd_nat @ A3 @ ( bit_se2002935070580805687sk_nat @ N3 ) ) ) ) ).

% take_bit_eq_mask
thf(fact_7838_or__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( bit_se1409905431419307370or_int @ A @ B )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        & ( B = zero_zero_int ) ) ) ).

% or_eq_0_iff
thf(fact_7839_or__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( bit_se1412395901928357646or_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% or_eq_0_iff
thf(fact_7840_bit_Odisj__zero__right,axiom,
    ! [X: int] :
      ( ( bit_se1409905431419307370or_int @ X @ zero_zero_int )
      = X ) ).

% bit.disj_zero_right
thf(fact_7841_or_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( bit_se1409905431419307370or_int @ B @ ( bit_se1409905431419307370or_int @ A @ C ) )
      = ( bit_se1409905431419307370or_int @ A @ ( bit_se1409905431419307370or_int @ B @ C ) ) ) ).

% or.left_commute
thf(fact_7842_or_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( bit_se1412395901928357646or_nat @ B @ ( bit_se1412395901928357646or_nat @ A @ C ) )
      = ( bit_se1412395901928357646or_nat @ A @ ( bit_se1412395901928357646or_nat @ B @ C ) ) ) ).

% or.left_commute
thf(fact_7843_or_Ocommute,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [A3: int,B3: int] : ( bit_se1409905431419307370or_int @ B3 @ A3 ) ) ) ).

% or.commute
thf(fact_7844_or_Ocommute,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [A3: nat,B3: nat] : ( bit_se1412395901928357646or_nat @ B3 @ A3 ) ) ) ).

% or.commute
thf(fact_7845_or_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( bit_se1409905431419307370or_int @ ( bit_se1409905431419307370or_int @ A @ B ) @ C )
      = ( bit_se1409905431419307370or_int @ A @ ( bit_se1409905431419307370or_int @ B @ C ) ) ) ).

% or.assoc
thf(fact_7846_or_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( bit_se1412395901928357646or_nat @ ( bit_se1412395901928357646or_nat @ A @ B ) @ C )
      = ( bit_se1412395901928357646or_nat @ A @ ( bit_se1412395901928357646or_nat @ B @ C ) ) ) ).

% or.assoc
thf(fact_7847_sum_Oneutral,axiom,
    ! [A2: set_int,G: int > int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( ( G @ X5 )
            = zero_zero_int ) )
     => ( ( groups4538972089207619220nt_int @ G @ A2 )
        = zero_zero_int ) ) ).

% sum.neutral
thf(fact_7848_sum_Oneutral,axiom,
    ! [A2: set_complex,G: complex > complex] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A2 )
         => ( ( G @ X5 )
            = zero_zero_complex ) )
     => ( ( groups7754918857620584856omplex @ G @ A2 )
        = zero_zero_complex ) ) ).

% sum.neutral
thf(fact_7849_sum_Oneutral,axiom,
    ! [A2: set_nat,G: nat > nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( ( G @ X5 )
            = zero_zero_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ A2 )
        = zero_zero_nat ) ) ).

% sum.neutral
thf(fact_7850_sum_Oneutral,axiom,
    ! [A2: set_nat,G: nat > real] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( ( G @ X5 )
            = zero_zero_real ) )
     => ( ( groups6591440286371151544t_real @ G @ A2 )
        = zero_zero_real ) ) ).

% sum.neutral
thf(fact_7851_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > complex,A2: set_real] :
      ( ( ( groups5754745047067104278omplex @ G @ A2 )
       != zero_zero_complex )
     => ~ ! [A5: real] :
            ( ( member_real @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_7852_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > complex,A2: set_nat] :
      ( ( ( groups2073611262835488442omplex @ G @ A2 )
       != zero_zero_complex )
     => ~ ! [A5: nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_7853_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > complex,A2: set_int] :
      ( ( ( groups3049146728041665814omplex @ G @ A2 )
       != zero_zero_complex )
     => ~ ! [A5: int] :
            ( ( member_int @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_7854_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: complex > real,A2: set_complex] :
      ( ( ( groups5808333547571424918x_real @ G @ A2 )
       != zero_zero_real )
     => ~ ! [A5: complex] :
            ( ( member_complex @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_7855_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > real,A2: set_real] :
      ( ( ( groups8097168146408367636l_real @ G @ A2 )
       != zero_zero_real )
     => ~ ! [A5: real] :
            ( ( member_real @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_7856_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > real,A2: set_int] :
      ( ( ( groups8778361861064173332t_real @ G @ A2 )
       != zero_zero_real )
     => ~ ! [A5: int] :
            ( ( member_int @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_7857_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: complex > rat,A2: set_complex] :
      ( ( ( groups5058264527183730370ex_rat @ G @ A2 )
       != zero_zero_rat )
     => ~ ! [A5: complex] :
            ( ( member_complex @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_7858_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > rat,A2: set_real] :
      ( ( ( groups1300246762558778688al_rat @ G @ A2 )
       != zero_zero_rat )
     => ~ ! [A5: real] :
            ( ( member_real @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_7859_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > rat,A2: set_nat] :
      ( ( ( groups2906978787729119204at_rat @ G @ A2 )
       != zero_zero_rat )
     => ~ ! [A5: nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_7860_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > rat,A2: set_int] :
      ( ( ( groups3906332499630173760nt_rat @ G @ A2 )
       != zero_zero_rat )
     => ~ ! [A5: int] :
            ( ( member_int @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_7861_take__bit__add,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ A @ B ) ) ) ).

% take_bit_add
thf(fact_7862_take__bit__add,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) )
      = ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ A @ B ) ) ) ).

% take_bit_add
thf(fact_7863_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).

% take_bit_nat_less_eq_self
thf(fact_7864_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N @ Q2 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_7865_take__bit__tightened,axiom,
    ! [N: nat,A: int,B: int,M: nat] :
      ( ( ( bit_se2923211474154528505it_int @ N @ A )
        = ( bit_se2923211474154528505it_int @ N @ B ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( bit_se2923211474154528505it_int @ M @ A )
          = ( bit_se2923211474154528505it_int @ M @ B ) ) ) ) ).

% take_bit_tightened
thf(fact_7866_take__bit__tightened,axiom,
    ! [N: nat,A: nat,B: nat,M: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ A )
        = ( bit_se2925701944663578781it_nat @ N @ B ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( bit_se2925701944663578781it_nat @ M @ A )
          = ( bit_se2925701944663578781it_nat @ M @ B ) ) ) ) ).

% take_bit_tightened
thf(fact_7867_take__bit__of__int,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( ring_1_of_int_int @ K ) )
      = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% take_bit_of_int
thf(fact_7868_of__int__mask__eq,axiom,
    ! [N: nat] :
      ( ( ring_1_of_int_int @ ( bit_se2000444600071755411sk_int @ N ) )
      = ( bit_se2000444600071755411sk_int @ N ) ) ).

% of_int_mask_eq
thf(fact_7869_of__int__or__eq,axiom,
    ! [K: int,L: int] :
      ( ( ring_1_of_int_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
      = ( bit_se1409905431419307370or_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L ) ) ) ).

% of_int_or_eq
thf(fact_7870_exp__not__eq__zero,axiom,
    ! [X: complex] :
      ( ( exp_complex @ X )
     != zero_zero_complex ) ).

% exp_not_eq_zero
thf(fact_7871_exp__not__eq__zero,axiom,
    ! [X: real] :
      ( ( exp_real @ X )
     != zero_zero_real ) ).

% exp_not_eq_zero
thf(fact_7872_take__bit__minus,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).

% take_bit_minus
thf(fact_7873_exp__times__arg__commute,axiom,
    ! [A2: complex] :
      ( ( times_times_complex @ ( exp_complex @ A2 ) @ A2 )
      = ( times_times_complex @ A2 @ ( exp_complex @ A2 ) ) ) ).

% exp_times_arg_commute
thf(fact_7874_exp__times__arg__commute,axiom,
    ! [A2: real] :
      ( ( times_times_real @ ( exp_real @ A2 ) @ A2 )
      = ( times_times_real @ A2 @ ( exp_real @ A2 ) ) ) ).

% exp_times_arg_commute
thf(fact_7875_take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ K @ L ) ) ) ).

% take_bit_mult
thf(fact_7876_take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ L ) ) ) ).

% take_bit_diff
thf(fact_7877_bit_Odisj__conj__distrib2,axiom,
    ! [Y: int,Z2: int,X: int] :
      ( ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ Y @ Z2 ) @ X )
      = ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ Y @ X ) @ ( bit_se1409905431419307370or_int @ Z2 @ X ) ) ) ).

% bit.disj_conj_distrib2
thf(fact_7878_bit_Oconj__disj__distrib2,axiom,
    ! [Y: int,Z2: int,X: int] :
      ( ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ Y @ Z2 ) @ X )
      = ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ Y @ X ) @ ( bit_se725231765392027082nd_int @ Z2 @ X ) ) ) ).

% bit.conj_disj_distrib2
thf(fact_7879_bit_Odisj__conj__distrib,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( bit_se1409905431419307370or_int @ X @ ( bit_se725231765392027082nd_int @ Y @ Z2 ) )
      = ( bit_se725231765392027082nd_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( bit_se1409905431419307370or_int @ X @ Z2 ) ) ) ).

% bit.disj_conj_distrib
thf(fact_7880_bit_Oconj__disj__distrib,axiom,
    ! [X: int,Y: int,Z2: int] :
      ( ( bit_se725231765392027082nd_int @ X @ ( bit_se1409905431419307370or_int @ Y @ Z2 ) )
      = ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ ( bit_se725231765392027082nd_int @ X @ Z2 ) ) ) ).

% bit.conj_disj_distrib
thf(fact_7881_concat__bit__take__bit__eq,axiom,
    ! [N: nat,B: int] :
      ( ( bit_concat_bit @ N @ ( bit_se2923211474154528505it_int @ N @ B ) )
      = ( bit_concat_bit @ N @ B ) ) ).

% concat_bit_take_bit_eq
thf(fact_7882_concat__bit__eq__iff,axiom,
    ! [N: nat,K: int,L: int,R2: int,S: int] :
      ( ( ( bit_concat_bit @ N @ K @ L )
        = ( bit_concat_bit @ N @ R2 @ S ) )
      = ( ( ( bit_se2923211474154528505it_int @ N @ K )
          = ( bit_se2923211474154528505it_int @ N @ R2 ) )
        & ( L = S ) ) ) ).

% concat_bit_eq_iff
thf(fact_7883_less__eq__mask,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).

% less_eq_mask
thf(fact_7884_sum__product,axiom,
    ! [F: int > int,A2: set_int,G: int > int,B2: set_int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ B2 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [I: int] :
            ( groups4538972089207619220nt_int
            @ ^ [J2: int] : ( times_times_int @ ( F @ I ) @ ( G @ J2 ) )
            @ B2 )
        @ A2 ) ) ).

% sum_product
thf(fact_7885_sum__product,axiom,
    ! [F: complex > complex,A2: set_complex,G: complex > complex,B2: set_complex] :
      ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ B2 ) )
      = ( groups7754918857620584856omplex
        @ ^ [I: complex] :
            ( groups7754918857620584856omplex
            @ ^ [J2: complex] : ( times_times_complex @ ( F @ I ) @ ( G @ J2 ) )
            @ B2 )
        @ A2 ) ) ).

% sum_product
thf(fact_7886_sum__product,axiom,
    ! [F: nat > nat,A2: set_nat,G: nat > nat,B2: set_nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ B2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [J2: nat] : ( times_times_nat @ ( F @ I ) @ ( G @ J2 ) )
            @ B2 )
        @ A2 ) ) ).

% sum_product
thf(fact_7887_sum__product,axiom,
    ! [F: nat > real,A2: set_nat,G: nat > real,B2: set_nat] :
      ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ B2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [I: nat] :
            ( groups6591440286371151544t_real
            @ ^ [J2: nat] : ( times_times_real @ ( F @ I ) @ ( G @ J2 ) )
            @ B2 )
        @ A2 ) ) ).

% sum_product
thf(fact_7888_sum__distrib__right,axiom,
    ! [F: int > int,A2: set_int,R2: int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ R2 )
      = ( groups4538972089207619220nt_int
        @ ^ [N3: int] : ( times_times_int @ ( F @ N3 ) @ R2 )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_7889_sum__distrib__right,axiom,
    ! [F: complex > complex,A2: set_complex,R2: complex] :
      ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R2 )
      = ( groups7754918857620584856omplex
        @ ^ [N3: complex] : ( times_times_complex @ ( F @ N3 ) @ R2 )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_7890_sum__distrib__right,axiom,
    ! [F: nat > nat,A2: set_nat,R2: nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ R2 )
      = ( groups3542108847815614940at_nat
        @ ^ [N3: nat] : ( times_times_nat @ ( F @ N3 ) @ R2 )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_7891_sum__distrib__right,axiom,
    ! [F: nat > real,A2: set_nat,R2: real] :
      ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R2 )
      = ( groups6591440286371151544t_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ R2 )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_7892_sum__distrib__left,axiom,
    ! [R2: int,F: int > int,A2: set_int] :
      ( ( times_times_int @ R2 @ ( groups4538972089207619220nt_int @ F @ A2 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [N3: int] : ( times_times_int @ R2 @ ( F @ N3 ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_7893_sum__distrib__left,axiom,
    ! [R2: complex,F: complex > complex,A2: set_complex] :
      ( ( times_times_complex @ R2 @ ( groups7754918857620584856omplex @ F @ A2 ) )
      = ( groups7754918857620584856omplex
        @ ^ [N3: complex] : ( times_times_complex @ R2 @ ( F @ N3 ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_7894_sum__distrib__left,axiom,
    ! [R2: nat,F: nat > nat,A2: set_nat] :
      ( ( times_times_nat @ R2 @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [N3: nat] : ( times_times_nat @ R2 @ ( F @ N3 ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_7895_sum__distrib__left,axiom,
    ! [R2: real,F: nat > real,A2: set_nat] :
      ( ( times_times_real @ R2 @ ( groups6591440286371151544t_real @ F @ A2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [N3: nat] : ( times_times_real @ R2 @ ( F @ N3 ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_7896_sum_Odistrib,axiom,
    ! [G: int > int,H2: int > int,A2: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X6: int] : ( plus_plus_int @ ( G @ X6 ) @ ( H2 @ X6 ) )
        @ A2 )
      = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A2 ) @ ( groups4538972089207619220nt_int @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_7897_sum_Odistrib,axiom,
    ! [G: complex > complex,H2: complex > complex,A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X6: complex] : ( plus_plus_complex @ ( G @ X6 ) @ ( H2 @ X6 ) )
        @ A2 )
      = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_7898_sum_Odistrib,axiom,
    ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X6: nat] : ( plus_plus_nat @ ( G @ X6 ) @ ( H2 @ X6 ) )
        @ A2 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_7899_sum_Odistrib,axiom,
    ! [G: nat > real,H2: nat > real,A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X6: nat] : ( plus_plus_real @ ( G @ X6 ) @ ( H2 @ X6 ) )
        @ A2 )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_7900_sum__divide__distrib,axiom,
    ! [F: complex > complex,A2: set_complex,R2: complex] :
      ( ( divide1717551699836669952omplex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R2 )
      = ( groups7754918857620584856omplex
        @ ^ [N3: complex] : ( divide1717551699836669952omplex @ ( F @ N3 ) @ R2 )
        @ A2 ) ) ).

% sum_divide_distrib
thf(fact_7901_sum__divide__distrib,axiom,
    ! [F: nat > real,A2: set_nat,R2: real] :
      ( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R2 )
      = ( groups6591440286371151544t_real
        @ ^ [N3: nat] : ( divide_divide_real @ ( F @ N3 ) @ R2 )
        @ A2 ) ) ).

% sum_divide_distrib
thf(fact_7902_mod__sum__eq,axiom,
    ! [F: int > int,A: int,A2: set_int] :
      ( ( modulo_modulo_int
        @ ( groups4538972089207619220nt_int
          @ ^ [I: int] : ( modulo_modulo_int @ ( F @ I ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ A ) ) ).

% mod_sum_eq
thf(fact_7903_mod__sum__eq,axiom,
    ! [F: nat > nat,A: nat,A2: set_nat] :
      ( ( modulo_modulo_nat
        @ ( groups3542108847815614940at_nat
          @ ^ [I: nat] : ( modulo_modulo_nat @ ( F @ I ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ A ) ) ).

% mod_sum_eq
thf(fact_7904_take__bit__eq__mask__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
        = ( bit_se2000444600071755411sk_int @ N ) )
      = ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
        = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
thf(fact_7905_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_7906_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_7907_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_7908_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_7909_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_7910_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_7911_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_7912_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_7913_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_7914_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_7915_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_7916_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_7917_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_7918_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_7919_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_7920_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_7921_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_7922_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_7923_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_7924_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_7925_not__exp__less__zero,axiom,
    ! [X: real] :
      ~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).

% not_exp_less_zero
thf(fact_7926_exp__gt__zero,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).

% exp_gt_zero
thf(fact_7927_exp__total,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ? [X5: real] :
          ( ( exp_real @ X5 )
          = Y ) ) ).

% exp_total
thf(fact_7928_not__exp__le__zero,axiom,
    ! [X: real] :
      ~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).

% not_exp_le_zero
thf(fact_7929_exp__ge__zero,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).

% exp_ge_zero
thf(fact_7930_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_7931_OR__lower,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X @ Y ) ) ) ) ).

% OR_lower
thf(fact_7932_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% take_bit_tightened_less_eq_int
thf(fact_7933_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( ( bit_ri631733984087533419it_int @ N @ A )
        = ( bit_ri631733984087533419it_int @ N @ B ) )
      = ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
        = ( bit_se2923211474154528505it_int @ ( suc @ N ) @ B ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
thf(fact_7934_take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% take_bit_int_less_eq_self_iff
thf(fact_7935_take__bit__nonnegative,axiom,
    ! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).

% take_bit_nonnegative
thf(fact_7936_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% take_bit_int_greater_self_iff
thf(fact_7937_not__take__bit__negative,axiom,
    ! [N: nat,K: int] :
      ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).

% not_take_bit_negative
thf(fact_7938_signed__take__bit__take__bit,axiom,
    ! [M: nat,N: nat,A: int] :
      ( ( bit_ri631733984087533419it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
      = ( if_int_int @ ( ord_less_eq_nat @ N @ M ) @ ( bit_se2923211474154528505it_int @ N ) @ ( bit_ri631733984087533419it_int @ M ) @ A ) ) ).

% signed_take_bit_take_bit
thf(fact_7939_mult__exp__exp,axiom,
    ! [X: complex,Y: complex] :
      ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) )
      = ( exp_complex @ ( plus_plus_complex @ X @ Y ) ) ) ).

% mult_exp_exp
thf(fact_7940_mult__exp__exp,axiom,
    ! [X: real,Y: real] :
      ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
      = ( exp_real @ ( plus_plus_real @ X @ Y ) ) ) ).

% mult_exp_exp
thf(fact_7941_exp__add__commuting,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( times_times_complex @ X @ Y )
        = ( times_times_complex @ Y @ X ) )
     => ( ( exp_complex @ ( plus_plus_complex @ X @ Y ) )
        = ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) ) ) ) ).

% exp_add_commuting
thf(fact_7942_exp__add__commuting,axiom,
    ! [X: real,Y: real] :
      ( ( ( times_times_real @ X @ Y )
        = ( times_times_real @ Y @ X ) )
     => ( ( exp_real @ ( plus_plus_real @ X @ Y ) )
        = ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ) ).

% exp_add_commuting
thf(fact_7943_exp__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( exp_complex @ ( minus_minus_complex @ X @ Y ) )
      = ( divide1717551699836669952omplex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) ) ) ).

% exp_diff
thf(fact_7944_exp__diff,axiom,
    ! [X: real,Y: real] :
      ( ( exp_real @ ( minus_minus_real @ X @ Y ) )
      = ( divide_divide_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).

% exp_diff
thf(fact_7945_take__bit__unset__bit__eq,axiom,
    ! [N: nat,M: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
          = ( bit_se2923211474154528505it_int @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
          = ( bit_se4203085406695923979it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).

% take_bit_unset_bit_eq
thf(fact_7946_take__bit__unset__bit__eq,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
          = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
          = ( bit_se4205575877204974255it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).

% take_bit_unset_bit_eq
thf(fact_7947_take__bit__set__bit__eq,axiom,
    ! [N: nat,M: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
          = ( bit_se2923211474154528505it_int @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
          = ( bit_se7879613467334960850it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).

% take_bit_set_bit_eq
thf(fact_7948_take__bit__set__bit__eq,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
          = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
          = ( bit_se7882103937844011126it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).

% take_bit_set_bit_eq
thf(fact_7949_take__bit__flip__bit__eq,axiom,
    ! [N: nat,M: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
          = ( bit_se2923211474154528505it_int @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
          = ( bit_se2159334234014336723it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).

% take_bit_flip_bit_eq
thf(fact_7950_take__bit__flip__bit__eq,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
          = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
          = ( bit_se2161824704523386999it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).

% take_bit_flip_bit_eq
thf(fact_7951_plus__and__or,axiom,
    ! [X: int,Y: int] :
      ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ ( bit_se1409905431419307370or_int @ X @ Y ) )
      = ( plus_plus_int @ X @ Y ) ) ).

% plus_and_or
thf(fact_7952_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: real > complex,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups5754745047067104278omplex @ G
          @ ( collect_real
            @ ^ [X6: real] :
                ( ( member_real @ X6 @ A2 )
                & ( P @ X6 ) ) ) )
        = ( groups5754745047067104278omplex
          @ ^ [X6: real] : ( if_complex @ ( P @ X6 ) @ ( G @ X6 ) @ zero_zero_complex )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7953_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: nat > complex,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups2073611262835488442omplex @ G
          @ ( collect_nat
            @ ^ [X6: nat] :
                ( ( member_nat @ X6 @ A2 )
                & ( P @ X6 ) ) ) )
        = ( groups2073611262835488442omplex
          @ ^ [X6: nat] : ( if_complex @ ( P @ X6 ) @ ( G @ X6 ) @ zero_zero_complex )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7954_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: int > complex,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3049146728041665814omplex @ G
          @ ( collect_int
            @ ^ [X6: int] :
                ( ( member_int @ X6 @ A2 )
                & ( P @ X6 ) ) ) )
        = ( groups3049146728041665814omplex
          @ ^ [X6: int] : ( if_complex @ ( P @ X6 ) @ ( G @ X6 ) @ zero_zero_complex )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7955_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: real > real,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups8097168146408367636l_real @ G
          @ ( collect_real
            @ ^ [X6: real] :
                ( ( member_real @ X6 @ A2 )
                & ( P @ X6 ) ) ) )
        = ( groups8097168146408367636l_real
          @ ^ [X6: real] : ( if_real @ ( P @ X6 ) @ ( G @ X6 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7956_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: int > real,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G
          @ ( collect_int
            @ ^ [X6: int] :
                ( ( member_int @ X6 @ A2 )
                & ( P @ X6 ) ) ) )
        = ( groups8778361861064173332t_real
          @ ^ [X6: int] : ( if_real @ ( P @ X6 ) @ ( G @ X6 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7957_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: complex > real,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( collect_complex
            @ ^ [X6: complex] :
                ( ( member_complex @ X6 @ A2 )
                & ( P @ X6 ) ) ) )
        = ( groups5808333547571424918x_real
          @ ^ [X6: complex] : ( if_real @ ( P @ X6 ) @ ( G @ X6 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7958_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: real > rat,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1300246762558778688al_rat @ G
          @ ( collect_real
            @ ^ [X6: real] :
                ( ( member_real @ X6 @ A2 )
                & ( P @ X6 ) ) ) )
        = ( groups1300246762558778688al_rat
          @ ^ [X6: real] : ( if_rat @ ( P @ X6 ) @ ( G @ X6 ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7959_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: nat > rat,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups2906978787729119204at_rat @ G
          @ ( collect_nat
            @ ^ [X6: nat] :
                ( ( member_nat @ X6 @ A2 )
                & ( P @ X6 ) ) ) )
        = ( groups2906978787729119204at_rat
          @ ^ [X6: nat] : ( if_rat @ ( P @ X6 ) @ ( G @ X6 ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7960_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: int > rat,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat @ G
          @ ( collect_int
            @ ^ [X6: int] :
                ( ( member_int @ X6 @ A2 )
                & ( P @ X6 ) ) ) )
        = ( groups3906332499630173760nt_rat
          @ ^ [X6: int] : ( if_rat @ ( P @ X6 ) @ ( G @ X6 ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7961_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: complex > rat,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G
          @ ( collect_complex
            @ ^ [X6: complex] :
                ( ( member_complex @ X6 @ A2 )
                & ( P @ X6 ) ) ) )
        = ( groups5058264527183730370ex_rat
          @ ^ [X6: complex] : ( if_rat @ ( P @ X6 ) @ ( G @ X6 ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7962_mask__nonnegative__int,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).

% mask_nonnegative_int
thf(fact_7963_not__mask__negative__int,axiom,
    ! [N: nat] :
      ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).

% not_mask_negative_int
thf(fact_7964_mask__Suc__exp,axiom,
    ! [N: nat] :
      ( ( bit_se2002935070580805687sk_nat @ ( suc @ N ) )
      = ( bit_se1412395901928357646or_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).

% mask_Suc_exp
thf(fact_7965_mask__Suc__exp,axiom,
    ! [N: nat] :
      ( ( bit_se2000444600071755411sk_int @ ( suc @ N ) )
      = ( bit_se1409905431419307370or_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).

% mask_Suc_exp
thf(fact_7966_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X6: real] :
                ( ( member_real @ X6 @ A2 )
               => ( ( F @ X6 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7967_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X6: int] :
                ( ( member_int @ X6 @ A2 )
               => ( ( F @ X6 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7968_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X6: complex] :
                ( ( member_complex @ X6 @ A2 )
               => ( ( F @ X6 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7969_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X6: real] :
                ( ( member_real @ X6 @ A2 )
               => ( ( F @ X6 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7970_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X6: nat] :
                ( ( member_nat @ X6 @ A2 )
               => ( ( F @ X6 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7971_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X6: int] :
                ( ( member_int @ X6 @ A2 )
               => ( ( F @ X6 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7972_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X6: complex] :
                ( ( member_complex @ X6 @ A2 )
               => ( ( F @ X6 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7973_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ A2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ A2 )
            = zero_zero_nat )
          = ( ! [X6: real] :
                ( ( member_real @ X6 @ A2 )
               => ( ( F @ X6 )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7974_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
            = zero_zero_nat )
          = ( ! [X6: int] :
                ( ( member_int @ X6 @ A2 )
               => ( ( F @ X6 )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7975_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
       => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
            = zero_zero_nat )
          = ( ! [X6: complex] :
                ( ( member_complex @ X6 @ A2 )
               => ( ( F @ X6 )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7976_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: int > real,I3: int > int,F: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I3 @ Xa )
                      = X5 )
                    & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7977_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: complex > real,I3: complex > int,F: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I3 @ Xa )
                      = X5 )
                    & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7978_sum__le__included,axiom,
    ! [S: set_complex,T: set_int,G: int > real,I3: int > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I3 @ Xa )
                      = X5 )
                    & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7979_sum__le__included,axiom,
    ! [S: set_complex,T: set_complex,G: complex > real,I3: complex > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I3 @ Xa )
                      = X5 )
                    & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7980_sum__le__included,axiom,
    ! [S: set_nat,T: set_nat,G: nat > rat,I3: nat > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I3 @ Xa )
                      = X5 )
                    & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7981_sum__le__included,axiom,
    ! [S: set_nat,T: set_int,G: int > rat,I3: int > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I3 @ Xa )
                      = X5 )
                    & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7982_sum__le__included,axiom,
    ! [S: set_nat,T: set_complex,G: complex > rat,I3: complex > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I3 @ Xa )
                      = X5 )
                    & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7983_sum__le__included,axiom,
    ! [S: set_int,T: set_nat,G: nat > rat,I3: nat > int,F: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I3 @ Xa )
                      = X5 )
                    & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7984_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: int > rat,I3: int > int,F: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I3 @ Xa )
                      = X5 )
                    & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7985_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: complex > rat,I3: complex > int,F: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I3 @ Xa )
                      = X5 )
                    & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7986_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A2 )
           => ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: int] :
              ( ( member_int @ X3 @ A2 )
              & ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7987_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A2 )
           => ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
              & ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7988_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: nat] :
              ( ( member_nat @ X3 @ A2 )
              & ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7989_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: int] :
              ( ( member_int @ X3 @ A2 )
              & ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7990_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
              & ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7991_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: int] :
              ( ( member_int @ X3 @ A2 )
              & ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7992_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
              & ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7993_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: nat > int,G: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ A2 )
           => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: nat] :
              ( ( member_nat @ X3 @ A2 )
              & ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7994_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > int,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A2 )
           => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
              & ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7995_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > int,G: int > int] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A2 )
           => ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X3: int] :
              ( ( member_int @ X3 @ A2 )
              & ( ord_less_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7996_sum_Orelated,axiom,
    ! [R3: complex > complex > $o,S3: set_nat,H2: nat > complex,G: nat > complex] :
      ( ( R3 @ zero_zero_complex @ zero_zero_complex )
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( ( R3 @ X1 @ X23 )
              & ( R3 @ Y1 @ Y23 ) )
           => ( R3 @ ( plus_plus_complex @ X1 @ Y1 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S3 )
               => ( R3 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R3 @ ( groups2073611262835488442omplex @ H2 @ S3 ) @ ( groups2073611262835488442omplex @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_7997_sum_Orelated,axiom,
    ! [R3: complex > complex > $o,S3: set_int,H2: int > complex,G: int > complex] :
      ( ( R3 @ zero_zero_complex @ zero_zero_complex )
     => ( ! [X1: complex,Y1: complex,X23: complex,Y23: complex] :
            ( ( ( R3 @ X1 @ X23 )
              & ( R3 @ Y1 @ Y23 ) )
           => ( R3 @ ( plus_plus_complex @ X1 @ Y1 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( R3 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R3 @ ( groups3049146728041665814omplex @ H2 @ S3 ) @ ( groups3049146728041665814omplex @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_7998_sum_Orelated,axiom,
    ! [R3: real > real > $o,S3: set_int,H2: int > real,G: int > real] :
      ( ( R3 @ zero_zero_real @ zero_zero_real )
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( ( R3 @ X1 @ X23 )
              & ( R3 @ Y1 @ Y23 ) )
           => ( R3 @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( R3 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R3 @ ( groups8778361861064173332t_real @ H2 @ S3 ) @ ( groups8778361861064173332t_real @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_7999_sum_Orelated,axiom,
    ! [R3: real > real > $o,S3: set_complex,H2: complex > real,G: complex > real] :
      ( ( R3 @ zero_zero_real @ zero_zero_real )
     => ( ! [X1: real,Y1: real,X23: real,Y23: real] :
            ( ( ( R3 @ X1 @ X23 )
              & ( R3 @ Y1 @ Y23 ) )
           => ( R3 @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( R3 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R3 @ ( groups5808333547571424918x_real @ H2 @ S3 ) @ ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_8000_sum_Orelated,axiom,
    ! [R3: rat > rat > $o,S3: set_nat,H2: nat > rat,G: nat > rat] :
      ( ( R3 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( ( R3 @ X1 @ X23 )
              & ( R3 @ Y1 @ Y23 ) )
           => ( R3 @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S3 )
               => ( R3 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R3 @ ( groups2906978787729119204at_rat @ H2 @ S3 ) @ ( groups2906978787729119204at_rat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_8001_sum_Orelated,axiom,
    ! [R3: rat > rat > $o,S3: set_int,H2: int > rat,G: int > rat] :
      ( ( R3 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( ( R3 @ X1 @ X23 )
              & ( R3 @ Y1 @ Y23 ) )
           => ( R3 @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( R3 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R3 @ ( groups3906332499630173760nt_rat @ H2 @ S3 ) @ ( groups3906332499630173760nt_rat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_8002_sum_Orelated,axiom,
    ! [R3: rat > rat > $o,S3: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( R3 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X1: rat,Y1: rat,X23: rat,Y23: rat] :
            ( ( ( R3 @ X1 @ X23 )
              & ( R3 @ Y1 @ Y23 ) )
           => ( R3 @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( R3 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R3 @ ( groups5058264527183730370ex_rat @ H2 @ S3 ) @ ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_8003_sum_Orelated,axiom,
    ! [R3: nat > nat > $o,S3: set_int,H2: int > nat,G: int > nat] :
      ( ( R3 @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
            ( ( ( R3 @ X1 @ X23 )
              & ( R3 @ Y1 @ Y23 ) )
           => ( R3 @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( R3 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R3 @ ( groups4541462559716669496nt_nat @ H2 @ S3 ) @ ( groups4541462559716669496nt_nat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_8004_sum_Orelated,axiom,
    ! [R3: nat > nat > $o,S3: set_complex,H2: complex > nat,G: complex > nat] :
      ( ( R3 @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
            ( ( ( R3 @ X1 @ X23 )
              & ( R3 @ Y1 @ Y23 ) )
           => ( R3 @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( R3 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R3 @ ( groups5693394587270226106ex_nat @ H2 @ S3 ) @ ( groups5693394587270226106ex_nat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_8005_sum_Orelated,axiom,
    ! [R3: int > int > $o,S3: set_nat,H2: nat > int,G: nat > int] :
      ( ( R3 @ zero_zero_int @ zero_zero_int )
     => ( ! [X1: int,Y1: int,X23: int,Y23: int] :
            ( ( ( R3 @ X1 @ X23 )
              & ( R3 @ Y1 @ Y23 ) )
           => ( R3 @ ( plus_plus_int @ X1 @ Y1 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S3 )
               => ( R3 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
           => ( R3 @ ( groups3539618377306564664at_int @ H2 @ S3 ) @ ( groups3539618377306564664at_int @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_8006_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ A2 )
             => ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8007_sum__strict__mono,axiom,
    ! [A2: set_real,F: real > real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ A2 )
             => ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8008_sum__strict__mono,axiom,
    ! [A2: set_o,F: $o > real,G: $o > real] :
      ( ( finite_finite_o @ A2 )
     => ( ( A2 != bot_bot_set_o )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ A2 )
             => ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_real @ ( groups8691415230153176458o_real @ F @ A2 ) @ ( groups8691415230153176458o_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8009_sum__strict__mono,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ A2 )
             => ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8010_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ A2 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8011_sum__strict__mono,axiom,
    ! [A2: set_real,F: real > rat,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ A2 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8012_sum__strict__mono,axiom,
    ! [A2: set_o,F: $o > rat,G: $o > rat] :
      ( ( finite_finite_o @ A2 )
     => ( ( A2 != bot_bot_set_o )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ A2 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups7872700643590313910_o_rat @ F @ A2 ) @ ( groups7872700643590313910_o_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8013_sum__strict__mono,axiom,
    ! [A2: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ A2 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8014_sum__strict__mono,axiom,
    ! [A2: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ A2 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8015_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ A2 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8016_sum_Oinsert__if,axiom,
    ! [A2: set_o,X: $o,G: $o > real] :
      ( ( finite_finite_o @ A2 )
     => ( ( ( member_o @ X @ A2 )
         => ( ( groups8691415230153176458o_real @ G @ ( insert_o @ X @ A2 ) )
            = ( groups8691415230153176458o_real @ G @ A2 ) ) )
        & ( ~ ( member_o @ X @ A2 )
         => ( ( groups8691415230153176458o_real @ G @ ( insert_o @ X @ A2 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups8691415230153176458o_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8017_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
            = ( groups8097168146408367636l_real @ G @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8018_sum_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X @ A2 )
         => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
            = ( groups8778361861064173332t_real @ G @ A2 ) ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8019_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
            = ( groups5808333547571424918x_real @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8020_sum_Oinsert__if,axiom,
    ! [A2: set_o,X: $o,G: $o > rat] :
      ( ( finite_finite_o @ A2 )
     => ( ( ( member_o @ X @ A2 )
         => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A2 ) )
            = ( groups7872700643590313910_o_rat @ G @ A2 ) ) )
        & ( ~ ( member_o @ X @ A2 )
         => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8021_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
            = ( groups1300246762558778688al_rat @ G @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8022_sum_Oinsert__if,axiom,
    ! [A2: set_nat,X: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat @ X @ A2 )
         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
            = ( groups2906978787729119204at_rat @ G @ A2 ) ) )
        & ( ~ ( member_nat @ X @ A2 )
         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8023_sum_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X @ A2 )
         => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
            = ( groups3906332499630173760nt_rat @ G @ A2 ) ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8024_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
            = ( groups5058264527183730370ex_rat @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8025_sum_Oinsert__if,axiom,
    ! [A2: set_o,X: $o,G: $o > nat] :
      ( ( finite_finite_o @ A2 )
     => ( ( ( member_o @ X @ A2 )
         => ( ( groups8507830703676809646_o_nat @ G @ ( insert_o @ X @ A2 ) )
            = ( groups8507830703676809646_o_nat @ G @ A2 ) ) )
        & ( ~ ( member_o @ X @ A2 )
         => ( ( groups8507830703676809646_o_nat @ G @ ( insert_o @ X @ A2 ) )
            = ( plus_plus_nat @ ( G @ X ) @ ( groups8507830703676809646_o_nat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8026_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_real,S3: set_real,I3: real > real,J: real > real,T3: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A5: real] :
              ( ( member_real @ A5 @ ( minus_minus_set_real @ S3 @ S5 ) )
             => ( ( I3 @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ S3 @ S5 ) )
               => ( member_real @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T5 ) ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ T3 @ T5 ) )
                 => ( ( J @ ( I3 @ B5 ) )
                    = B5 ) )
             => ( ! [B5: real] :
                    ( ( member_real @ B5 @ ( minus_minus_set_real @ T3 @ T5 ) )
                   => ( member_real @ ( I3 @ B5 ) @ ( minus_minus_set_real @ S3 @ S5 ) ) )
               => ( ! [A5: real] :
                      ( ( member_real @ A5 @ S5 )
                     => ( ( G @ A5 )
                        = zero_zero_complex ) )
                 => ( ! [B5: real] :
                        ( ( member_real @ B5 @ T5 )
                       => ( ( H2 @ B5 )
                          = zero_zero_complex ) )
                   => ( ! [A5: real] :
                          ( ( member_real @ A5 @ S3 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups5754745047067104278omplex @ G @ S3 )
                        = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8027_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_int,S3: set_real,I3: int > real,J: real > int,T3: set_int,G: real > complex,H2: int > complex] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A5: real] :
              ( ( member_real @ A5 @ ( minus_minus_set_real @ S3 @ S5 ) )
             => ( ( I3 @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ S3 @ S5 ) )
               => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T5 ) )
                 => ( ( J @ ( I3 @ B5 ) )
                    = B5 ) )
             => ( ! [B5: int] :
                    ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T5 ) )
                   => ( member_real @ ( I3 @ B5 ) @ ( minus_minus_set_real @ S3 @ S5 ) ) )
               => ( ! [A5: real] :
                      ( ( member_real @ A5 @ S5 )
                     => ( ( G @ A5 )
                        = zero_zero_complex ) )
                 => ( ! [B5: int] :
                        ( ( member_int @ B5 @ T5 )
                       => ( ( H2 @ B5 )
                          = zero_zero_complex ) )
                   => ( ! [A5: real] :
                          ( ( member_real @ A5 @ S3 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups5754745047067104278omplex @ G @ S3 )
                        = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8028_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T5: set_real,S3: set_int,I3: real > int,J: int > real,T3: set_real,G: int > complex,H2: real > complex] :
      ( ( finite_finite_int @ S5 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A5: int] :
              ( ( member_int @ A5 @ ( minus_minus_set_int @ S3 @ S5 ) )
             => ( ( I3 @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ S3 @ S5 ) )
               => ( member_real @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T5 ) ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ T3 @ T5 ) )
                 => ( ( J @ ( I3 @ B5 ) )
                    = B5 ) )
             => ( ! [B5: real] :
                    ( ( member_real @ B5 @ ( minus_minus_set_real @ T3 @ T5 ) )
                   => ( member_int @ ( I3 @ B5 ) @ ( minus_minus_set_int @ S3 @ S5 ) ) )
               => ( ! [A5: int] :
                      ( ( member_int @ A5 @ S5 )
                     => ( ( G @ A5 )
                        = zero_zero_complex ) )
                 => ( ! [B5: real] :
                        ( ( member_real @ B5 @ T5 )
                       => ( ( H2 @ B5 )
                          = zero_zero_complex ) )
                   => ( ! [A5: int] :
                          ( ( member_int @ A5 @ S3 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups3049146728041665814omplex @ G @ S3 )
                        = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8029_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T5: set_int,S3: set_int,I3: int > int,J: int > int,T3: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ S5 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A5: int] :
              ( ( member_int @ A5 @ ( minus_minus_set_int @ S3 @ S5 ) )
             => ( ( I3 @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ S3 @ S5 ) )
               => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T5 ) )
                 => ( ( J @ ( I3 @ B5 ) )
                    = B5 ) )
             => ( ! [B5: int] :
                    ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T5 ) )
                   => ( member_int @ ( I3 @ B5 ) @ ( minus_minus_set_int @ S3 @ S5 ) ) )
               => ( ! [A5: int] :
                      ( ( member_int @ A5 @ S5 )
                     => ( ( G @ A5 )
                        = zero_zero_complex ) )
                 => ( ! [B5: int] :
                        ( ( member_int @ B5 @ T5 )
                       => ( ( H2 @ B5 )
                          = zero_zero_complex ) )
                   => ( ! [A5: int] :
                          ( ( member_int @ A5 @ S3 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups3049146728041665814omplex @ G @ S3 )
                        = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8030_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_real,S3: set_real,I3: real > real,J: real > real,T3: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A5: real] :
              ( ( member_real @ A5 @ ( minus_minus_set_real @ S3 @ S5 ) )
             => ( ( I3 @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ S3 @ S5 ) )
               => ( member_real @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T5 ) ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ T3 @ T5 ) )
                 => ( ( J @ ( I3 @ B5 ) )
                    = B5 ) )
             => ( ! [B5: real] :
                    ( ( member_real @ B5 @ ( minus_minus_set_real @ T3 @ T5 ) )
                   => ( member_real @ ( I3 @ B5 ) @ ( minus_minus_set_real @ S3 @ S5 ) ) )
               => ( ! [A5: real] :
                      ( ( member_real @ A5 @ S5 )
                     => ( ( G @ A5 )
                        = zero_zero_real ) )
                 => ( ! [B5: real] :
                        ( ( member_real @ B5 @ T5 )
                       => ( ( H2 @ B5 )
                          = zero_zero_real ) )
                   => ( ! [A5: real] :
                          ( ( member_real @ A5 @ S3 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups8097168146408367636l_real @ G @ S3 )
                        = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8031_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_int,S3: set_real,I3: int > real,J: real > int,T3: set_int,G: real > real,H2: int > real] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A5: real] :
              ( ( member_real @ A5 @ ( minus_minus_set_real @ S3 @ S5 ) )
             => ( ( I3 @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ S3 @ S5 ) )
               => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T5 ) )
                 => ( ( J @ ( I3 @ B5 ) )
                    = B5 ) )
             => ( ! [B5: int] :
                    ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T5 ) )
                   => ( member_real @ ( I3 @ B5 ) @ ( minus_minus_set_real @ S3 @ S5 ) ) )
               => ( ! [A5: real] :
                      ( ( member_real @ A5 @ S5 )
                     => ( ( G @ A5 )
                        = zero_zero_real ) )
                 => ( ! [B5: int] :
                        ( ( member_int @ B5 @ T5 )
                       => ( ( H2 @ B5 )
                          = zero_zero_real ) )
                   => ( ! [A5: real] :
                          ( ( member_real @ A5 @ S3 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups8097168146408367636l_real @ G @ S3 )
                        = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8032_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T5: set_complex,S3: set_real,I3: complex > real,J: real > complex,T3: set_complex,G: real > real,H2: complex > real] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite3207457112153483333omplex @ T5 )
       => ( ! [A5: real] :
              ( ( member_real @ A5 @ ( minus_minus_set_real @ S3 @ S5 ) )
             => ( ( I3 @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ S3 @ S5 ) )
               => ( member_complex @ ( J @ A5 ) @ ( minus_811609699411566653omplex @ T3 @ T5 ) ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                 => ( ( J @ ( I3 @ B5 ) )
                    = B5 ) )
             => ( ! [B5: complex] :
                    ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                   => ( member_real @ ( I3 @ B5 ) @ ( minus_minus_set_real @ S3 @ S5 ) ) )
               => ( ! [A5: real] :
                      ( ( member_real @ A5 @ S5 )
                     => ( ( G @ A5 )
                        = zero_zero_real ) )
                 => ( ! [B5: complex] :
                        ( ( member_complex @ B5 @ T5 )
                       => ( ( H2 @ B5 )
                          = zero_zero_real ) )
                   => ( ! [A5: real] :
                          ( ( member_real @ A5 @ S3 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups8097168146408367636l_real @ G @ S3 )
                        = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8033_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T5: set_real,S3: set_int,I3: real > int,J: int > real,T3: set_real,G: int > real,H2: real > real] :
      ( ( finite_finite_int @ S5 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A5: int] :
              ( ( member_int @ A5 @ ( minus_minus_set_int @ S3 @ S5 ) )
             => ( ( I3 @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ S3 @ S5 ) )
               => ( member_real @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T5 ) ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ T3 @ T5 ) )
                 => ( ( J @ ( I3 @ B5 ) )
                    = B5 ) )
             => ( ! [B5: real] :
                    ( ( member_real @ B5 @ ( minus_minus_set_real @ T3 @ T5 ) )
                   => ( member_int @ ( I3 @ B5 ) @ ( minus_minus_set_int @ S3 @ S5 ) ) )
               => ( ! [A5: int] :
                      ( ( member_int @ A5 @ S5 )
                     => ( ( G @ A5 )
                        = zero_zero_real ) )
                 => ( ! [B5: real] :
                        ( ( member_real @ B5 @ T5 )
                       => ( ( H2 @ B5 )
                          = zero_zero_real ) )
                   => ( ! [A5: int] :
                          ( ( member_int @ A5 @ S3 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups8778361861064173332t_real @ G @ S3 )
                        = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8034_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T5: set_int,S3: set_int,I3: int > int,J: int > int,T3: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ S5 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A5: int] :
              ( ( member_int @ A5 @ ( minus_minus_set_int @ S3 @ S5 ) )
             => ( ( I3 @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ S3 @ S5 ) )
               => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T5 ) )
                 => ( ( J @ ( I3 @ B5 ) )
                    = B5 ) )
             => ( ! [B5: int] :
                    ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T5 ) )
                   => ( member_int @ ( I3 @ B5 ) @ ( minus_minus_set_int @ S3 @ S5 ) ) )
               => ( ! [A5: int] :
                      ( ( member_int @ A5 @ S5 )
                     => ( ( G @ A5 )
                        = zero_zero_real ) )
                 => ( ! [B5: int] :
                        ( ( member_int @ B5 @ T5 )
                       => ( ( H2 @ B5 )
                          = zero_zero_real ) )
                   => ( ! [A5: int] :
                          ( ( member_int @ A5 @ S3 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups8778361861064173332t_real @ G @ S3 )
                        = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8035_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T5: set_complex,S3: set_int,I3: complex > int,J: int > complex,T3: set_complex,G: int > real,H2: complex > real] :
      ( ( finite_finite_int @ S5 )
     => ( ( finite3207457112153483333omplex @ T5 )
       => ( ! [A5: int] :
              ( ( member_int @ A5 @ ( minus_minus_set_int @ S3 @ S5 ) )
             => ( ( I3 @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ S3 @ S5 ) )
               => ( member_complex @ ( J @ A5 ) @ ( minus_811609699411566653omplex @ T3 @ T5 ) ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                 => ( ( J @ ( I3 @ B5 ) )
                    = B5 ) )
             => ( ! [B5: complex] :
                    ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
                   => ( member_int @ ( I3 @ B5 ) @ ( minus_minus_set_int @ S3 @ S5 ) ) )
               => ( ! [A5: int] :
                      ( ( member_int @ A5 @ S5 )
                     => ( ( G @ A5 )
                        = zero_zero_real ) )
                 => ( ! [B5: complex] :
                        ( ( member_complex @ B5 @ T5 )
                       => ( ( H2 @ B5 )
                          = zero_zero_real ) )
                   => ( ! [A5: int] :
                          ( ( member_int @ A5 @ S3 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups8778361861064173332t_real @ G @ S3 )
                        = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8036_exp__gt__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).

% exp_gt_one
thf(fact_8037_take__bit__signed__take__bit,axiom,
    ! [M: nat,N: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri631733984087533419it_int @ N @ A ) )
        = ( bit_se2923211474154528505it_int @ M @ A ) ) ) ).

% take_bit_signed_take_bit
thf(fact_8038_exp__minus__inverse,axiom,
    ! [X: real] :
      ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) )
      = one_one_real ) ).

% exp_minus_inverse
thf(fact_8039_exp__minus__inverse,axiom,
    ! [X: complex] :
      ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) )
      = one_one_complex ) ).

% exp_minus_inverse
thf(fact_8040_take__bit__decr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
       != zero_zero_int )
     => ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
        = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).

% take_bit_decr_eq
thf(fact_8041_mask__Suc__double,axiom,
    ! [N: nat] :
      ( ( bit_se2002935070580805687sk_nat @ ( suc @ N ) )
      = ( bit_se1412395901928357646or_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ) ).

% mask_Suc_double
thf(fact_8042_mask__Suc__double,axiom,
    ! [N: nat] :
      ( ( bit_se2000444600071755411sk_int @ ( suc @ N ) )
      = ( bit_se1409905431419307370or_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N ) ) ) ) ).

% mask_Suc_double
thf(fact_8043_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > real,I3: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S )
            = zero_zero_real )
         => ( ( member_real @ I3 @ S )
           => ( ( F @ I3 )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8044_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > real,I3: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S )
            = zero_zero_real )
         => ( ( member_int @ I3 @ S )
           => ( ( F @ I3 )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8045_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > real,I3: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = zero_zero_real )
         => ( ( member_complex @ I3 @ S )
           => ( ( F @ I3 )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8046_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > rat,I3: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_real @ I3 @ S )
           => ( ( F @ I3 )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8047_sum__nonneg__0,axiom,
    ! [S: set_nat,F: nat > rat,I3: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_nat @ I3 @ S )
           => ( ( F @ I3 )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8048_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > rat,I3: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_int @ I3 @ S )
           => ( ( F @ I3 )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8049_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > rat,I3: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_complex @ I3 @ S )
           => ( ( F @ I3 )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8050_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > nat,I3: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ S )
            = zero_zero_nat )
         => ( ( member_real @ I3 @ S )
           => ( ( F @ I3 )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8051_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > nat,I3: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F @ S )
            = zero_zero_nat )
         => ( ( member_int @ I3 @ S )
           => ( ( F @ I3 )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8052_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > nat,I3: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
       => ( ( ( groups5693394587270226106ex_nat @ F @ S )
            = zero_zero_nat )
         => ( ( member_complex @ I3 @ S )
           => ( ( F @ I3 )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8053_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > real,B2: real,I3: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S )
            = B2 )
         => ( ( member_real @ I3 @ S )
           => ( ord_less_eq_real @ ( F @ I3 ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8054_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > real,B2: real,I3: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S )
            = B2 )
         => ( ( member_int @ I3 @ S )
           => ( ord_less_eq_real @ ( F @ I3 ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8055_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > real,B2: real,I3: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = B2 )
         => ( ( member_complex @ I3 @ S )
           => ( ord_less_eq_real @ ( F @ I3 ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8056_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > rat,B2: rat,I3: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S )
            = B2 )
         => ( ( member_real @ I3 @ S )
           => ( ord_less_eq_rat @ ( F @ I3 ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8057_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: nat > rat,B2: rat,I3: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S )
            = B2 )
         => ( ( member_nat @ I3 @ S )
           => ( ord_less_eq_rat @ ( F @ I3 ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8058_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > rat,B2: rat,I3: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S )
            = B2 )
         => ( ( member_int @ I3 @ S )
           => ( ord_less_eq_rat @ ( F @ I3 ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8059_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > rat,B2: rat,I3: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S )
            = B2 )
         => ( ( member_complex @ I3 @ S )
           => ( ord_less_eq_rat @ ( F @ I3 ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8060_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > nat,B2: nat,I3: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ S )
            = B2 )
         => ( ( member_real @ I3 @ S )
           => ( ord_less_eq_nat @ ( F @ I3 ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8061_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > nat,B2: nat,I3: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F @ S )
            = B2 )
         => ( ( member_int @ I3 @ S )
           => ( ord_less_eq_nat @ ( F @ I3 ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8062_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > nat,B2: nat,I3: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
       => ( ( ( groups5693394587270226106ex_nat @ F @ S )
            = B2 )
         => ( ( member_complex @ I3 @ S )
           => ( ord_less_eq_nat @ ( F @ I3 ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8063_sum_Ointer__restrict,axiom,
    ! [A2: set_real,G: real > complex,B2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups5754745047067104278omplex @ G @ ( inf_inf_set_real @ A2 @ B2 ) )
        = ( groups5754745047067104278omplex
          @ ^ [X6: real] : ( if_complex @ ( member_real @ X6 @ B2 ) @ ( G @ X6 ) @ zero_zero_complex )
          @ A2 ) ) ) ).

% sum.inter_restrict
thf(fact_8064_sum_Ointer__restrict,axiom,
    ! [A2: set_int,G: int > complex,B2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3049146728041665814omplex @ G @ ( inf_inf_set_int @ A2 @ B2 ) )
        = ( groups3049146728041665814omplex
          @ ^ [X6: int] : ( if_complex @ ( member_int @ X6 @ B2 ) @ ( G @ X6 ) @ zero_zero_complex )
          @ A2 ) ) ) ).

% sum.inter_restrict
thf(fact_8065_sum_Ointer__restrict,axiom,
    ! [A2: set_real,G: real > real,B2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups8097168146408367636l_real @ G @ ( inf_inf_set_real @ A2 @ B2 ) )
        = ( groups8097168146408367636l_real
          @ ^ [X6: real] : ( if_real @ ( member_real @ X6 @ B2 ) @ ( G @ X6 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_restrict
thf(fact_8066_sum_Ointer__restrict,axiom,
    ! [A2: set_int,G: int > real,B2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G @ ( inf_inf_set_int @ A2 @ B2 ) )
        = ( groups8778361861064173332t_real
          @ ^ [X6: int] : ( if_real @ ( member_int @ X6 @ B2 ) @ ( G @ X6 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_restrict
thf(fact_8067_sum_Ointer__restrict,axiom,
    ! [A2: set_complex,G: complex > real,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G @ ( inf_inf_set_complex @ A2 @ B2 ) )
        = ( groups5808333547571424918x_real
          @ ^ [X6: complex] : ( if_real @ ( member_complex @ X6 @ B2 ) @ ( G @ X6 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_restrict
thf(fact_8068_sum_Ointer__restrict,axiom,
    ! [A2: set_real,G: real > rat,B2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1300246762558778688al_rat @ G @ ( inf_inf_set_real @ A2 @ B2 ) )
        = ( groups1300246762558778688al_rat
          @ ^ [X6: real] : ( if_rat @ ( member_real @ X6 @ B2 ) @ ( G @ X6 ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_restrict
thf(fact_8069_sum_Ointer__restrict,axiom,
    ! [A2: set_int,G: int > rat,B2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat @ G @ ( inf_inf_set_int @ A2 @ B2 ) )
        = ( groups3906332499630173760nt_rat
          @ ^ [X6: int] : ( if_rat @ ( member_int @ X6 @ B2 ) @ ( G @ X6 ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_restrict
thf(fact_8070_sum_Ointer__restrict,axiom,
    ! [A2: set_complex,G: complex > rat,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G @ ( inf_inf_set_complex @ A2 @ B2 ) )
        = ( groups5058264527183730370ex_rat
          @ ^ [X6: complex] : ( if_rat @ ( member_complex @ X6 @ B2 ) @ ( G @ X6 ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_restrict
thf(fact_8071_sum_Ointer__restrict,axiom,
    ! [A2: set_real,G: real > nat,B2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1935376822645274424al_nat @ G @ ( inf_inf_set_real @ A2 @ B2 ) )
        = ( groups1935376822645274424al_nat
          @ ^ [X6: real] : ( if_nat @ ( member_real @ X6 @ B2 ) @ ( G @ X6 ) @ zero_zero_nat )
          @ A2 ) ) ) ).

% sum.inter_restrict
thf(fact_8072_sum_Ointer__restrict,axiom,
    ! [A2: set_int,G: int > nat,B2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat @ G @ ( inf_inf_set_int @ A2 @ B2 ) )
        = ( groups4541462559716669496nt_nat
          @ ^ [X6: int] : ( if_nat @ ( member_int @ X6 @ B2 ) @ ( G @ X6 ) @ zero_zero_nat )
          @ A2 ) ) ) ).

% sum.inter_restrict
thf(fact_8073_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: real > complex] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups5754745047067104278omplex @ G
          @ ( minus_minus_set_real @ A2
            @ ( collect_real
              @ ^ [X6: real] :
                  ( ( G @ X6 )
                  = zero_zero_complex ) ) ) )
        = ( groups5754745047067104278omplex @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8074_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > complex] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3049146728041665814omplex @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X6: int] :
                  ( ( G @ X6 )
                  = zero_zero_complex ) ) ) )
        = ( groups3049146728041665814omplex @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8075_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups8097168146408367636l_real @ G
          @ ( minus_minus_set_real @ A2
            @ ( collect_real
              @ ^ [X6: real] :
                  ( ( G @ X6 )
                  = zero_zero_real ) ) ) )
        = ( groups8097168146408367636l_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8076_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X6: int] :
                  ( ( G @ X6 )
                  = zero_zero_real ) ) ) )
        = ( groups8778361861064173332t_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8077_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X6: complex] :
                  ( ( G @ X6 )
                  = zero_zero_real ) ) ) )
        = ( groups5808333547571424918x_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8078_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1300246762558778688al_rat @ G
          @ ( minus_minus_set_real @ A2
            @ ( collect_real
              @ ^ [X6: real] :
                  ( ( G @ X6 )
                  = zero_zero_rat ) ) ) )
        = ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8079_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X6: int] :
                  ( ( G @ X6 )
                  = zero_zero_rat ) ) ) )
        = ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8080_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X6: complex] :
                  ( ( G @ X6 )
                  = zero_zero_rat ) ) ) )
        = ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8081_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1935376822645274424al_nat @ G
          @ ( minus_minus_set_real @ A2
            @ ( collect_real
              @ ^ [X6: real] :
                  ( ( G @ X6 )
                  = zero_zero_nat ) ) ) )
        = ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8082_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X6: int] :
                  ( ( G @ X6 )
                  = zero_zero_nat ) ) ) )
        = ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8083_less__mask,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).

% less_mask
thf(fact_8084_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
        = ( bit_se2000444600071755411sk_int @ N ) )
      = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).

% take_bit_eq_mask_iff_exp_dvd
thf(fact_8085_sum__pos2,axiom,
    ! [I6: set_real,I3: real,F: real > real] :
      ( ( finite_finite_real @ I6 )
     => ( ( member_real @ I3 @ I6 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I3 ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I6 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8086_sum__pos2,axiom,
    ! [I6: set_int,I3: int,F: int > real] :
      ( ( finite_finite_int @ I6 )
     => ( ( member_int @ I3 @ I6 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I3 ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I6 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8087_sum__pos2,axiom,
    ! [I6: set_complex,I3: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( member_complex @ I3 @ I6 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I3 ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I6 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8088_sum__pos2,axiom,
    ! [I6: set_real,I3: real,F: real > extended_enat] :
      ( ( finite_finite_real @ I6 )
     => ( ( member_real @ I3 @ I6 )
       => ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I6 )
               => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I2 ) ) )
           => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups2800946370649118462d_enat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8089_sum__pos2,axiom,
    ! [I6: set_nat,I3: nat,F: nat > extended_enat] :
      ( ( finite_finite_nat @ I6 )
     => ( ( member_nat @ I3 @ I6 )
       => ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I6 )
               => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I2 ) ) )
           => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups7108830773950497114d_enat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8090_sum__pos2,axiom,
    ! [I6: set_int,I3: int,F: int > extended_enat] :
      ( ( finite_finite_int @ I6 )
     => ( ( member_int @ I3 @ I6 )
       => ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I6 )
               => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I2 ) ) )
           => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups4225252721152677374d_enat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8091_sum__pos2,axiom,
    ! [I6: set_complex,I3: complex,F: complex > extended_enat] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( member_complex @ I3 @ I6 )
       => ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( F @ I3 ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I6 )
               => ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( F @ I2 ) ) )
           => ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( groups1752964319039525884d_enat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8092_sum__pos2,axiom,
    ! [I6: set_real,I3: real,F: real > rat] :
      ( ( finite_finite_real @ I6 )
     => ( ( member_real @ I3 @ I6 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I6 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8093_sum__pos2,axiom,
    ! [I6: set_nat,I3: nat,F: nat > rat] :
      ( ( finite_finite_nat @ I6 )
     => ( ( member_nat @ I3 @ I6 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I6 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8094_sum__pos2,axiom,
    ! [I6: set_int,I3: int,F: int > rat] :
      ( ( finite_finite_int @ I6 )
     => ( ( member_int @ I3 @ I6 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I6 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I6 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8095_sum__pos,axiom,
    ! [I6: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I6 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_8096_sum__pos,axiom,
    ! [I6: set_real,F: real > real] :
      ( ( finite_finite_real @ I6 )
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I6 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_8097_sum__pos,axiom,
    ! [I6: set_o,F: $o > real] :
      ( ( finite_finite_o @ I6 )
     => ( ( I6 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I6 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8691415230153176458o_real @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_8098_sum__pos,axiom,
    ! [I6: set_int,F: int > real] :
      ( ( finite_finite_int @ I6 )
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I6 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_8099_sum__pos,axiom,
    ! [I6: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I6 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_8100_sum__pos,axiom,
    ! [I6: set_real,F: real > rat] :
      ( ( finite_finite_real @ I6 )
     => ( ( I6 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I6 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_8101_sum__pos,axiom,
    ! [I6: set_o,F: $o > rat] :
      ( ( finite_finite_o @ I6 )
     => ( ( I6 != bot_bot_set_o )
       => ( ! [I2: $o] :
              ( ( member_o @ I2 @ I6 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups7872700643590313910_o_rat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_8102_sum__pos,axiom,
    ! [I6: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ I6 )
     => ( ( I6 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I6 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_8103_sum__pos,axiom,
    ! [I6: set_int,F: int > rat] :
      ( ( finite_finite_int @ I6 )
     => ( ( I6 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I6 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_8104_sum__pos,axiom,
    ! [I6: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ I6 )
     => ( ( I6 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I6 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I6 ) ) ) ) ) ).

% sum_pos
thf(fact_8105_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5754745047067104278omplex @ G @ T3 )
              = ( groups5754745047067104278omplex @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8106_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups3049146728041665814omplex @ G @ T3 )
              = ( groups3049146728041665814omplex @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8107_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ T3 )
              = ( groups8097168146408367636l_real @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8108_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8778361861064173332t_real @ G @ T3 )
              = ( groups8778361861064173332t_real @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8109_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ T3 )
              = ( groups5808333547571424918x_real @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8110_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ T3 )
              = ( groups1300246762558778688al_rat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8111_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups3906332499630173760nt_rat @ G @ T3 )
              = ( groups3906332499630173760nt_rat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8112_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5058264527183730370ex_rat @ G @ T3 )
              = ( groups5058264527183730370ex_rat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8113_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1935376822645274424al_nat @ G @ T3 )
              = ( groups1935376822645274424al_nat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8114_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > nat,H2: int > nat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4541462559716669496nt_nat @ G @ T3 )
              = ( groups4541462559716669496nt_nat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8115_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H2: real > complex,G: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_complex ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5754745047067104278omplex @ G @ S3 )
              = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8116_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S3: set_int,H2: int > complex,G: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_complex ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups3049146728041665814omplex @ G @ S3 )
              = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8117_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H2: real > real,G: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ S3 )
              = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8118_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S3: set_int,H2: int > real,G: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups8778361861064173332t_real @ G @ S3 )
              = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8119_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S3: set_complex,H2: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ S3 )
              = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8120_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H2: real > rat,G: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ S3 )
              = ( groups1300246762558778688al_rat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8121_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S3: set_int,H2: int > rat,G: int > rat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups3906332499630173760nt_rat @ G @ S3 )
              = ( groups3906332499630173760nt_rat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8122_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S3: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups5058264527183730370ex_rat @ G @ S3 )
              = ( groups5058264527183730370ex_rat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8123_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H2: real > nat,G: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups1935376822645274424al_nat @ G @ S3 )
              = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8124_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S3: set_int,H2: int > nat,G: int > nat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( H2 @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H2 @ X5 ) ) )
           => ( ( groups4541462559716669496nt_nat @ G @ S3 )
              = ( groups4541462559716669496nt_nat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8125_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ( groups3049146728041665814omplex @ G @ T3 )
            = ( groups3049146728041665814omplex @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8126_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups8778361861064173332t_real @ G @ T3 )
            = ( groups8778361861064173332t_real @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8127_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ T3 )
            = ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8128_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > rat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G @ T3 )
            = ( groups3906332499630173760nt_rat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8129_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ T3 )
            = ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8130_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > nat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups4541462559716669496nt_nat @ G @ T3 )
            = ( groups4541462559716669496nt_nat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8131_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ T3 )
            = ( groups5693394587270226106ex_nat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8132_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ T3 )
            = ( groups5690904116761175830ex_int @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8133_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ( groups2073611262835488442omplex @ G @ T3 )
            = ( groups2073611262835488442omplex @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8134_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups2906978787729119204at_rat @ G @ T3 )
            = ( groups2906978787729119204at_rat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8135_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S3: set_int,G: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ( groups3049146728041665814omplex @ G @ S3 )
            = ( groups3049146728041665814omplex @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8136_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S3: set_int,G: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups8778361861064173332t_real @ G @ S3 )
            = ( groups8778361861064173332t_real @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8137_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ S3 )
            = ( groups5808333547571424918x_real @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8138_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S3: set_int,G: int > rat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G @ S3 )
            = ( groups3906332499630173760nt_rat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8139_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ S3 )
            = ( groups5058264527183730370ex_rat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8140_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S3: set_int,G: int > nat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups4541462559716669496nt_nat @ G @ S3 )
            = ( groups4541462559716669496nt_nat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8141_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ S3 )
            = ( groups5693394587270226106ex_nat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8142_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ S3 )
            = ( groups5690904116761175830ex_int @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8143_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ( groups2073611262835488442omplex @ G @ S3 )
            = ( groups2073611262835488442omplex @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8144_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups2906978787729119204at_rat @ G @ S3 )
            = ( groups2906978787729119204at_rat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8145_sum_Osame__carrierI,axiom,
    ! [C5: set_real,A2: set_real,B2: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C5 )
     => ( ( ord_less_eq_set_real @ A2 @ C5 )
       => ( ( ord_less_eq_set_real @ B2 @ C5 )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ C5 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_complex ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ C5 @ B2 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_complex ) )
             => ( ( ( groups5754745047067104278omplex @ G @ C5 )
                  = ( groups5754745047067104278omplex @ H2 @ C5 ) )
               => ( ( groups5754745047067104278omplex @ G @ A2 )
                  = ( groups5754745047067104278omplex @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8146_sum_Osame__carrierI,axiom,
    ! [C5: set_int,A2: set_int,B2: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ C5 )
     => ( ( ord_less_eq_set_int @ A2 @ C5 )
       => ( ( ord_less_eq_set_int @ B2 @ C5 )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ C5 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_complex ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ C5 @ B2 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_complex ) )
             => ( ( ( groups3049146728041665814omplex @ G @ C5 )
                  = ( groups3049146728041665814omplex @ H2 @ C5 ) )
               => ( ( groups3049146728041665814omplex @ G @ A2 )
                  = ( groups3049146728041665814omplex @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8147_sum_Osame__carrierI,axiom,
    ! [C5: set_real,A2: set_real,B2: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C5 )
     => ( ( ord_less_eq_set_real @ A2 @ C5 )
       => ( ( ord_less_eq_set_real @ B2 @ C5 )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ C5 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_real ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ C5 @ B2 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ C5 )
                  = ( groups8097168146408367636l_real @ H2 @ C5 ) )
               => ( ( groups8097168146408367636l_real @ G @ A2 )
                  = ( groups8097168146408367636l_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8148_sum_Osame__carrierI,axiom,
    ! [C5: set_int,A2: set_int,B2: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ C5 )
     => ( ( ord_less_eq_set_int @ A2 @ C5 )
       => ( ( ord_less_eq_set_int @ B2 @ C5 )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ C5 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_real ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ C5 @ B2 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_real ) )
             => ( ( ( groups8778361861064173332t_real @ G @ C5 )
                  = ( groups8778361861064173332t_real @ H2 @ C5 ) )
               => ( ( groups8778361861064173332t_real @ G @ A2 )
                  = ( groups8778361861064173332t_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8149_sum_Osame__carrierI,axiom,
    ! [C5: set_complex,A2: set_complex,B2: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C5 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C5 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C5 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_real ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ C5 @ B2 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ C5 )
                  = ( groups5808333547571424918x_real @ H2 @ C5 ) )
               => ( ( groups5808333547571424918x_real @ G @ A2 )
                  = ( groups5808333547571424918x_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8150_sum_Osame__carrierI,axiom,
    ! [C5: set_real,A2: set_real,B2: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C5 )
     => ( ( ord_less_eq_set_real @ A2 @ C5 )
       => ( ( ord_less_eq_set_real @ B2 @ C5 )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ C5 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_rat ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ C5 @ B2 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ C5 )
                  = ( groups1300246762558778688al_rat @ H2 @ C5 ) )
               => ( ( groups1300246762558778688al_rat @ G @ A2 )
                  = ( groups1300246762558778688al_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8151_sum_Osame__carrierI,axiom,
    ! [C5: set_int,A2: set_int,B2: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ C5 )
     => ( ( ord_less_eq_set_int @ A2 @ C5 )
       => ( ( ord_less_eq_set_int @ B2 @ C5 )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ C5 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_rat ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ C5 @ B2 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_rat ) )
             => ( ( ( groups3906332499630173760nt_rat @ G @ C5 )
                  = ( groups3906332499630173760nt_rat @ H2 @ C5 ) )
               => ( ( groups3906332499630173760nt_rat @ G @ A2 )
                  = ( groups3906332499630173760nt_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8152_sum_Osame__carrierI,axiom,
    ! [C5: set_complex,A2: set_complex,B2: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ C5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C5 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C5 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C5 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_rat ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ C5 @ B2 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_rat ) )
             => ( ( ( groups5058264527183730370ex_rat @ G @ C5 )
                  = ( groups5058264527183730370ex_rat @ H2 @ C5 ) )
               => ( ( groups5058264527183730370ex_rat @ G @ A2 )
                  = ( groups5058264527183730370ex_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8153_sum_Osame__carrierI,axiom,
    ! [C5: set_real,A2: set_real,B2: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C5 )
     => ( ( ord_less_eq_set_real @ A2 @ C5 )
       => ( ( ord_less_eq_set_real @ B2 @ C5 )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ C5 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_nat ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ C5 @ B2 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_nat ) )
             => ( ( ( groups1935376822645274424al_nat @ G @ C5 )
                  = ( groups1935376822645274424al_nat @ H2 @ C5 ) )
               => ( ( groups1935376822645274424al_nat @ G @ A2 )
                  = ( groups1935376822645274424al_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8154_sum_Osame__carrierI,axiom,
    ! [C5: set_int,A2: set_int,B2: set_int,G: int > nat,H2: int > nat] :
      ( ( finite_finite_int @ C5 )
     => ( ( ord_less_eq_set_int @ A2 @ C5 )
       => ( ( ord_less_eq_set_int @ B2 @ C5 )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ C5 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_nat ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ C5 @ B2 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_nat ) )
             => ( ( ( groups4541462559716669496nt_nat @ G @ C5 )
                  = ( groups4541462559716669496nt_nat @ H2 @ C5 ) )
               => ( ( groups4541462559716669496nt_nat @ G @ A2 )
                  = ( groups4541462559716669496nt_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8155_sum_Osame__carrier,axiom,
    ! [C5: set_real,A2: set_real,B2: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C5 )
     => ( ( ord_less_eq_set_real @ A2 @ C5 )
       => ( ( ord_less_eq_set_real @ B2 @ C5 )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ C5 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_complex ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ C5 @ B2 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_complex ) )
             => ( ( ( groups5754745047067104278omplex @ G @ A2 )
                  = ( groups5754745047067104278omplex @ H2 @ B2 ) )
                = ( ( groups5754745047067104278omplex @ G @ C5 )
                  = ( groups5754745047067104278omplex @ H2 @ C5 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8156_sum_Osame__carrier,axiom,
    ! [C5: set_int,A2: set_int,B2: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ C5 )
     => ( ( ord_less_eq_set_int @ A2 @ C5 )
       => ( ( ord_less_eq_set_int @ B2 @ C5 )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ C5 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_complex ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ C5 @ B2 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_complex ) )
             => ( ( ( groups3049146728041665814omplex @ G @ A2 )
                  = ( groups3049146728041665814omplex @ H2 @ B2 ) )
                = ( ( groups3049146728041665814omplex @ G @ C5 )
                  = ( groups3049146728041665814omplex @ H2 @ C5 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8157_sum_Osame__carrier,axiom,
    ! [C5: set_real,A2: set_real,B2: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C5 )
     => ( ( ord_less_eq_set_real @ A2 @ C5 )
       => ( ( ord_less_eq_set_real @ B2 @ C5 )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ C5 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_real ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ C5 @ B2 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ A2 )
                  = ( groups8097168146408367636l_real @ H2 @ B2 ) )
                = ( ( groups8097168146408367636l_real @ G @ C5 )
                  = ( groups8097168146408367636l_real @ H2 @ C5 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8158_sum_Osame__carrier,axiom,
    ! [C5: set_int,A2: set_int,B2: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ C5 )
     => ( ( ord_less_eq_set_int @ A2 @ C5 )
       => ( ( ord_less_eq_set_int @ B2 @ C5 )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ C5 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_real ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ C5 @ B2 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_real ) )
             => ( ( ( groups8778361861064173332t_real @ G @ A2 )
                  = ( groups8778361861064173332t_real @ H2 @ B2 ) )
                = ( ( groups8778361861064173332t_real @ G @ C5 )
                  = ( groups8778361861064173332t_real @ H2 @ C5 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8159_sum_Osame__carrier,axiom,
    ! [C5: set_complex,A2: set_complex,B2: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C5 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C5 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C5 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_real ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ C5 @ B2 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ A2 )
                  = ( groups5808333547571424918x_real @ H2 @ B2 ) )
                = ( ( groups5808333547571424918x_real @ G @ C5 )
                  = ( groups5808333547571424918x_real @ H2 @ C5 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8160_sum_Osame__carrier,axiom,
    ! [C5: set_real,A2: set_real,B2: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C5 )
     => ( ( ord_less_eq_set_real @ A2 @ C5 )
       => ( ( ord_less_eq_set_real @ B2 @ C5 )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ C5 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_rat ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ C5 @ B2 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ A2 )
                  = ( groups1300246762558778688al_rat @ H2 @ B2 ) )
                = ( ( groups1300246762558778688al_rat @ G @ C5 )
                  = ( groups1300246762558778688al_rat @ H2 @ C5 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8161_sum_Osame__carrier,axiom,
    ! [C5: set_int,A2: set_int,B2: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ C5 )
     => ( ( ord_less_eq_set_int @ A2 @ C5 )
       => ( ( ord_less_eq_set_int @ B2 @ C5 )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ C5 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_rat ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ C5 @ B2 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_rat ) )
             => ( ( ( groups3906332499630173760nt_rat @ G @ A2 )
                  = ( groups3906332499630173760nt_rat @ H2 @ B2 ) )
                = ( ( groups3906332499630173760nt_rat @ G @ C5 )
                  = ( groups3906332499630173760nt_rat @ H2 @ C5 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8162_sum_Osame__carrier,axiom,
    ! [C5: set_complex,A2: set_complex,B2: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ C5 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C5 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C5 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C5 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_rat ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ C5 @ B2 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_rat ) )
             => ( ( ( groups5058264527183730370ex_rat @ G @ A2 )
                  = ( groups5058264527183730370ex_rat @ H2 @ B2 ) )
                = ( ( groups5058264527183730370ex_rat @ G @ C5 )
                  = ( groups5058264527183730370ex_rat @ H2 @ C5 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8163_sum_Osame__carrier,axiom,
    ! [C5: set_real,A2: set_real,B2: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C5 )
     => ( ( ord_less_eq_set_real @ A2 @ C5 )
       => ( ( ord_less_eq_set_real @ B2 @ C5 )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ C5 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_nat ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ C5 @ B2 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_nat ) )
             => ( ( ( groups1935376822645274424al_nat @ G @ A2 )
                  = ( groups1935376822645274424al_nat @ H2 @ B2 ) )
                = ( ( groups1935376822645274424al_nat @ G @ C5 )
                  = ( groups1935376822645274424al_nat @ H2 @ C5 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8164_sum_Osame__carrier,axiom,
    ! [C5: set_int,A2: set_int,B2: set_int,G: int > nat,H2: int > nat] :
      ( ( finite_finite_int @ C5 )
     => ( ( ord_less_eq_set_int @ A2 @ C5 )
       => ( ( ord_less_eq_set_int @ B2 @ C5 )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ C5 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_nat ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ C5 @ B2 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_nat ) )
             => ( ( ( groups4541462559716669496nt_nat @ G @ A2 )
                  = ( groups4541462559716669496nt_nat @ H2 @ B2 ) )
                = ( ( groups4541462559716669496nt_nat @ G @ C5 )
                  = ( groups4541462559716669496nt_nat @ H2 @ C5 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8165_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: int > real] :
      ( ( ord_less_eq_set_int @ B2 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups8778361861064173332t_real @ G @ A2 )
          = ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups8778361861064173332t_real @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8166_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > real] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ A2 )
          = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5808333547571424918x_real @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8167_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: int > rat] :
      ( ( ord_less_eq_set_int @ B2 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups3906332499630173760nt_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8168_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > rat] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5058264527183730370ex_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8169_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: int > nat] :
      ( ( ord_less_eq_set_int @ B2 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups4541462559716669496nt_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8170_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5693394587270226106ex_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8171_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > int] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5690904116761175830ex_int @ G @ A2 )
          = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5690904116761175830ex_int @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8172_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups2906978787729119204at_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups2906978787729119204at_rat @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8173_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups3539618377306564664at_int @ G @ A2 )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups3539618377306564664at_int @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8174_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: int > int] :
      ( ( ord_less_eq_set_int @ B2 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups4538972089207619220nt_int @ G @ A2 )
          = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups4538972089207619220nt_int @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8175_even__or__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1080825931792720795nteger @ A @ B ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_or_iff
thf(fact_8176_even__or__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_or_iff
thf(fact_8177_even__or__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_or_iff
thf(fact_8178_sum_Omono__neutral__cong,axiom,
    ! [T3: set_real,S3: set_real,H2: real > complex,G: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( finite_finite_real @ S3 )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ I2 )
                = zero_zero_complex ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ ( minus_minus_set_real @ S3 @ T3 ) )
               => ( ( G @ I2 )
                  = zero_zero_complex ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ ( inf_inf_set_real @ S3 @ T3 ) )
                 => ( ( G @ X5 )
                    = ( H2 @ X5 ) ) )
             => ( ( groups5754745047067104278omplex @ G @ S3 )
                = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_8179_sum_Omono__neutral__cong,axiom,
    ! [T3: set_int,S3: set_int,H2: int > complex,G: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( finite_finite_int @ S3 )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( H2 @ I2 )
                = zero_zero_complex ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ ( minus_minus_set_int @ S3 @ T3 ) )
               => ( ( G @ I2 )
                  = zero_zero_complex ) )
           => ( ! [X5: int] :
                  ( ( member_int @ X5 @ ( inf_inf_set_int @ S3 @ T3 ) )
                 => ( ( G @ X5 )
                    = ( H2 @ X5 ) ) )
             => ( ( groups3049146728041665814omplex @ G @ S3 )
                = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_8180_sum_Omono__neutral__cong,axiom,
    ! [T3: set_real,S3: set_real,H2: real > real,G: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( finite_finite_real @ S3 )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ I2 )
                = zero_zero_real ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ ( minus_minus_set_real @ S3 @ T3 ) )
               => ( ( G @ I2 )
                  = zero_zero_real ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ ( inf_inf_set_real @ S3 @ T3 ) )
                 => ( ( G @ X5 )
                    = ( H2 @ X5 ) ) )
             => ( ( groups8097168146408367636l_real @ G @ S3 )
                = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_8181_sum_Omono__neutral__cong,axiom,
    ! [T3: set_int,S3: set_int,H2: int > real,G: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( finite_finite_int @ S3 )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( H2 @ I2 )
                = zero_zero_real ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ ( minus_minus_set_int @ S3 @ T3 ) )
               => ( ( G @ I2 )
                  = zero_zero_real ) )
           => ( ! [X5: int] :
                  ( ( member_int @ X5 @ ( inf_inf_set_int @ S3 @ T3 ) )
                 => ( ( G @ X5 )
                    = ( H2 @ X5 ) ) )
             => ( ( groups8778361861064173332t_real @ G @ S3 )
                = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_8182_sum_Omono__neutral__cong,axiom,
    ! [T3: set_complex,S3: set_complex,H2: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( finite3207457112153483333omplex @ S3 )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( H2 @ I2 )
                = zero_zero_real ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ ( minus_811609699411566653omplex @ S3 @ T3 ) )
               => ( ( G @ I2 )
                  = zero_zero_real ) )
           => ( ! [X5: complex] :
                  ( ( member_complex @ X5 @ ( inf_inf_set_complex @ S3 @ T3 ) )
                 => ( ( G @ X5 )
                    = ( H2 @ X5 ) ) )
             => ( ( groups5808333547571424918x_real @ G @ S3 )
                = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_8183_sum_Omono__neutral__cong,axiom,
    ! [T3: set_real,S3: set_real,H2: real > rat,G: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( finite_finite_real @ S3 )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ I2 )
                = zero_zero_rat ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ ( minus_minus_set_real @ S3 @ T3 ) )
               => ( ( G @ I2 )
                  = zero_zero_rat ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ ( inf_inf_set_real @ S3 @ T3 ) )
                 => ( ( G @ X5 )
                    = ( H2 @ X5 ) ) )
             => ( ( groups1300246762558778688al_rat @ G @ S3 )
                = ( groups1300246762558778688al_rat @ H2 @ T3 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_8184_sum_Omono__neutral__cong,axiom,
    ! [T3: set_int,S3: set_int,H2: int > rat,G: int > rat] :
      ( ( finite_finite_int @ T3 )
     => ( ( finite_finite_int @ S3 )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( H2 @ I2 )
                = zero_zero_rat ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ ( minus_minus_set_int @ S3 @ T3 ) )
               => ( ( G @ I2 )
                  = zero_zero_rat ) )
           => ( ! [X5: int] :
                  ( ( member_int @ X5 @ ( inf_inf_set_int @ S3 @ T3 ) )
                 => ( ( G @ X5 )
                    = ( H2 @ X5 ) ) )
             => ( ( groups3906332499630173760nt_rat @ G @ S3 )
                = ( groups3906332499630173760nt_rat @ H2 @ T3 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_8185_sum_Omono__neutral__cong,axiom,
    ! [T3: set_complex,S3: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( finite3207457112153483333omplex @ S3 )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( H2 @ I2 )
                = zero_zero_rat ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ ( minus_811609699411566653omplex @ S3 @ T3 ) )
               => ( ( G @ I2 )
                  = zero_zero_rat ) )
           => ( ! [X5: complex] :
                  ( ( member_complex @ X5 @ ( inf_inf_set_complex @ S3 @ T3 ) )
                 => ( ( G @ X5 )
                    = ( H2 @ X5 ) ) )
             => ( ( groups5058264527183730370ex_rat @ G @ S3 )
                = ( groups5058264527183730370ex_rat @ H2 @ T3 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_8186_sum_Omono__neutral__cong,axiom,
    ! [T3: set_real,S3: set_real,H2: real > nat,G: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( finite_finite_real @ S3 )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ I2 )
                = zero_zero_nat ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ ( minus_minus_set_real @ S3 @ T3 ) )
               => ( ( G @ I2 )
                  = zero_zero_nat ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ ( inf_inf_set_real @ S3 @ T3 ) )
                 => ( ( G @ X5 )
                    = ( H2 @ X5 ) ) )
             => ( ( groups1935376822645274424al_nat @ G @ S3 )
                = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_8187_sum_Omono__neutral__cong,axiom,
    ! [T3: set_int,S3: set_int,H2: int > nat,G: int > nat] :
      ( ( finite_finite_int @ T3 )
     => ( ( finite_finite_int @ S3 )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( H2 @ I2 )
                = zero_zero_nat ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ ( minus_minus_set_int @ S3 @ T3 ) )
               => ( ( G @ I2 )
                  = zero_zero_nat ) )
           => ( ! [X5: int] :
                  ( ( member_int @ X5 @ ( inf_inf_set_int @ S3 @ T3 ) )
                 => ( ( G @ X5 )
                    = ( H2 @ X5 ) ) )
             => ( ( groups4541462559716669496nt_nat @ G @ S3 )
                = ( groups4541462559716669496nt_nat @ H2 @ T3 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_8188_sum_Ounion__inter,axiom,
    ! [A2: set_int,B2: set_int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( sup_sup_set_int @ A2 @ B2 ) ) @ ( groups8778361861064173332t_real @ G @ ( inf_inf_set_int @ A2 @ B2 ) ) )
          = ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ A2 ) @ ( groups8778361861064173332t_real @ G @ B2 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8189_sum_Ounion__inter,axiom,
    ! [A2: set_complex,B2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( sup_sup_set_complex @ A2 @ B2 ) ) @ ( groups5808333547571424918x_real @ G @ ( inf_inf_set_complex @ A2 @ B2 ) ) )
          = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ A2 ) @ ( groups5808333547571424918x_real @ G @ B2 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8190_sum_Ounion__inter,axiom,
    ! [A2: set_int,B2: set_int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( sup_sup_set_int @ A2 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ G @ ( inf_inf_set_int @ A2 @ B2 ) ) )
          = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ B2 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8191_sum_Ounion__inter,axiom,
    ! [A2: set_complex,B2: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( sup_sup_set_complex @ A2 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ G @ ( inf_inf_set_complex @ A2 @ B2 ) ) )
          = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ B2 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8192_sum_Ounion__inter,axiom,
    ! [A2: set_int,B2: set_int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( sup_sup_set_int @ A2 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ G @ ( inf_inf_set_int @ A2 @ B2 ) ) )
          = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ B2 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8193_sum_Ounion__inter,axiom,
    ! [A2: set_complex,B2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( sup_sup_set_complex @ A2 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ G @ ( inf_inf_set_complex @ A2 @ B2 ) ) )
          = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ B2 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8194_sum_Ounion__inter,axiom,
    ! [A2: set_complex,B2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( sup_sup_set_complex @ A2 @ B2 ) ) @ ( groups5690904116761175830ex_int @ G @ ( inf_inf_set_complex @ A2 @ B2 ) ) )
          = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ A2 ) @ ( groups5690904116761175830ex_int @ G @ B2 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8195_sum_Ounion__inter,axiom,
    ! [A2: set_nat,B2: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( inf_inf_set_nat @ A2 @ B2 ) ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ A2 ) @ ( groups2906978787729119204at_rat @ G @ B2 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8196_sum_Ounion__inter,axiom,
    ! [A2: set_nat,B2: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ ( groups3539618377306564664at_int @ G @ ( inf_inf_set_nat @ A2 @ B2 ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ A2 ) @ ( groups3539618377306564664at_int @ G @ B2 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8197_sum_Ounion__inter,axiom,
    ! [A2: set_int,B2: set_int,G: int > int] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( sup_sup_set_int @ A2 @ B2 ) ) @ ( groups4538972089207619220nt_int @ G @ ( inf_inf_set_int @ A2 @ B2 ) ) )
          = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A2 ) @ ( groups4538972089207619220nt_int @ G @ B2 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8198_sum_OInt__Diff,axiom,
    ! [A2: set_int,G: int > real,B2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G @ A2 )
        = ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( inf_inf_set_int @ A2 @ B2 ) ) @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8199_sum_OInt__Diff,axiom,
    ! [A2: set_complex,G: complex > real,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G @ A2 )
        = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( inf_inf_set_complex @ A2 @ B2 ) ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8200_sum_OInt__Diff,axiom,
    ! [A2: set_int,G: int > rat,B2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat @ G @ A2 )
        = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( inf_inf_set_int @ A2 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8201_sum_OInt__Diff,axiom,
    ! [A2: set_complex,G: complex > rat,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G @ A2 )
        = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( inf_inf_set_complex @ A2 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8202_sum_OInt__Diff,axiom,
    ! [A2: set_int,G: int > nat,B2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat @ G @ A2 )
        = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( inf_inf_set_int @ A2 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8203_sum_OInt__Diff,axiom,
    ! [A2: set_complex,G: complex > nat,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G @ A2 )
        = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( inf_inf_set_complex @ A2 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8204_sum_OInt__Diff,axiom,
    ! [A2: set_complex,G: complex > int,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5690904116761175830ex_int @ G @ A2 )
        = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( inf_inf_set_complex @ A2 @ B2 ) ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8205_sum_OInt__Diff,axiom,
    ! [A2: set_nat,G: nat > rat,B2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups2906978787729119204at_rat @ G @ A2 )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( inf_inf_set_nat @ A2 @ B2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8206_sum_OInt__Diff,axiom,
    ! [A2: set_nat,G: nat > int,B2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups3539618377306564664at_int @ G @ A2 )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( inf_inf_set_nat @ A2 @ B2 ) ) @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8207_sum_OInt__Diff,axiom,
    ! [A2: set_int,G: int > int,B2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups4538972089207619220nt_int @ G @ A2 )
        = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( inf_inf_set_int @ A2 @ B2 ) ) @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8208_bit_Ocomplement__unique,axiom,
    ! [A: code_integer,X: code_integer,Y: code_integer] :
      ( ( ( bit_se3949692690581998587nteger @ A @ X )
        = zero_z3403309356797280102nteger )
     => ( ( ( bit_se1080825931792720795nteger @ A @ X )
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
       => ( ( ( bit_se3949692690581998587nteger @ A @ Y )
            = zero_z3403309356797280102nteger )
         => ( ( ( bit_se1080825931792720795nteger @ A @ Y )
              = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
           => ( X = Y ) ) ) ) ) ).

% bit.complement_unique
thf(fact_8209_bit_Ocomplement__unique,axiom,
    ! [A: int,X: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ A @ X )
        = zero_zero_int )
     => ( ( ( bit_se1409905431419307370or_int @ A @ X )
          = ( uminus_uminus_int @ one_one_int ) )
       => ( ( ( bit_se725231765392027082nd_int @ A @ Y )
            = zero_zero_int )
         => ( ( ( bit_se1409905431419307370or_int @ A @ Y )
              = ( uminus_uminus_int @ one_one_int ) )
           => ( X = Y ) ) ) ) ) ).

% bit.complement_unique
thf(fact_8210_exp__ge__add__one__self__aux,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ) ).

% exp_ge_add_one_self_aux
thf(fact_8211_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_8212_ln__ge__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ Y @ ( ln_ln_real @ X ) )
        = ( ord_less_eq_real @ ( exp_real @ Y ) @ X ) ) ) ).

% ln_ge_iff
thf(fact_8213_sum_OIf__cases,axiom,
    ! [A2: set_real,P: real > $o,H2: real > real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups8097168146408367636l_real
          @ ^ [X6: real] : ( if_real @ ( P @ X6 ) @ ( H2 @ X6 ) @ ( G @ X6 ) )
          @ A2 )
        = ( plus_plus_real @ ( groups8097168146408367636l_real @ H2 @ ( inf_inf_set_real @ A2 @ ( collect_real @ P ) ) ) @ ( groups8097168146408367636l_real @ G @ ( inf_inf_set_real @ A2 @ ( uminus612125837232591019t_real @ ( collect_real @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8214_sum_OIf__cases,axiom,
    ! [A2: set_int,P: int > $o,H2: int > real,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real
          @ ^ [X6: int] : ( if_real @ ( P @ X6 ) @ ( H2 @ X6 ) @ ( G @ X6 ) )
          @ A2 )
        = ( plus_plus_real @ ( groups8778361861064173332t_real @ H2 @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) @ ( groups8778361861064173332t_real @ G @ ( inf_inf_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8215_sum_OIf__cases,axiom,
    ! [A2: set_complex,P: complex > $o,H2: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real
          @ ^ [X6: complex] : ( if_real @ ( P @ X6 ) @ ( H2 @ X6 ) @ ( G @ X6 ) )
          @ A2 )
        = ( plus_plus_real @ ( groups5808333547571424918x_real @ H2 @ ( inf_inf_set_complex @ A2 @ ( collect_complex @ P ) ) ) @ ( groups5808333547571424918x_real @ G @ ( inf_inf_set_complex @ A2 @ ( uminus8566677241136511917omplex @ ( collect_complex @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8216_sum_OIf__cases,axiom,
    ! [A2: set_real,P: real > $o,H2: real > rat,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1300246762558778688al_rat
          @ ^ [X6: real] : ( if_rat @ ( P @ X6 ) @ ( H2 @ X6 ) @ ( G @ X6 ) )
          @ A2 )
        = ( plus_plus_rat @ ( groups1300246762558778688al_rat @ H2 @ ( inf_inf_set_real @ A2 @ ( collect_real @ P ) ) ) @ ( groups1300246762558778688al_rat @ G @ ( inf_inf_set_real @ A2 @ ( uminus612125837232591019t_real @ ( collect_real @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8217_sum_OIf__cases,axiom,
    ! [A2: set_int,P: int > $o,H2: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat
          @ ^ [X6: int] : ( if_rat @ ( P @ X6 ) @ ( H2 @ X6 ) @ ( G @ X6 ) )
          @ A2 )
        = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ H2 @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) @ ( groups3906332499630173760nt_rat @ G @ ( inf_inf_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8218_sum_OIf__cases,axiom,
    ! [A2: set_complex,P: complex > $o,H2: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat
          @ ^ [X6: complex] : ( if_rat @ ( P @ X6 ) @ ( H2 @ X6 ) @ ( G @ X6 ) )
          @ A2 )
        = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ H2 @ ( inf_inf_set_complex @ A2 @ ( collect_complex @ P ) ) ) @ ( groups5058264527183730370ex_rat @ G @ ( inf_inf_set_complex @ A2 @ ( uminus8566677241136511917omplex @ ( collect_complex @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8219_sum_OIf__cases,axiom,
    ! [A2: set_real,P: real > $o,H2: real > nat,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1935376822645274424al_nat
          @ ^ [X6: real] : ( if_nat @ ( P @ X6 ) @ ( H2 @ X6 ) @ ( G @ X6 ) )
          @ A2 )
        = ( plus_plus_nat @ ( groups1935376822645274424al_nat @ H2 @ ( inf_inf_set_real @ A2 @ ( collect_real @ P ) ) ) @ ( groups1935376822645274424al_nat @ G @ ( inf_inf_set_real @ A2 @ ( uminus612125837232591019t_real @ ( collect_real @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8220_sum_OIf__cases,axiom,
    ! [A2: set_int,P: int > $o,H2: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X6: int] : ( if_nat @ ( P @ X6 ) @ ( H2 @ X6 ) @ ( G @ X6 ) )
          @ A2 )
        = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ H2 @ ( inf_inf_set_int @ A2 @ ( collect_int @ P ) ) ) @ ( groups4541462559716669496nt_nat @ G @ ( inf_inf_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8221_sum_OIf__cases,axiom,
    ! [A2: set_complex,P: complex > $o,H2: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat
          @ ^ [X6: complex] : ( if_nat @ ( P @ X6 ) @ ( H2 @ X6 ) @ ( G @ X6 ) )
          @ A2 )
        = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ H2 @ ( inf_inf_set_complex @ A2 @ ( collect_complex @ P ) ) ) @ ( groups5693394587270226106ex_nat @ G @ ( inf_inf_set_complex @ A2 @ ( uminus8566677241136511917omplex @ ( collect_complex @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8222_sum_OIf__cases,axiom,
    ! [A2: set_real,P: real > $o,H2: real > int,G: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1932886352136224148al_int
          @ ^ [X6: real] : ( if_int @ ( P @ X6 ) @ ( H2 @ X6 ) @ ( G @ X6 ) )
          @ A2 )
        = ( plus_plus_int @ ( groups1932886352136224148al_int @ H2 @ ( inf_inf_set_real @ A2 @ ( collect_real @ P ) ) ) @ ( groups1932886352136224148al_int @ G @ ( inf_inf_set_real @ A2 @ ( uminus612125837232591019t_real @ ( collect_real @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8223_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_8224_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_8225_take__bit__eq__mod,axiom,
    ( bit_se1745604003318907178nteger
    = ( ^ [N3: nat,A3: code_integer] : ( modulo364778990260209775nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% take_bit_eq_mod
thf(fact_8226_take__bit__eq__mod,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N3: nat,A3: int] : ( modulo_modulo_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% take_bit_eq_mod
thf(fact_8227_take__bit__eq__mod,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N3: nat,A3: nat] : ( modulo_modulo_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% take_bit_eq_mod
thf(fact_8228_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: real > real] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ! [B5: real] :
              ( ( member_real @ B5 @ ( minus_minus_set_real @ B2 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B5 ) ) )
         => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_8229_sum__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: int > real] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ! [B5: int] :
              ( ( member_int @ B5 @ ( minus_minus_set_int @ B2 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B5 ) ) )
         => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_8230_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [B5: complex] :
              ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B5 ) ) )
         => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_8231_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: real > rat] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ! [B5: real] :
              ( ( member_real @ B5 @ ( minus_minus_set_real @ B2 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B5 ) ) )
         => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_8232_sum__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: int > rat] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ! [B5: int] :
              ( ( member_int @ B5 @ ( minus_minus_set_int @ B2 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B5 ) ) )
         => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_8233_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [B5: complex] :
              ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B5 ) ) )
         => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_8234_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: real > nat] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ! [B5: real] :
              ( ( member_real @ B5 @ ( minus_minus_set_real @ B2 @ A2 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B5 ) ) )
         => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_8235_sum__mono2,axiom,
    ! [B2: set_int,A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ! [B5: int] :
              ( ( member_int @ B5 @ ( minus_minus_set_int @ B2 @ A2 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B5 ) ) )
         => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_8236_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [B5: complex] :
              ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B5 ) ) )
         => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_8237_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: real > int] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ! [B5: real] :
              ( ( member_real @ B5 @ ( minus_minus_set_real @ B2 @ A2 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ B5 ) ) )
         => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_8238_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ M )
        = M )
      = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_nat_eq_self_iff
thf(fact_8239_take__bit__nat__less__exp,axiom,
    ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% take_bit_nat_less_exp
thf(fact_8240_take__bit__nat__eq__self,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( bit_se2925701944663578781it_nat @ N @ M )
        = M ) ) ).

% take_bit_nat_eq_self
thf(fact_8241_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_int,B2: set_int,G: int > complex] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( inf_inf_set_int @ A2 @ B2 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ( groups3049146728041665814omplex @ G @ ( sup_sup_set_int @ A2 @ B2 ) )
            = ( plus_plus_complex @ ( groups3049146728041665814omplex @ G @ A2 ) @ ( groups3049146728041665814omplex @ G @ B2 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8242_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_int,B2: set_int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( inf_inf_set_int @ A2 @ B2 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups8778361861064173332t_real @ G @ ( sup_sup_set_int @ A2 @ B2 ) )
            = ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ A2 ) @ ( groups8778361861064173332t_real @ G @ B2 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8243_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_complex,B2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( inf_inf_set_complex @ A2 @ B2 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ ( sup_sup_set_complex @ A2 @ B2 ) )
            = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ A2 ) @ ( groups5808333547571424918x_real @ G @ B2 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8244_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_int,B2: set_int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( inf_inf_set_int @ A2 @ B2 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G @ ( sup_sup_set_int @ A2 @ B2 ) )
            = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ B2 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8245_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_complex,B2: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( inf_inf_set_complex @ A2 @ B2 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ ( sup_sup_set_complex @ A2 @ B2 ) )
            = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ B2 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8246_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_int,B2: set_int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( inf_inf_set_int @ A2 @ B2 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups4541462559716669496nt_nat @ G @ ( sup_sup_set_int @ A2 @ B2 ) )
            = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ B2 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8247_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_complex,B2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( inf_inf_set_complex @ A2 @ B2 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ ( sup_sup_set_complex @ A2 @ B2 ) )
            = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ B2 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8248_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_complex,B2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( inf_inf_set_complex @ A2 @ B2 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ ( sup_sup_set_complex @ A2 @ B2 ) )
            = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ A2 ) @ ( groups5690904116761175830ex_int @ G @ B2 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8249_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_nat,B2: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( inf_inf_set_nat @ A2 @ B2 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ( groups2073611262835488442omplex @ G @ ( sup_sup_set_nat @ A2 @ B2 ) )
            = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ A2 ) @ ( groups2073611262835488442omplex @ G @ B2 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8250_sum_Ounion__inter__neutral,axiom,
    ! [A2: set_nat,B2: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( inf_inf_set_nat @ A2 @ B2 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups2906978787729119204at_rat @ G @ ( sup_sup_set_nat @ A2 @ B2 ) )
            = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ A2 ) @ ( groups2906978787729119204at_rat @ G @ B2 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8251_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > real,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8252_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > rat,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8253_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > nat,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8254_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > int,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5690904116761175830ex_int @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_int @ ( G @ X ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8255_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: real > real,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
        = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8256_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: real > rat,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8257_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: real > nat,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X @ A2 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8258_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: real > int,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1932886352136224148al_int @ G @ ( insert_real @ X @ A2 ) )
        = ( plus_plus_int @ ( G @ X ) @ ( groups1932886352136224148al_int @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8259_sum_Oinsert__remove,axiom,
    ! [A2: set_o,G: $o > real,X: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( groups8691415230153176458o_real @ G @ ( insert_o @ X @ A2 ) )
        = ( plus_plus_real @ ( G @ X ) @ ( groups8691415230153176458o_real @ G @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8260_sum_Oinsert__remove,axiom,
    ! [A2: set_o,G: $o > rat,X: $o] :
      ( ( finite_finite_o @ A2 )
     => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X @ A2 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8261_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8262_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5058264527183730370ex_rat @ G @ A2 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8263_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5693394587270226106ex_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8264_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5690904116761175830ex_int @ G @ A2 )
          = ( plus_plus_int @ ( G @ X ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8265_sum_Oremove,axiom,
    ! [A2: set_real,X: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ( groups8097168146408367636l_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8266_sum_Oremove,axiom,
    ! [A2: set_real,X: real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ( groups1300246762558778688al_rat @ G @ A2 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8267_sum_Oremove,axiom,
    ! [A2: set_real,X: real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ( groups1935376822645274424al_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8268_sum_Oremove,axiom,
    ! [A2: set_real,X: real,G: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ( groups1932886352136224148al_int @ G @ A2 )
          = ( plus_plus_int @ ( G @ X ) @ ( groups1932886352136224148al_int @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8269_sum_Oremove,axiom,
    ! [A2: set_o,X: $o,G: $o > real] :
      ( ( finite_finite_o @ A2 )
     => ( ( member_o @ X @ A2 )
       => ( ( groups8691415230153176458o_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X ) @ ( groups8691415230153176458o_real @ G @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8270_sum_Oremove,axiom,
    ! [A2: set_o,X: $o,G: $o > rat] :
      ( ( finite_finite_o @ A2 )
     => ( ( member_o @ X @ A2 )
       => ( ( groups7872700643590313910_o_rat @ G @ A2 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups7872700643590313910_o_rat @ G @ ( minus_minus_set_o @ A2 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8271_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ A @ A2 )
         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_complex @ A @ A2 )
         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( groups5808333547571424918x_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8272_sum__diff1,axiom,
    ! [A2: set_real,A: real,F: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ A @ A2 )
         => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( minus_minus_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_real @ A @ A2 )
         => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( groups8097168146408367636l_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8273_sum__diff1,axiom,
    ! [A2: set_o,A: $o,F: $o > real] :
      ( ( finite_finite_o @ A2 )
     => ( ( ( member_o @ A @ A2 )
         => ( ( groups8691415230153176458o_real @ F @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
            = ( minus_minus_real @ ( groups8691415230153176458o_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_o @ A @ A2 )
         => ( ( groups8691415230153176458o_real @ F @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
            = ( groups8691415230153176458o_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8274_sum__diff1,axiom,
    ! [A2: set_int,A: int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ A @ A2 )
         => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_int @ A @ A2 )
         => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( groups8778361861064173332t_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8275_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ A @ A2 )
         => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_complex @ A @ A2 )
         => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8276_sum__diff1,axiom,
    ! [A2: set_real,A: real,F: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ A @ A2 )
         => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( minus_minus_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_real @ A @ A2 )
         => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8277_sum__diff1,axiom,
    ! [A2: set_o,A: $o,F: $o > rat] :
      ( ( finite_finite_o @ A2 )
     => ( ( ( member_o @ A @ A2 )
         => ( ( groups7872700643590313910_o_rat @ F @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
            = ( minus_minus_rat @ ( groups7872700643590313910_o_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_o @ A @ A2 )
         => ( ( groups7872700643590313910_o_rat @ F @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
            = ( groups7872700643590313910_o_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8278_sum__diff1,axiom,
    ! [A2: set_int,A: int,F: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ A @ A2 )
         => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_int @ A @ A2 )
         => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8279_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ A @ A2 )
         => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_complex @ A @ A2 )
         => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( groups5690904116761175830ex_int @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8280_sum__diff1,axiom,
    ! [A2: set_real,A: real,F: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ A @ A2 )
         => ( ( groups1932886352136224148al_int @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( minus_minus_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_real @ A @ A2 )
         => ( ( groups1932886352136224148al_int @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( groups1932886352136224148al_int @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8281_sum__Un,axiom,
    ! [A2: set_int,B2: set_int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ( groups8778361861064173332t_real @ F @ ( sup_sup_set_int @ A2 @ B2 ) )
          = ( minus_minus_real @ ( plus_plus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B2 ) ) @ ( groups8778361861064173332t_real @ F @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8282_sum__Un,axiom,
    ! [A2: set_complex,B2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups5808333547571424918x_real @ F @ ( sup_sup_set_complex @ A2 @ B2 ) )
          = ( minus_minus_real @ ( plus_plus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) @ ( groups5808333547571424918x_real @ F @ ( inf_inf_set_complex @ A2 @ B2 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8283_sum__Un,axiom,
    ! [A2: set_int,B2: set_int,F: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ( groups3906332499630173760nt_rat @ F @ ( sup_sup_set_int @ A2 @ B2 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B2 ) ) @ ( groups3906332499630173760nt_rat @ F @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8284_sum__Un,axiom,
    ! [A2: set_complex,B2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups5058264527183730370ex_rat @ F @ ( sup_sup_set_complex @ A2 @ B2 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) @ ( groups5058264527183730370ex_rat @ F @ ( inf_inf_set_complex @ A2 @ B2 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8285_sum__Un,axiom,
    ! [A2: set_nat,B2: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups2906978787729119204at_rat @ F @ ( sup_sup_set_nat @ A2 @ B2 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) @ ( groups2906978787729119204at_rat @ F @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8286_sum__Un,axiom,
    ! [A2: set_complex,B2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups5690904116761175830ex_int @ F @ ( sup_sup_set_complex @ A2 @ B2 ) )
          = ( minus_minus_int @ ( plus_plus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B2 ) ) @ ( groups5690904116761175830ex_int @ F @ ( inf_inf_set_complex @ A2 @ B2 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8287_sum__Un,axiom,
    ! [A2: set_nat,B2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups3539618377306564664at_int @ F @ ( sup_sup_set_nat @ A2 @ B2 ) )
          = ( minus_minus_int @ ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) @ ( groups3539618377306564664at_int @ F @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8288_sum__Un,axiom,
    ! [A2: set_int,B2: set_int,F: int > int] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ( groups4538972089207619220nt_int @ F @ ( sup_sup_set_int @ A2 @ B2 ) )
          = ( minus_minus_int @ ( plus_plus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ F @ B2 ) ) @ ( groups4538972089207619220nt_int @ F @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8289_sum__Un,axiom,
    ! [A2: set_complex,B2: set_complex,F: complex > complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups7754918857620584856omplex @ F @ ( sup_sup_set_complex @ A2 @ B2 ) )
          = ( minus_minus_complex @ ( plus_plus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ F @ B2 ) ) @ ( groups7754918857620584856omplex @ F @ ( inf_inf_set_complex @ A2 @ B2 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8290_sum__Un,axiom,
    ! [A2: set_nat,B2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups6591440286371151544t_real @ F @ ( sup_sup_set_nat @ A2 @ B2 ) )
          = ( minus_minus_real @ ( plus_plus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ F @ B2 ) ) @ ( groups6591440286371151544t_real @ F @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8291_sum_Ounion__disjoint,axiom,
    ! [A2: set_complex,B2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( ( inf_inf_set_complex @ A2 @ B2 )
            = bot_bot_set_complex )
         => ( ( groups5808333547571424918x_real @ G @ ( sup_sup_set_complex @ A2 @ B2 ) )
            = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ A2 ) @ ( groups5808333547571424918x_real @ G @ B2 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8292_sum_Ounion__disjoint,axiom,
    ! [A2: set_complex,B2: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( ( inf_inf_set_complex @ A2 @ B2 )
            = bot_bot_set_complex )
         => ( ( groups5058264527183730370ex_rat @ G @ ( sup_sup_set_complex @ A2 @ B2 ) )
            = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ B2 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8293_sum_Ounion__disjoint,axiom,
    ! [A2: set_complex,B2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( ( inf_inf_set_complex @ A2 @ B2 )
            = bot_bot_set_complex )
         => ( ( groups5693394587270226106ex_nat @ G @ ( sup_sup_set_complex @ A2 @ B2 ) )
            = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ B2 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8294_sum_Ounion__disjoint,axiom,
    ! [A2: set_complex,B2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( ( inf_inf_set_complex @ A2 @ B2 )
            = bot_bot_set_complex )
         => ( ( groups5690904116761175830ex_int @ G @ ( sup_sup_set_complex @ A2 @ B2 ) )
            = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ A2 ) @ ( groups5690904116761175830ex_int @ G @ B2 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8295_sum_Ounion__disjoint,axiom,
    ! [A2: set_real,B2: set_real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_finite_real @ B2 )
       => ( ( ( inf_inf_set_real @ A2 @ B2 )
            = bot_bot_set_real )
         => ( ( groups8097168146408367636l_real @ G @ ( sup_sup_set_real @ A2 @ B2 ) )
            = ( plus_plus_real @ ( groups8097168146408367636l_real @ G @ A2 ) @ ( groups8097168146408367636l_real @ G @ B2 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8296_sum_Ounion__disjoint,axiom,
    ! [A2: set_real,B2: set_real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_finite_real @ B2 )
       => ( ( ( inf_inf_set_real @ A2 @ B2 )
            = bot_bot_set_real )
         => ( ( groups1300246762558778688al_rat @ G @ ( sup_sup_set_real @ A2 @ B2 ) )
            = ( plus_plus_rat @ ( groups1300246762558778688al_rat @ G @ A2 ) @ ( groups1300246762558778688al_rat @ G @ B2 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8297_sum_Ounion__disjoint,axiom,
    ! [A2: set_real,B2: set_real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_finite_real @ B2 )
       => ( ( ( inf_inf_set_real @ A2 @ B2 )
            = bot_bot_set_real )
         => ( ( groups1935376822645274424al_nat @ G @ ( sup_sup_set_real @ A2 @ B2 ) )
            = ( plus_plus_nat @ ( groups1935376822645274424al_nat @ G @ A2 ) @ ( groups1935376822645274424al_nat @ G @ B2 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8298_sum_Ounion__disjoint,axiom,
    ! [A2: set_real,B2: set_real,G: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_finite_real @ B2 )
       => ( ( ( inf_inf_set_real @ A2 @ B2 )
            = bot_bot_set_real )
         => ( ( groups1932886352136224148al_int @ G @ ( sup_sup_set_real @ A2 @ B2 ) )
            = ( plus_plus_int @ ( groups1932886352136224148al_int @ G @ A2 ) @ ( groups1932886352136224148al_int @ G @ B2 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8299_sum_Ounion__disjoint,axiom,
    ! [A2: set_o,B2: set_o,G: $o > real] :
      ( ( finite_finite_o @ A2 )
     => ( ( finite_finite_o @ B2 )
       => ( ( ( inf_inf_set_o @ A2 @ B2 )
            = bot_bot_set_o )
         => ( ( groups8691415230153176458o_real @ G @ ( sup_sup_set_o @ A2 @ B2 ) )
            = ( plus_plus_real @ ( groups8691415230153176458o_real @ G @ A2 ) @ ( groups8691415230153176458o_real @ G @ B2 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8300_sum_Ounion__disjoint,axiom,
    ! [A2: set_o,B2: set_o,G: $o > rat] :
      ( ( finite_finite_o @ A2 )
     => ( ( finite_finite_o @ B2 )
       => ( ( ( inf_inf_set_o @ A2 @ B2 )
            = bot_bot_set_o )
         => ( ( groups7872700643590313910_o_rat @ G @ ( sup_sup_set_o @ A2 @ B2 ) )
            = ( plus_plus_rat @ ( groups7872700643590313910_o_rat @ G @ A2 ) @ ( groups7872700643590313910_o_rat @ G @ B2 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8301_sum_Ounion__diff2,axiom,
    ! [A2: set_int,B2: set_int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ( groups8778361861064173332t_real @ G @ ( sup_sup_set_int @ A2 @ B2 ) )
          = ( plus_plus_real @ ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ B2 @ A2 ) ) ) @ ( groups8778361861064173332t_real @ G @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8302_sum_Ounion__diff2,axiom,
    ! [A2: set_complex,B2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups5808333547571424918x_real @ G @ ( sup_sup_set_complex @ A2 @ B2 ) )
          = ( plus_plus_real @ ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ B2 @ A2 ) ) ) @ ( groups5808333547571424918x_real @ G @ ( inf_inf_set_complex @ A2 @ B2 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8303_sum_Ounion__diff2,axiom,
    ! [A2: set_int,B2: set_int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ( groups3906332499630173760nt_rat @ G @ ( sup_sup_set_int @ A2 @ B2 ) )
          = ( plus_plus_rat @ ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ B2 @ A2 ) ) ) @ ( groups3906332499630173760nt_rat @ G @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8304_sum_Ounion__diff2,axiom,
    ! [A2: set_complex,B2: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups5058264527183730370ex_rat @ G @ ( sup_sup_set_complex @ A2 @ B2 ) )
          = ( plus_plus_rat @ ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ B2 @ A2 ) ) ) @ ( groups5058264527183730370ex_rat @ G @ ( inf_inf_set_complex @ A2 @ B2 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8305_sum_Ounion__diff2,axiom,
    ! [A2: set_int,B2: set_int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ( groups4541462559716669496nt_nat @ G @ ( sup_sup_set_int @ A2 @ B2 ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ B2 @ A2 ) ) ) @ ( groups4541462559716669496nt_nat @ G @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8306_sum_Ounion__diff2,axiom,
    ! [A2: set_complex,B2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups5693394587270226106ex_nat @ G @ ( sup_sup_set_complex @ A2 @ B2 ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ B2 @ A2 ) ) ) @ ( groups5693394587270226106ex_nat @ G @ ( inf_inf_set_complex @ A2 @ B2 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8307_sum_Ounion__diff2,axiom,
    ! [A2: set_complex,B2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups5690904116761175830ex_int @ G @ ( sup_sup_set_complex @ A2 @ B2 ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ B2 @ A2 ) ) ) @ ( groups5690904116761175830ex_int @ G @ ( inf_inf_set_complex @ A2 @ B2 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8308_sum_Ounion__diff2,axiom,
    ! [A2: set_nat,B2: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups2906978787729119204at_rat @ G @ ( sup_sup_set_nat @ A2 @ B2 ) )
          = ( plus_plus_rat @ ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) @ ( groups2906978787729119204at_rat @ G @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8309_sum_Ounion__diff2,axiom,
    ! [A2: set_nat,B2: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups3539618377306564664at_int @ G @ ( sup_sup_set_nat @ A2 @ B2 ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) @ ( groups3539618377306564664at_int @ G @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8310_sum_Ounion__diff2,axiom,
    ! [A2: set_int,B2: set_int,G: int > int] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ( groups4538972089207619220nt_int @ G @ ( sup_sup_set_int @ A2 @ B2 ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ B2 @ A2 ) ) ) @ ( groups4538972089207619220nt_int @ G @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8311_sum__Un2,axiom,
    ! [A2: set_int,B2: set_int,F: int > real] :
      ( ( finite_finite_int @ ( sup_sup_set_int @ A2 @ B2 ) )
     => ( ( groups8778361861064173332t_real @ F @ ( sup_sup_set_int @ A2 @ B2 ) )
        = ( plus_plus_real @ ( plus_plus_real @ ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ B2 @ A2 ) ) ) @ ( groups8778361861064173332t_real @ F @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ).

% sum_Un2
thf(fact_8312_sum__Un2,axiom,
    ! [A2: set_complex,B2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ A2 @ B2 ) )
     => ( ( groups5808333547571424918x_real @ F @ ( sup_sup_set_complex @ A2 @ B2 ) )
        = ( plus_plus_real @ ( plus_plus_real @ ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ B2 @ A2 ) ) ) @ ( groups5808333547571424918x_real @ F @ ( inf_inf_set_complex @ A2 @ B2 ) ) ) ) ) ).

% sum_Un2
thf(fact_8313_sum__Un2,axiom,
    ! [A2: set_int,B2: set_int,F: int > rat] :
      ( ( finite_finite_int @ ( sup_sup_set_int @ A2 @ B2 ) )
     => ( ( groups3906332499630173760nt_rat @ F @ ( sup_sup_set_int @ A2 @ B2 ) )
        = ( plus_plus_rat @ ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ B2 @ A2 ) ) ) @ ( groups3906332499630173760nt_rat @ F @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ).

% sum_Un2
thf(fact_8314_sum__Un2,axiom,
    ! [A2: set_complex,B2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ A2 @ B2 ) )
     => ( ( groups5058264527183730370ex_rat @ F @ ( sup_sup_set_complex @ A2 @ B2 ) )
        = ( plus_plus_rat @ ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ B2 @ A2 ) ) ) @ ( groups5058264527183730370ex_rat @ F @ ( inf_inf_set_complex @ A2 @ B2 ) ) ) ) ) ).

% sum_Un2
thf(fact_8315_sum__Un2,axiom,
    ! [A2: set_int,B2: set_int,F: int > nat] :
      ( ( finite_finite_int @ ( sup_sup_set_int @ A2 @ B2 ) )
     => ( ( groups4541462559716669496nt_nat @ F @ ( sup_sup_set_int @ A2 @ B2 ) )
        = ( plus_plus_nat @ ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ B2 @ A2 ) ) ) @ ( groups4541462559716669496nt_nat @ F @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ).

% sum_Un2
thf(fact_8316_sum__Un2,axiom,
    ! [A2: set_complex,B2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ A2 @ B2 ) )
     => ( ( groups5693394587270226106ex_nat @ F @ ( sup_sup_set_complex @ A2 @ B2 ) )
        = ( plus_plus_nat @ ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ B2 @ A2 ) ) ) @ ( groups5693394587270226106ex_nat @ F @ ( inf_inf_set_complex @ A2 @ B2 ) ) ) ) ) ).

% sum_Un2
thf(fact_8317_sum__Un2,axiom,
    ! [A2: set_complex,B2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ A2 @ B2 ) )
     => ( ( groups5690904116761175830ex_int @ F @ ( sup_sup_set_complex @ A2 @ B2 ) )
        = ( plus_plus_int @ ( plus_plus_int @ ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ B2 @ A2 ) ) ) @ ( groups5690904116761175830ex_int @ F @ ( inf_inf_set_complex @ A2 @ B2 ) ) ) ) ) ).

% sum_Un2
thf(fact_8318_sum__Un2,axiom,
    ! [A2: set_nat,B2: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ A2 @ B2 ) )
     => ( ( groups2906978787729119204at_rat @ F @ ( sup_sup_set_nat @ A2 @ B2 ) )
        = ( plus_plus_rat @ ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) @ ( groups2906978787729119204at_rat @ F @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ).

% sum_Un2
thf(fact_8319_sum__Un2,axiom,
    ! [A2: set_nat,B2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ A2 @ B2 ) )
     => ( ( groups3539618377306564664at_int @ F @ ( sup_sup_set_nat @ A2 @ B2 ) )
        = ( plus_plus_int @ ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) @ ( groups3539618377306564664at_int @ F @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ).

% sum_Un2
thf(fact_8320_sum__Un2,axiom,
    ! [A2: set_int,B2: set_int,F: int > int] :
      ( ( finite_finite_int @ ( sup_sup_set_int @ A2 @ B2 ) )
     => ( ( groups4538972089207619220nt_int @ F @ ( sup_sup_set_int @ A2 @ B2 ) )
        = ( plus_plus_int @ ( plus_plus_int @ ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ B2 @ A2 ) ) ) @ ( groups4538972089207619220nt_int @ F @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ).

% sum_Un2
thf(fact_8321_take__bit__nat__def,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N3: nat,M4: nat] : ( modulo_modulo_nat @ M4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% take_bit_nat_def
thf(fact_8322_exp__le,axiom,
    ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).

% exp_le
thf(fact_8323_take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% take_bit_int_less_exp
thf(fact_8324_take__bit__int__def,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N3: nat,K3: int] : ( modulo_modulo_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% take_bit_int_def
thf(fact_8325_sum__div__partition,axiom,
    ! [A2: set_real,F: real > code_integer,B: code_integer] :
      ( ( finite_finite_real @ A2 )
     => ( ( divide6298287555418463151nteger @ ( groups7713935264441627589nteger @ F @ A2 ) @ B )
        = ( plus_p5714425477246183910nteger
          @ ( groups7713935264441627589nteger
            @ ^ [A3: real] : ( divide6298287555418463151nteger @ ( F @ A3 ) @ B )
            @ ( inf_inf_set_real @ A2
              @ ( collect_real
                @ ^ [A3: real] : ( dvd_dvd_Code_integer @ B @ ( F @ A3 ) ) ) ) )
          @ ( divide6298287555418463151nteger
            @ ( groups7713935264441627589nteger @ F
              @ ( inf_inf_set_real @ A2
                @ ( collect_real
                  @ ^ [A3: real] :
                      ~ ( dvd_dvd_Code_integer @ B @ ( F @ A3 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8326_sum__div__partition,axiom,
    ! [A2: set_int,F: int > code_integer,B: code_integer] :
      ( ( finite_finite_int @ A2 )
     => ( ( divide6298287555418463151nteger @ ( groups7873554091576472773nteger @ F @ A2 ) @ B )
        = ( plus_p5714425477246183910nteger
          @ ( groups7873554091576472773nteger
            @ ^ [A3: int] : ( divide6298287555418463151nteger @ ( F @ A3 ) @ B )
            @ ( inf_inf_set_int @ A2
              @ ( collect_int
                @ ^ [A3: int] : ( dvd_dvd_Code_integer @ B @ ( F @ A3 ) ) ) ) )
          @ ( divide6298287555418463151nteger
            @ ( groups7873554091576472773nteger @ F
              @ ( inf_inf_set_int @ A2
                @ ( collect_int
                  @ ^ [A3: int] :
                      ~ ( dvd_dvd_Code_integer @ B @ ( F @ A3 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8327_sum__div__partition,axiom,
    ! [A2: set_complex,F: complex > code_integer,B: code_integer] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( divide6298287555418463151nteger @ ( groups6621422865394947399nteger @ F @ A2 ) @ B )
        = ( plus_p5714425477246183910nteger
          @ ( groups6621422865394947399nteger
            @ ^ [A3: complex] : ( divide6298287555418463151nteger @ ( F @ A3 ) @ B )
            @ ( inf_inf_set_complex @ A2
              @ ( collect_complex
                @ ^ [A3: complex] : ( dvd_dvd_Code_integer @ B @ ( F @ A3 ) ) ) ) )
          @ ( divide6298287555418463151nteger
            @ ( groups6621422865394947399nteger @ F
              @ ( inf_inf_set_complex @ A2
                @ ( collect_complex
                  @ ^ [A3: complex] :
                      ~ ( dvd_dvd_Code_integer @ B @ ( F @ A3 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8328_sum__div__partition,axiom,
    ! [A2: set_nat,F: nat > code_integer,B: code_integer] :
      ( ( finite_finite_nat @ A2 )
     => ( ( divide6298287555418463151nteger @ ( groups7501900531339628137nteger @ F @ A2 ) @ B )
        = ( plus_p5714425477246183910nteger
          @ ( groups7501900531339628137nteger
            @ ^ [A3: nat] : ( divide6298287555418463151nteger @ ( F @ A3 ) @ B )
            @ ( inf_inf_set_nat @ A2
              @ ( collect_nat
                @ ^ [A3: nat] : ( dvd_dvd_Code_integer @ B @ ( F @ A3 ) ) ) ) )
          @ ( divide6298287555418463151nteger
            @ ( groups7501900531339628137nteger @ F
              @ ( inf_inf_set_nat @ A2
                @ ( collect_nat
                  @ ^ [A3: nat] :
                      ~ ( dvd_dvd_Code_integer @ B @ ( F @ A3 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8329_sum__div__partition,axiom,
    ! [A2: set_real,F: real > nat,B: nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( divide_divide_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ B )
        = ( plus_plus_nat
          @ ( groups1935376822645274424al_nat
            @ ^ [A3: real] : ( divide_divide_nat @ ( F @ A3 ) @ B )
            @ ( inf_inf_set_real @ A2
              @ ( collect_real
                @ ^ [A3: real] : ( dvd_dvd_nat @ B @ ( F @ A3 ) ) ) ) )
          @ ( divide_divide_nat
            @ ( groups1935376822645274424al_nat @ F
              @ ( inf_inf_set_real @ A2
                @ ( collect_real
                  @ ^ [A3: real] :
                      ~ ( dvd_dvd_nat @ B @ ( F @ A3 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8330_sum__div__partition,axiom,
    ! [A2: set_int,F: int > nat,B: nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( divide_divide_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ B )
        = ( plus_plus_nat
          @ ( groups4541462559716669496nt_nat
            @ ^ [A3: int] : ( divide_divide_nat @ ( F @ A3 ) @ B )
            @ ( inf_inf_set_int @ A2
              @ ( collect_int
                @ ^ [A3: int] : ( dvd_dvd_nat @ B @ ( F @ A3 ) ) ) ) )
          @ ( divide_divide_nat
            @ ( groups4541462559716669496nt_nat @ F
              @ ( inf_inf_set_int @ A2
                @ ( collect_int
                  @ ^ [A3: int] :
                      ~ ( dvd_dvd_nat @ B @ ( F @ A3 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8331_sum__div__partition,axiom,
    ! [A2: set_complex,F: complex > nat,B: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( divide_divide_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ B )
        = ( plus_plus_nat
          @ ( groups5693394587270226106ex_nat
            @ ^ [A3: complex] : ( divide_divide_nat @ ( F @ A3 ) @ B )
            @ ( inf_inf_set_complex @ A2
              @ ( collect_complex
                @ ^ [A3: complex] : ( dvd_dvd_nat @ B @ ( F @ A3 ) ) ) ) )
          @ ( divide_divide_nat
            @ ( groups5693394587270226106ex_nat @ F
              @ ( inf_inf_set_complex @ A2
                @ ( collect_complex
                  @ ^ [A3: complex] :
                      ~ ( dvd_dvd_nat @ B @ ( F @ A3 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8332_sum__div__partition,axiom,
    ! [A2: set_real,F: real > int,B: int] :
      ( ( finite_finite_real @ A2 )
     => ( ( divide_divide_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ B )
        = ( plus_plus_int
          @ ( groups1932886352136224148al_int
            @ ^ [A3: real] : ( divide_divide_int @ ( F @ A3 ) @ B )
            @ ( inf_inf_set_real @ A2
              @ ( collect_real
                @ ^ [A3: real] : ( dvd_dvd_int @ B @ ( F @ A3 ) ) ) ) )
          @ ( divide_divide_int
            @ ( groups1932886352136224148al_int @ F
              @ ( inf_inf_set_real @ A2
                @ ( collect_real
                  @ ^ [A3: real] :
                      ~ ( dvd_dvd_int @ B @ ( F @ A3 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8333_sum__div__partition,axiom,
    ! [A2: set_complex,F: complex > int,B: int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( divide_divide_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ B )
        = ( plus_plus_int
          @ ( groups5690904116761175830ex_int
            @ ^ [A3: complex] : ( divide_divide_int @ ( F @ A3 ) @ B )
            @ ( inf_inf_set_complex @ A2
              @ ( collect_complex
                @ ^ [A3: complex] : ( dvd_dvd_int @ B @ ( F @ A3 ) ) ) ) )
          @ ( divide_divide_int
            @ ( groups5690904116761175830ex_int @ F
              @ ( inf_inf_set_complex @ A2
                @ ( collect_complex
                  @ ^ [A3: complex] :
                      ~ ( dvd_dvd_int @ B @ ( F @ A3 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8334_sum__div__partition,axiom,
    ! [A2: set_nat,F: nat > int,B: int] :
      ( ( finite_finite_nat @ A2 )
     => ( ( divide_divide_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ B )
        = ( plus_plus_int
          @ ( groups3539618377306564664at_int
            @ ^ [A3: nat] : ( divide_divide_int @ ( F @ A3 ) @ B )
            @ ( inf_inf_set_nat @ A2
              @ ( collect_nat
                @ ^ [A3: nat] : ( dvd_dvd_int @ B @ ( F @ A3 ) ) ) ) )
          @ ( divide_divide_int
            @ ( groups3539618377306564664at_int @ F
              @ ( inf_inf_set_nat @ A2
                @ ( collect_nat
                  @ ^ [A3: nat] :
                      ~ ( dvd_dvd_int @ B @ ( F @ A3 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8335_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > real,C: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8336_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > rat,C: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8337_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > nat,C: complex > nat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K3: complex] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K3: complex] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8338_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > int,C: complex > int] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5690904116761175830ex_int
              @ ^ [K3: complex] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_int @ ( B @ A ) @ ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5690904116761175830ex_int
              @ ^ [K3: complex] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8339_sum_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: real > real,C: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups8097168146408367636l_real @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups8097168146408367636l_real @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8340_sum_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: real > rat,C: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups1300246762558778688al_rat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups1300246762558778688al_rat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8341_sum_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: real > nat,C: real > nat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K3: real] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups1935376822645274424al_nat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K3: real] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups1935376822645274424al_nat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8342_sum_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: real > int,C: real > int] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1932886352136224148al_int
              @ ^ [K3: real] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_int @ ( B @ A ) @ ( groups1932886352136224148al_int @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1932886352136224148al_int
              @ ^ [K3: real] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups1932886352136224148al_int @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8343_sum_Odelta__remove,axiom,
    ! [S3: set_o,A: $o,B: $o > real,C: $o > real] :
      ( ( finite_finite_o @ S3 )
     => ( ( ( member_o @ A @ S3 )
         => ( ( groups8691415230153176458o_real
              @ ^ [K3: $o] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups8691415230153176458o_real @ C @ ( minus_minus_set_o @ S3 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
        & ( ~ ( member_o @ A @ S3 )
         => ( ( groups8691415230153176458o_real
              @ ^ [K3: $o] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups8691415230153176458o_real @ C @ ( minus_minus_set_o @ S3 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8344_sum_Odelta__remove,axiom,
    ! [S3: set_o,A: $o,B: $o > rat,C: $o > rat] :
      ( ( finite_finite_o @ S3 )
     => ( ( ( member_o @ A @ S3 )
         => ( ( groups7872700643590313910_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups7872700643590313910_o_rat @ C @ ( minus_minus_set_o @ S3 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
        & ( ~ ( member_o @ A @ S3 )
         => ( ( groups7872700643590313910_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups7872700643590313910_o_rat @ C @ ( minus_minus_set_o @ S3 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8345_tanh__altdef,axiom,
    ( tanh_real
    = ( ^ [X6: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X6 ) @ ( exp_real @ ( uminus_uminus_real @ X6 ) ) ) @ ( plus_plus_real @ ( exp_real @ X6 ) @ ( exp_real @ ( uminus_uminus_real @ X6 ) ) ) ) ) ) ).

% tanh_altdef
thf(fact_8346_tanh__altdef,axiom,
    ( tanh_complex
    = ( ^ [X6: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X6 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X6 ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X6 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X6 ) ) ) ) ) ) ).

% tanh_altdef
thf(fact_8347_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( bit_se1745604003318907178nteger @ N @ A )
        = zero_z3403309356797280102nteger )
      = ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A ) ) ).

% take_bit_eq_0_iff
thf(fact_8348_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ A )
        = zero_zero_int )
      = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A ) ) ).

% take_bit_eq_0_iff
thf(fact_8349_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ A )
        = zero_zero_nat )
      = ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A ) ) ).

% take_bit_eq_0_iff
thf(fact_8350_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: real > real] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ B2 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
             => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8351_sum__strict__mono2,axiom,
    ! [B2: set_int,A2: set_int,B: int,F: int > real] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ( member_int @ B @ ( minus_minus_set_int @ B2 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X5: int] :
                  ( ( member_int @ X5 @ B2 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
             => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8352_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X5: complex] :
                  ( ( member_complex @ X5 @ B2 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
             => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8353_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: real > rat] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ B2 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
             => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8354_sum__strict__mono2,axiom,
    ! [B2: set_int,A2: set_int,B: int,F: int > rat] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ( member_int @ B @ ( minus_minus_set_int @ B2 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X5: int] :
                  ( ( member_int @ X5 @ B2 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
             => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8355_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X5: complex] :
                  ( ( member_complex @ X5 @ B2 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
             => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8356_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: real > nat] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ B2 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
             => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8357_sum__strict__mono2,axiom,
    ! [B2: set_int,A2: set_int,B: int,F: int > nat] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( ( member_int @ B @ ( minus_minus_set_int @ B2 @ A2 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X5: int] :
                  ( ( member_int @ X5 @ B2 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
             => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8358_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X5: complex] :
                  ( ( member_complex @ X5 @ B2 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
             => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8359_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: real > int] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ B2 )
                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
             => ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8360_member__le__sum,axiom,
    ! [I3: complex,A2: set_complex,F: complex > real] :
      ( ( member_complex @ I3 @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I3 @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( finite3207457112153483333omplex @ A2 )
         => ( ord_less_eq_real @ ( F @ I3 ) @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8361_member__le__sum,axiom,
    ! [I3: real,A2: set_real,F: real > real] :
      ( ( member_real @ I3 @ A2 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I3 @ bot_bot_set_real ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( finite_finite_real @ A2 )
         => ( ord_less_eq_real @ ( F @ I3 ) @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8362_member__le__sum,axiom,
    ! [I3: $o,A2: set_o,F: $o > real] :
      ( ( member_o @ I3 @ A2 )
     => ( ! [X5: $o] :
            ( ( member_o @ X5 @ ( minus_minus_set_o @ A2 @ ( insert_o @ I3 @ bot_bot_set_o ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( finite_finite_o @ A2 )
         => ( ord_less_eq_real @ ( F @ I3 ) @ ( groups8691415230153176458o_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8363_member__le__sum,axiom,
    ! [I3: int,A2: set_int,F: int > real] :
      ( ( member_int @ I3 @ A2 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I3 @ bot_bot_set_int ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( finite_finite_int @ A2 )
         => ( ord_less_eq_real @ ( F @ I3 ) @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8364_member__le__sum,axiom,
    ! [I3: complex,A2: set_complex,F: complex > rat] :
      ( ( member_complex @ I3 @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I3 @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( finite3207457112153483333omplex @ A2 )
         => ( ord_less_eq_rat @ ( F @ I3 ) @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8365_member__le__sum,axiom,
    ! [I3: real,A2: set_real,F: real > rat] :
      ( ( member_real @ I3 @ A2 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I3 @ bot_bot_set_real ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( finite_finite_real @ A2 )
         => ( ord_less_eq_rat @ ( F @ I3 ) @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8366_member__le__sum,axiom,
    ! [I3: $o,A2: set_o,F: $o > rat] :
      ( ( member_o @ I3 @ A2 )
     => ( ! [X5: $o] :
            ( ( member_o @ X5 @ ( minus_minus_set_o @ A2 @ ( insert_o @ I3 @ bot_bot_set_o ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( finite_finite_o @ A2 )
         => ( ord_less_eq_rat @ ( F @ I3 ) @ ( groups7872700643590313910_o_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8367_member__le__sum,axiom,
    ! [I3: int,A2: set_int,F: int > rat] :
      ( ( member_int @ I3 @ A2 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I3 @ bot_bot_set_int ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( finite_finite_int @ A2 )
         => ( ord_less_eq_rat @ ( F @ I3 ) @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8368_member__le__sum,axiom,
    ! [I3: nat,A2: set_nat,F: nat > rat] :
      ( ( member_nat @ I3 @ A2 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ I3 @ bot_bot_set_nat ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( finite_finite_nat @ A2 )
         => ( ord_less_eq_rat @ ( F @ I3 ) @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8369_member__le__sum,axiom,
    ! [I3: complex,A2: set_complex,F: complex > nat] :
      ( ( member_complex @ I3 @ A2 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I3 @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
       => ( ( finite3207457112153483333omplex @ A2 )
         => ( ord_less_eq_nat @ ( F @ I3 ) @ ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8370_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_8371_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_8372_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).

% take_bit_nat_less_self_iff
thf(fact_8373_Suc__mask__eq__exp,axiom,
    ! [N: nat] :
      ( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% Suc_mask_eq_exp
thf(fact_8374_mask__nat__less__exp,axiom,
    ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% mask_nat_less_exp
thf(fact_8375_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_8376_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_minus_bit0
thf(fact_8377_convex__sum__bound__le,axiom,
    ! [I6: set_complex,X: complex > code_integer,A: complex > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ I6 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
     => ( ( ( groups6621422865394947399nteger @ X @ I6 )
          = one_one_Code_integer )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I6 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups6621422865394947399nteger
                  @ ^ [I: complex] : ( times_3573771949741848930nteger @ ( A @ I ) @ ( X @ I ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8378_convex__sum__bound__le,axiom,
    ! [I6: set_real,X: real > code_integer,A: real > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I6 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
     => ( ( ( groups7713935264441627589nteger @ X @ I6 )
          = one_one_Code_integer )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I6 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7713935264441627589nteger
                  @ ^ [I: real] : ( times_3573771949741848930nteger @ ( A @ I ) @ ( X @ I ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8379_convex__sum__bound__le,axiom,
    ! [I6: set_nat,X: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I6 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
     => ( ( ( groups7501900531339628137nteger @ X @ I6 )
          = one_one_Code_integer )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I6 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7501900531339628137nteger
                  @ ^ [I: nat] : ( times_3573771949741848930nteger @ ( A @ I ) @ ( X @ I ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8380_convex__sum__bound__le,axiom,
    ! [I6: set_int,X: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I6 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
     => ( ( ( groups7873554091576472773nteger @ X @ I6 )
          = one_one_Code_integer )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I6 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7873554091576472773nteger
                  @ ^ [I: int] : ( times_3573771949741848930nteger @ ( A @ I ) @ ( X @ I ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8381_convex__sum__bound__le,axiom,
    ! [I6: set_complex,X: complex > real,A: complex > real,B: real,Delta: real] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ I6 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
     => ( ( ( groups5808333547571424918x_real @ X @ I6 )
          = one_one_real )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I6 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups5808333547571424918x_real
                  @ ^ [I: complex] : ( times_times_real @ ( A @ I ) @ ( X @ I ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8382_convex__sum__bound__le,axiom,
    ! [I6: set_real,X: real > real,A: real > real,B: real,Delta: real] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I6 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
     => ( ( ( groups8097168146408367636l_real @ X @ I6 )
          = one_one_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I6 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8097168146408367636l_real
                  @ ^ [I: real] : ( times_times_real @ ( A @ I ) @ ( X @ I ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8383_convex__sum__bound__le,axiom,
    ! [I6: set_int,X: int > real,A: int > real,B: real,Delta: real] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I6 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
     => ( ( ( groups8778361861064173332t_real @ X @ I6 )
          = one_one_real )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I6 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8778361861064173332t_real
                  @ ^ [I: int] : ( times_times_real @ ( A @ I ) @ ( X @ I ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8384_convex__sum__bound__le,axiom,
    ! [I6: set_complex,X: complex > rat,A: complex > rat,B: rat,Delta: rat] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ I6 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
     => ( ( ( groups5058264527183730370ex_rat @ X @ I6 )
          = one_one_rat )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I6 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups5058264527183730370ex_rat
                  @ ^ [I: complex] : ( times_times_rat @ ( A @ I ) @ ( X @ I ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8385_convex__sum__bound__le,axiom,
    ! [I6: set_real,X: real > rat,A: real > rat,B: rat,Delta: rat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I6 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
     => ( ( ( groups1300246762558778688al_rat @ X @ I6 )
          = one_one_rat )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I6 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups1300246762558778688al_rat
                  @ ^ [I: real] : ( times_times_rat @ ( A @ I ) @ ( X @ I ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8386_convex__sum__bound__le,axiom,
    ! [I6: set_nat,X: nat > rat,A: nat > rat,B: rat,Delta: rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I6 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
     => ( ( ( groups2906978787729119204at_rat @ X @ I6 )
          = one_one_rat )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I6 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [I: nat] : ( times_times_rat @ ( A @ I ) @ ( X @ I ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8387_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).

% take_bit_int_less_self_iff
thf(fact_8388_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_int_greater_eq_self_iff
thf(fact_8389_exp__double,axiom,
    ! [Z2: complex] :
      ( ( exp_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 ) )
      = ( power_power_complex @ ( exp_complex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% exp_double
thf(fact_8390_exp__double,axiom,
    ! [Z2: real] :
      ( ( exp_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 ) )
      = ( power_power_real @ ( exp_real @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% exp_double
thf(fact_8391_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ N ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
thf(fact_8392_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
thf(fact_8393_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
thf(fact_8394_or__one__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se1080825931792720795nteger @ A @ one_one_Code_integer )
      = ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% or_one_eq
thf(fact_8395_or__one__eq,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ A @ one_one_int )
      = ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% or_one_eq
thf(fact_8396_or__one__eq,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ A @ one_one_nat )
      = ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% or_one_eq
thf(fact_8397_one__or__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se1080825931792720795nteger @ one_one_Code_integer @ A )
      = ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_or_eq
thf(fact_8398_one__or__eq,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ A )
      = ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_or_eq
thf(fact_8399_one__or__eq,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ one_one_nat @ A )
      = ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_or_eq
thf(fact_8400_OR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
         => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% OR_upper
thf(fact_8401_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
        = K )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% take_bit_int_eq_self_iff
thf(fact_8402_take__bit__int__eq__self,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_se2923211474154528505it_int @ N @ K )
          = K ) ) ) ).

% take_bit_int_eq_self
thf(fact_8403_mask__nat__def,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).

% mask_nat_def
thf(fact_8404_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_8405_mask__half__int,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).

% mask_half_int
thf(fact_8406_take__bit__incr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
       != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
     => ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
        = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).

% take_bit_incr_eq
thf(fact_8407_mask__int__def,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N3: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ) ).

% mask_int_def
thf(fact_8408_take__bit__Suc__minus__1__eq,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_Code_integer ) ) ).

% take_bit_Suc_minus_1_eq
thf(fact_8409_take__bit__Suc__minus__1__eq,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_1_eq
thf(fact_8410_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_bit1
thf(fact_8411_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).

% take_bit_Suc_bit1
thf(fact_8412_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_8413_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_8414_take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_8415_take__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_8416_take__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ A )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_8417_mask__eq__exp__minus__1,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_8418_mask__eq__exp__minus__1,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N3: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_8419_exp__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% exp_bound
thf(fact_8420_take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% take_bit_int_less_eq
thf(fact_8421_take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% take_bit_int_greater_eq
thf(fact_8422_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_8423_signed__take__bit__eq__take__bit__shift,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N3: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% signed_take_bit_eq_take_bit_shift
thf(fact_8424_stable__imp__take__bit__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se1745604003318907178nteger @ N @ A )
            = zero_z3403309356797280102nteger ) )
        & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se1745604003318907178nteger @ N @ A )
            = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_8425_stable__imp__take__bit__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2923211474154528505it_int @ N @ A )
            = zero_zero_int ) )
        & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2923211474154528505it_int @ N @ A )
            = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_8426_stable__imp__take__bit__eq,axiom,
    ! [A: nat,N: nat] :
      ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2925701944663578781it_nat @ N @ A )
            = zero_zero_nat ) )
        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2925701944663578781it_nat @ N @ A )
            = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_8427_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_8428_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_8429_real__exp__bound__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ) ) ) ).

% real_exp_bound_lemma
thf(fact_8430_take__bit__minus__small__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
          = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).

% take_bit_minus_small_eq
thf(fact_8431_exp__lower__Taylor__quadratic,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( divide_divide_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X ) ) ) ).

% exp_lower_Taylor_quadratic
thf(fact_8432_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_8433_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_bit1
thf(fact_8434_or__minus__numerals_I1_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(1)
thf(fact_8435_or__minus__numerals_I5_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(5)
thf(fact_8436_log__base__10__eq1,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
        = ( 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 @ X ) ) ) ) ).

% log_base_10_eq1
thf(fact_8437_log__one,axiom,
    ! [A: real] :
      ( ( log @ A @ one_one_real )
      = zero_zero_real ) ).

% log_one
thf(fact_8438_pred__numeral__inc,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( inc @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% pred_numeral_inc
thf(fact_8439_or__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_nat_numerals(4)
thf(fact_8440_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_8441_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_8442_log__less__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ zero_zero_real @ Y )
         => ( ( ord_less_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
            = ( ord_less_real @ X @ Y ) ) ) ) ) ).

% log_less_cancel_iff
thf(fact_8443_log__less__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( log @ A @ X ) @ one_one_real )
          = ( ord_less_real @ X @ A ) ) ) ) ).

% log_less_one_cancel_iff
thf(fact_8444_one__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ one_one_real @ ( log @ A @ X ) )
          = ( ord_less_real @ A @ X ) ) ) ) ).

% one_less_log_cancel_iff
thf(fact_8445_log__less__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( log @ A @ X ) @ zero_zero_real )
          = ( ord_less_real @ X @ one_one_real ) ) ) ) ).

% log_less_zero_cancel_iff
thf(fact_8446_zero__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X ) )
          = ( ord_less_real @ one_one_real @ X ) ) ) ) ).

% zero_less_log_cancel_iff
thf(fact_8447_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_8448_or__nat__numerals_I3_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_nat_numerals(3)
thf(fact_8449_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_8450_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_8451_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_8452_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_8453_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_8454_log__le__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
            = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).

% log_le_cancel_iff
thf(fact_8455_log__le__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( log @ A @ X ) @ one_one_real )
          = ( ord_less_eq_real @ X @ A ) ) ) ) ).

% log_le_one_cancel_iff
thf(fact_8456_one__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X ) )
          = ( ord_less_eq_real @ A @ X ) ) ) ) ).

% one_le_log_cancel_iff
thf(fact_8457_log__le__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( log @ A @ X ) @ zero_zero_real )
          = ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).

% log_le_zero_cancel_iff
thf(fact_8458_zero__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X ) )
          = ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).

% zero_le_log_cancel_iff
thf(fact_8459_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_8460_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_8461_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_8462_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_8463_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_8464_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_8465_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_8466_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_8467_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_8468_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_8469_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_8470_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_8471_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_8472_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_8473_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_8474_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ N ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_8475_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_8476_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( numera6690914467698888265omplex @ N ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_8477_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_8478_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_8479_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > complex] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I: nat] : ( times_times_complex @ ( C @ I ) @ ( power_power_complex @ zero_zero_complex @ I ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I: nat] : ( times_times_complex @ ( C @ I ) @ ( power_power_complex @ zero_zero_complex @ I ) )
            @ A2 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power
thf(fact_8480_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > rat] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I: nat] : ( times_times_rat @ ( C @ I ) @ ( power_power_rat @ zero_zero_rat @ I ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I: nat] : ( times_times_rat @ ( C @ I ) @ ( power_power_rat @ zero_zero_rat @ I ) )
            @ A2 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power
thf(fact_8481_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > real] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I: nat] : ( times_times_real @ ( C @ I ) @ ( power_power_real @ zero_zero_real @ I ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I: nat] : ( times_times_real @ ( C @ I ) @ ( power_power_real @ zero_zero_real @ I ) )
            @ A2 )
          = zero_zero_real ) ) ) ).

% sum_zero_power
thf(fact_8482_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: nat > complex,D: nat > complex] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I ) @ ( power_power_complex @ zero_zero_complex @ I ) ) @ ( D @ I ) )
            @ A2 )
          = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I ) @ ( power_power_complex @ zero_zero_complex @ I ) ) @ ( D @ I ) )
            @ A2 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power'
thf(fact_8483_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: nat > rat,D: nat > rat] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I ) @ ( power_power_rat @ zero_zero_rat @ I ) ) @ ( D @ I ) )
            @ A2 )
          = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I ) @ ( power_power_rat @ zero_zero_rat @ I ) ) @ ( D @ I ) )
            @ A2 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power'
thf(fact_8484_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: nat > real,D: nat > real] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I ) @ ( power_power_real @ zero_zero_real @ I ) ) @ ( D @ I ) )
            @ A2 )
          = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I ) @ ( power_power_real @ zero_zero_real @ I ) ) @ ( D @ I ) )
            @ A2 )
          = zero_zero_real ) ) ) ).

% sum_zero_power'
thf(fact_8485_or__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).

% or_minus_numerals(4)
thf(fact_8486_or__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).

% or_minus_numerals(8)
thf(fact_8487_or__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(3)
thf(fact_8488_or__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(7)
thf(fact_8489_or__not__num__neg_Osimps_I1_J,axiom,
    ( ( bit_or_not_num_neg @ one @ one )
    = one ) ).

% or_not_num_neg.simps(1)
thf(fact_8490_num__induct,axiom,
    ! [P: num > $o,X: num] :
      ( ( P @ one )
     => ( ! [X5: num] :
            ( ( P @ X5 )
           => ( P @ ( inc @ X5 ) ) )
       => ( P @ X ) ) ) ).

% num_induct
thf(fact_8491_add__inc,axiom,
    ! [X: num,Y: num] :
      ( ( plus_plus_num @ X @ ( inc @ Y ) )
      = ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).

% add_inc
thf(fact_8492_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_8493_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_8494_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( G @ ( suc @ I ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_8495_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > real,M: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I: nat] : ( G @ ( suc @ I ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_8496_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( G @ ( plus_plus_nat @ I @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_8497_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > real,M: nat,K: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I: nat] : ( G @ ( plus_plus_nat @ I @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_8498_or__not__num__neg_Osimps_I4_J,axiom,
    ! [N: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one )
      = ( bit0 @ one ) ) ).

% or_not_num_neg.simps(4)
thf(fact_8499_or__not__num__neg_Osimps_I6_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
      = ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).

% or_not_num_neg.simps(6)
thf(fact_8500_or__not__num__neg_Osimps_I7_J,axiom,
    ! [N: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one )
      = one ) ).

% or_not_num_neg.simps(7)
thf(fact_8501_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_8502_inc_Osimps_I1_J,axiom,
    ( ( inc @ one )
    = ( bit0 @ one ) ) ).

% inc.simps(1)
thf(fact_8503_inc_Osimps_I3_J,axiom,
    ! [X: num] :
      ( ( inc @ ( bit1 @ X ) )
      = ( bit0 @ ( inc @ X ) ) ) ).

% inc.simps(3)
thf(fact_8504_inc_Osimps_I2_J,axiom,
    ! [X: num] :
      ( ( inc @ ( bit0 @ X ) )
      = ( bit1 @ X ) ) ).

% inc.simps(2)
thf(fact_8505_or__not__num__neg_Osimps_I5_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).

% or_not_num_neg.simps(5)
thf(fact_8506_sum__eq__Suc0__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X6: int] :
              ( ( member_int @ X6 @ A2 )
              & ( ( F @ X6 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y6: int] :
                  ( ( member_int @ Y6 @ A2 )
                 => ( ( X6 != Y6 )
                   => ( ( F @ Y6 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_8507_sum__eq__Suc0__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X6: complex] :
              ( ( member_complex @ X6 @ A2 )
              & ( ( F @ X6 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y6: complex] :
                  ( ( member_complex @ Y6 @ A2 )
                 => ( ( X6 != Y6 )
                   => ( ( F @ Y6 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_8508_sum__eq__Suc0__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X6: nat] :
              ( ( member_nat @ X6 @ A2 )
              & ( ( F @ X6 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y6: nat] :
                  ( ( member_nat @ Y6 @ A2 )
                 => ( ( X6 != Y6 )
                   => ( ( F @ Y6 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_8509_sum__SucD,axiom,
    ! [F: nat > nat,A2: set_nat,N: nat] :
      ( ( ( groups3542108847815614940at_nat @ F @ A2 )
        = ( suc @ N ) )
     => ? [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
          & ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) ) ) ).

% sum_SucD
thf(fact_8510_or__not__num__neg_Osimps_I9_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit1 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).

% or_not_num_neg.simps(9)
thf(fact_8511_sum__eq__1__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X6: int] :
              ( ( member_int @ X6 @ A2 )
              & ( ( F @ X6 )
                = one_one_nat )
              & ! [Y6: int] :
                  ( ( member_int @ Y6 @ A2 )
                 => ( ( X6 != Y6 )
                   => ( ( F @ Y6 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_8512_sum__eq__1__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X6: complex] :
              ( ( member_complex @ X6 @ A2 )
              & ( ( F @ X6 )
                = one_one_nat )
              & ! [Y6: complex] :
                  ( ( member_complex @ Y6 @ A2 )
                 => ( ( X6 != Y6 )
                   => ( ( F @ Y6 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_8513_sum__eq__1__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X6: nat] :
              ( ( member_nat @ X6 @ A2 )
              & ( ( F @ X6 )
                = one_one_nat )
              & ! [Y6: nat] :
                  ( ( member_nat @ Y6 @ A2 )
                 => ( ( X6 != Y6 )
                   => ( ( F @ Y6 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_8514_add__One,axiom,
    ! [X: num] :
      ( ( plus_plus_num @ X @ one )
      = ( inc @ X ) ) ).

% add_One
thf(fact_8515_inc__BitM__eq,axiom,
    ! [N: num] :
      ( ( inc @ ( bitM @ N ) )
      = ( bit0 @ N ) ) ).

% inc_BitM_eq
thf(fact_8516_BitM__inc__eq,axiom,
    ! [N: num] :
      ( ( bitM @ ( inc @ N ) )
      = ( bit1 @ N ) ) ).

% BitM_inc_eq
thf(fact_8517_mult__inc,axiom,
    ! [X: num,Y: num] :
      ( ( times_times_num @ X @ ( inc @ Y ) )
      = ( plus_plus_num @ ( times_times_num @ X @ Y ) @ X ) ) ).

% mult_inc
thf(fact_8518_sum__power__add,axiom,
    ! [X: complex,M: nat,I6: set_nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [I: nat] : ( power_power_complex @ X @ ( plus_plus_nat @ M @ I ) )
        @ I6 )
      = ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ I6 ) ) ) ).

% sum_power_add
thf(fact_8519_sum__power__add,axiom,
    ! [X: rat,M: nat,I6: set_nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I: nat] : ( power_power_rat @ X @ ( plus_plus_nat @ M @ I ) )
        @ I6 )
      = ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ I6 ) ) ) ).

% sum_power_add
thf(fact_8520_sum__power__add,axiom,
    ! [X: int,M: nat,I6: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M @ I ) )
        @ I6 )
      = ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I6 ) ) ) ).

% sum_power_add
thf(fact_8521_sum__power__add,axiom,
    ! [X: real,M: nat,I6: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I: nat] : ( power_power_real @ X @ ( plus_plus_nat @ M @ I ) )
        @ I6 )
      = ( times_times_real @ ( power_power_real @ X @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ I6 ) ) ) ).

% sum_power_add
thf(fact_8522_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N: nat,M: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I ) )
        @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_8523_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > real,N: nat,M: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [I: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I ) )
        @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_8524_sum__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( groups7754918857620584856omplex
          @ ^ [X6: complex] : X6
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( power_power_complex @ Z3 @ N )
                = C ) ) )
        = zero_zero_complex ) ) ).

% sum_nth_roots
thf(fact_8525_sum__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( groups7754918857620584856omplex
          @ ^ [X6: complex] : X6
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( power_power_complex @ Z3 @ N )
                = one_one_complex ) ) )
        = zero_zero_complex ) ) ).

% sum_roots_unity
thf(fact_8526_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_8527_log__base__change,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ B @ X )
          = ( divide_divide_real @ ( log @ A @ X ) @ ( log @ A @ B ) ) ) ) ) ).

% log_base_change
thf(fact_8528_sum__diff1__nat,axiom,
    ! [A: complex,A2: set_complex,F: complex > nat] :
      ( ( ( member_complex @ A @ A2 )
       => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
          = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_complex @ A @ A2 )
       => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
          = ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8529_sum__diff1__nat,axiom,
    ! [A: set_nat,A2: set_set_nat,F: set_nat > nat] :
      ( ( ( member_set_nat @ A @ A2 )
       => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
          = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_set_nat @ A @ A2 )
       => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
          = ( groups8294997508430121362at_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8530_sum__diff1__nat,axiom,
    ! [A: real,A2: set_real,F: real > nat] :
      ( ( ( member_real @ A @ A2 )
       => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
          = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_real @ A @ A2 )
       => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
          = ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8531_sum__diff1__nat,axiom,
    ! [A: $o,A2: set_o,F: $o > nat] :
      ( ( ( member_o @ A @ A2 )
       => ( ( groups8507830703676809646_o_nat @ F @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
          = ( minus_minus_nat @ ( groups8507830703676809646_o_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_o @ A @ A2 )
       => ( ( groups8507830703676809646_o_nat @ F @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
          = ( groups8507830703676809646_o_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8532_sum__diff1__nat,axiom,
    ! [A: int,A2: set_int,F: int > nat] :
      ( ( ( member_int @ A @ A2 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_int @ A @ A2 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
          = ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8533_sum__diff1__nat,axiom,
    ! [A: nat,A2: set_nat,F: nat > nat] :
      ( ( ( member_nat @ A @ A2 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_nat @ A @ A2 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
          = ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8534_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_8535_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_8536_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_8537_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_8538_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_8539_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_8540_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_8541_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_8542_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_8543_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_8544_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_8545_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_8546_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_8547_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( plus_plus_rat @ ( G @ ( suc @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_8548_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( plus_plus_int @ ( G @ ( suc @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_8549_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( plus_plus_nat @ ( G @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_8550_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( plus_plus_real @ ( G @ ( suc @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_8551_numeral__inc,axiom,
    ! [X: num] :
      ( ( numera1916890842035813515d_enat @ ( inc @ X ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat ) ) ).

% numeral_inc
thf(fact_8552_numeral__inc,axiom,
    ! [X: num] :
      ( ( numera6690914467698888265omplex @ ( inc @ X ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).

% numeral_inc
thf(fact_8553_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_real @ ( inc @ X ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).

% numeral_inc
thf(fact_8554_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_nat @ ( inc @ X ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).

% numeral_inc
thf(fact_8555_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_int @ ( inc @ X ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).

% numeral_inc
thf(fact_8556_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_rat @ ( inc @ X ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).

% numeral_inc
thf(fact_8557_or__not__num__neg_Osimps_I8_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).

% or_not_num_neg.simps(8)
thf(fact_8558_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( plus_plus_rat @ ( G @ M )
          @ ( groups2906978787729119204at_rat
            @ ^ [I: nat] : ( G @ ( suc @ I ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_8559_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( plus_plus_int @ ( G @ M )
          @ ( groups3539618377306564664at_int
            @ ^ [I: nat] : ( G @ ( suc @ I ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_8560_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( plus_plus_nat @ ( G @ M )
          @ ( groups3542108847815614940at_nat
            @ ^ [I: nat] : ( G @ ( suc @ I ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_8561_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( plus_plus_real @ ( G @ M )
          @ ( groups6591440286371151544t_real
            @ ^ [I: nat] : ( G @ ( suc @ I ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_8562_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I: nat] : ( minus_minus_rat @ ( F @ ( suc @ I ) ) @ ( F @ I ) )
          @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_8563_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups3539618377306564664at_int
          @ ^ [I: nat] : ( minus_minus_int @ ( F @ ( suc @ I ) ) @ ( F @ I ) )
          @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_8564_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups6591440286371151544t_real
          @ ^ [I: nat] : ( minus_minus_real @ ( F @ ( suc @ I ) ) @ ( F @ I ) )
          @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_8565_sum__Un__nat,axiom,
    ! [A2: set_int,B2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( sup_sup_set_int @ A2 @ B2 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B2 ) ) @ ( groups4541462559716669496nt_nat @ F @ ( inf_inf_set_int @ A2 @ B2 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_8566_sum__Un__nat,axiom,
    ! [A2: set_complex,B2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups5693394587270226106ex_nat @ F @ ( sup_sup_set_complex @ A2 @ B2 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) @ ( groups5693394587270226106ex_nat @ F @ ( inf_inf_set_complex @ A2 @ B2 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_8567_sum__Un__nat,axiom,
    ! [A2: set_nat,B2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups3542108847815614940at_nat @ F @ ( sup_sup_set_nat @ A2 @ B2 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B2 ) ) @ ( groups3542108847815614940at_nat @ F @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_8568_log__mult,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ zero_zero_real @ Y )
           => ( ( log @ A @ ( times_times_real @ X @ Y ) )
              = ( plus_plus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).

% log_mult
thf(fact_8569_log__divide,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ zero_zero_real @ Y )
           => ( ( log @ A @ ( divide_divide_real @ X @ Y ) )
              = ( minus_minus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).

% log_divide
thf(fact_8570_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: nat > rat,P2: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P2 ) ) )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_8571_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: nat > int,P2: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P2 ) ) )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_8572_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: nat > nat,P2: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P2 ) ) )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_8573_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: nat > real,P2: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P2 ) ) )
        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_8574_set__encode__def,axiom,
    ( nat_set_encode
    = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% set_encode_def
thf(fact_8575_or__not__num__neg_Oelims,axiom,
    ! [X: num,Xa2: num,Y: num] :
      ( ( ( bit_or_not_num_neg @ X @ Xa2 )
        = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y != one ) ) )
       => ( ( ( X = one )
           => ! [M3: num] :
                ( ( Xa2
                  = ( bit0 @ M3 ) )
               => ( Y
                 != ( bit1 @ M3 ) ) ) )
         => ( ( ( X = one )
             => ! [M3: num] :
                  ( ( Xa2
                    = ( bit1 @ M3 ) )
                 => ( Y
                   != ( bit1 @ M3 ) ) ) )
           => ( ( ? [N2: num] :
                    ( X
                    = ( bit0 @ N2 ) )
               => ( ( Xa2 = one )
                 => ( Y
                   != ( bit0 @ one ) ) ) )
             => ( ! [N2: num] :
                    ( ( X
                      = ( bit0 @ N2 ) )
                   => ! [M3: num] :
                        ( ( Xa2
                          = ( bit0 @ M3 ) )
                       => ( Y
                         != ( bitM @ ( bit_or_not_num_neg @ N2 @ M3 ) ) ) ) )
               => ( ! [N2: num] :
                      ( ( X
                        = ( bit0 @ N2 ) )
                     => ! [M3: num] :
                          ( ( Xa2
                            = ( bit1 @ M3 ) )
                         => ( Y
                           != ( bit0 @ ( bit_or_not_num_neg @ N2 @ M3 ) ) ) ) )
                 => ( ( ? [N2: num] :
                          ( X
                          = ( bit1 @ N2 ) )
                     => ( ( Xa2 = one )
                       => ( Y != one ) ) )
                   => ( ! [N2: num] :
                          ( ( X
                            = ( bit1 @ N2 ) )
                         => ! [M3: num] :
                              ( ( Xa2
                                = ( bit0 @ M3 ) )
                             => ( Y
                               != ( bitM @ ( bit_or_not_num_neg @ N2 @ M3 ) ) ) ) )
                     => ~ ! [N2: num] :
                            ( ( X
                              = ( bit1 @ N2 ) )
                           => ! [M3: num] :
                                ( ( Xa2
                                  = ( bit1 @ M3 ) )
                               => ( Y
                                 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
thf(fact_8576_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,X: 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 @ X )
             => ( ( log @ A @ X )
                = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X ) ) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
thf(fact_8577_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: nat > complex] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( groups2073611262835488442omplex
            @ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( minus_minus_complex @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( groups2073611262835488442omplex
            @ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_complex ) ) ) ).

% sum_natinterval_diff
thf(fact_8578_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: nat > rat] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
thf(fact_8579_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: nat > int] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( groups3539618377306564664at_int
            @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( groups3539618377306564664at_int
            @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_int ) ) ) ).

% sum_natinterval_diff
thf(fact_8580_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: nat > real] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( groups6591440286371151544t_real
            @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( groups6591440286371151544t_real
            @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_real ) ) ) ).

% sum_natinterval_diff
thf(fact_8581_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups2906978787729119204at_rat
          @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
        = ( minus_minus_rat @ ( F @ N ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_8582_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3539618377306564664at_int
          @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
        = ( minus_minus_int @ ( F @ N ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_8583_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups6591440286371151544t_real
          @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
        = ( minus_minus_real @ ( F @ N ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_8584_mask__eq__sum__exp,axiom,
    ! [N: nat] :
      ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int )
      = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_8585_mask__eq__sum__exp,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_8586_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: complex] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
        = ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_8587_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
        = ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_8588_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
        = ( minus_minus_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_8589_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
        = ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_8590_sum_Oin__pairs,axiom,
    ! [G: nat > rat,M: nat,N: 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 ) ) @ N ) ) ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.in_pairs
thf(fact_8591_sum_Oin__pairs,axiom,
    ! [G: nat > int,M: nat,N: 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 ) ) @ N ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.in_pairs
thf(fact_8592_sum_Oin__pairs,axiom,
    ! [G: nat > nat,M: nat,N: 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 ) ) @ N ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.in_pairs
thf(fact_8593_sum_Oin__pairs,axiom,
    ! [G: nat > real,M: nat,N: 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 ) ) @ N ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.in_pairs
thf(fact_8594_Suc__0__or__eq,axiom,
    ! [N: nat] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% Suc_0_or_eq
thf(fact_8595_or__Suc__0__eq,axiom,
    ! [N: nat] :
      ( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% or_Suc_0_eq
thf(fact_8596_or__nat__rec,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M4: nat,N3: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 )
              | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_8597_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N ) ) ) ) ).

% mask_eq_sum_exp_nat
thf(fact_8598_gauss__sum__nat,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X6: nat] : X6
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_nat
thf(fact_8599_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M4: nat,N3: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M4 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_8600_arith__series__nat,axiom,
    ! [A: nat,D: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series_nat
thf(fact_8601_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X6: nat] : X6
        @ ( set_or1269000886237332187st_nat @ M @ N ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Icc_nat
thf(fact_8602_log__base__10__eq2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
        = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X ) ) ) ) ).

% log_base_10_eq2
thf(fact_8603_sum__gp,axiom,
    ! [N: nat,M: nat,X: rat] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( ( X = one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
          & ( ( X != one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_8604_sum__gp,axiom,
    ! [N: nat,M: nat,X: complex] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( ( X = one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
          & ( ( X != one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_8605_sum__gp,axiom,
    ! [N: nat,M: nat,X: real] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( ( X = one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
          & ( ( X != one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_8606_signed__take__bit__eq__take__bit__minus,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N3: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N3 ) ) ) ) ) ) ).

% signed_take_bit_eq_take_bit_minus
thf(fact_8607_log2__of__power__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log2_of_power_le
thf(fact_8608_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_8609_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_8610_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_8611_arctan__half,axiom,
    ( arctan
    = ( ^ [X6: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X6 @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% arctan_half
thf(fact_8612_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( semiri5074537144036343181t_real @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_8613_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_8614_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_8615_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri8010041392384452111omplex @ M )
        = ( semiri8010041392384452111omplex @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_8616_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri4939895301339042750nteger @ M )
        = ( semiri4939895301339042750nteger @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_8617_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_8618_bit__0__eq,axiom,
    ( ( bit_se1146084159140164899it_int @ zero_zero_int )
    = bot_bot_nat_o ) ).

% bit_0_eq
thf(fact_8619_bit__0__eq,axiom,
    ( ( bit_se1148574629649215175it_nat @ zero_zero_nat )
    = bot_bot_nat_o ) ).

% bit_0_eq
thf(fact_8620_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
      = ( semiri681578069525770553at_rat @ N ) ) ).

% abs_of_nat
thf(fact_8621_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( semiri5074537144036343181t_real @ N ) ) ).

% abs_of_nat
thf(fact_8622_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% abs_of_nat
thf(fact_8623_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
      = ( semiri4939895301339042750nteger @ N ) ) ).

% abs_of_nat
thf(fact_8624_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
        = ( semiri1314217659103216013at_int @ M ) )
      = ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
thf(fact_8625_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [X: real] :
      ( ( ( sqrt @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
thf(fact_8626_real__sqrt__zero,axiom,
    ( ( sqrt @ zero_zero_real )
    = zero_zero_real ) ).

% real_sqrt_zero
thf(fact_8627_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_8628_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_8629_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_8630_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_8631_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri8010041392384452111omplex @ M )
        = zero_zero_complex )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_8632_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri4939895301339042750nteger @ M )
        = zero_z3403309356797280102nteger )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_8633_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_rat
        = ( semiri681578069525770553at_rat @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_8634_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_real
        = ( semiri5074537144036343181t_real @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_8635_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_int
        = ( semiri1314217659103216013at_int @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_8636_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_nat
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_8637_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_complex
        = ( semiri8010041392384452111omplex @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_8638_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_z3403309356797280102nteger
        = ( semiri4939895301339042750nteger @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_8639_of__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% of_nat_0
thf(fact_8640_of__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% of_nat_0
thf(fact_8641_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_8642_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_8643_of__nat__0,axiom,
    ( ( semiri8010041392384452111omplex @ zero_zero_nat )
    = zero_zero_complex ) ).

% of_nat_0
thf(fact_8644_of__nat__0,axiom,
    ( ( semiri4939895301339042750nteger @ zero_zero_nat )
    = zero_z3403309356797280102nteger ) ).

% of_nat_0
thf(fact_8645_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_8646_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le72135733267957522d_enat @ ( semiri4216267220026989637d_enat @ M ) @ ( semiri4216267220026989637d_enat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_8647_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_8648_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_8649_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_8650_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_8651_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri4216267220026989637d_enat @ ( numeral_numeral_nat @ N ) )
      = ( numera1916890842035813515d_enat @ N ) ) ).

% of_nat_numeral
thf(fact_8652_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_rat @ N ) ) ).

% of_nat_numeral
thf(fact_8653_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_real @ N ) ) ).

% of_nat_numeral
thf(fact_8654_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% of_nat_numeral
thf(fact_8655_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ N ) ) ).

% of_nat_numeral
thf(fact_8656_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
      = ( numera6690914467698888265omplex @ N ) ) ).

% of_nat_numeral
thf(fact_8657_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri4939895301339042750nteger @ ( numeral_numeral_nat @ N ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% of_nat_numeral
thf(fact_8658_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_8659_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_8660_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_8661_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_8662_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_8663_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% of_nat_add
thf(fact_8664_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% of_nat_add
thf(fact_8665_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_add
thf(fact_8666_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_add
thf(fact_8667_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).

% of_nat_add
thf(fact_8668_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( plus_plus_nat @ M @ N ) )
      = ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% of_nat_add
thf(fact_8669_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% of_nat_mult
thf(fact_8670_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% of_nat_mult
thf(fact_8671_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_mult
thf(fact_8672_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_mult
thf(fact_8673_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).

% of_nat_mult
thf(fact_8674_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( times_times_nat @ M @ N ) )
      = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% of_nat_mult
thf(fact_8675_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri681578069525770553at_rat @ N )
        = one_one_rat )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_8676_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri5074537144036343181t_real @ N )
        = one_one_real )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_8677_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri1314217659103216013at_int @ N )
        = one_one_int )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_8678_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ N )
        = one_one_nat )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_8679_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri8010041392384452111omplex @ N )
        = one_one_complex )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_8680_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri4939895301339042750nteger @ N )
        = one_one_Code_integer )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_8681_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_rat
        = ( semiri681578069525770553at_rat @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_8682_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_real
        = ( semiri5074537144036343181t_real @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_8683_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_int
        = ( semiri1314217659103216013at_int @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_8684_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_nat
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_8685_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_complex
        = ( semiri8010041392384452111omplex @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_8686_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_Code_integer
        = ( semiri4939895301339042750nteger @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_8687_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_8688_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_8689_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_8690_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_8691_of__nat__1,axiom,
    ( ( semiri8010041392384452111omplex @ one_one_nat )
    = one_one_complex ) ).

% of_nat_1
thf(fact_8692_of__nat__1,axiom,
    ( ( semiri4939895301339042750nteger @ one_one_nat )
    = one_one_Code_integer ) ).

% of_nat_1
thf(fact_8693_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri5074537144036343181t_real @ X )
        = ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_8694_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri1314217659103216013at_int @ X )
        = ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_8695_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri1316708129612266289at_nat @ X )
        = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_8696_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri8010041392384452111omplex @ X )
        = ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_8697_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri4939895301339042750nteger @ X )
        = ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_8698_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
        = ( semiri5074537144036343181t_real @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_8699_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
        = ( semiri1314217659103216013at_int @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_8700_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
        = ( semiri1316708129612266289at_nat @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_8701_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
        = ( semiri8010041392384452111omplex @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_8702_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W )
        = ( semiri4939895301339042750nteger @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_8703_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
      = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).

% of_nat_power
thf(fact_8704_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
      = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).

% of_nat_power
thf(fact_8705_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
      = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).

% of_nat_power
thf(fact_8706_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
      = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).

% of_nat_power
thf(fact_8707_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( power_power_nat @ M @ N ) )
      = ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ M ) @ N ) ) ).

% of_nat_power
thf(fact_8708_negative__zless,axiom,
    ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zless
thf(fact_8709_real__sqrt__lt__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( sqrt @ X ) @ zero_zero_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% real_sqrt_lt_0_iff
thf(fact_8710_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_8711_real__sqrt__le__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% real_sqrt_le_0_iff
thf(fact_8712_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_8713_real__sqrt__abs2,axiom,
    ! [X: real] :
      ( ( sqrt @ ( times_times_real @ X @ X ) )
      = ( abs_abs_real @ X ) ) ).

% real_sqrt_abs2
thf(fact_8714_real__sqrt__mult__self,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
      = ( abs_abs_real @ A ) ) ).

% real_sqrt_mult_self
thf(fact_8715_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% of_nat_of_bool
thf(fact_8716_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri8010041392384452111omplex @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n1201886186963655149omplex @ P ) ) ).

% of_nat_of_bool
thf(fact_8717_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2687167440665602831ol_nat @ P ) ) ).

% of_nat_of_bool
thf(fact_8718_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% of_nat_of_bool
thf(fact_8719_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% of_nat_of_bool
thf(fact_8720_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_8721_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ M ) @ zero_z3403309356797280102nteger )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_8722_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_8723_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_8724_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_8725_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).

% of_nat_Suc
thf(fact_8726_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).

% of_nat_Suc
thf(fact_8727_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% of_nat_Suc
thf(fact_8728_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).

% of_nat_Suc
thf(fact_8729_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
      = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).

% of_nat_Suc
thf(fact_8730_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri4939895301339042750nteger @ ( suc @ M ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( semiri4939895301339042750nteger @ M ) ) ) ).

% of_nat_Suc
thf(fact_8731_bit__numeral__Bit0__Suc__iff,axiom,
    ! [M: num,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( suc @ N ) )
      = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).

% bit_numeral_Bit0_Suc_iff
thf(fact_8732_bit__numeral__Bit0__Suc__iff,axiom,
    ! [M: num,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( suc @ N ) )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).

% bit_numeral_Bit0_Suc_iff
thf(fact_8733_bit__numeral__Bit1__Suc__iff,axiom,
    ! [M: num,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( suc @ N ) )
      = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).

% bit_numeral_Bit1_Suc_iff
thf(fact_8734_bit__numeral__Bit1__Suc__iff,axiom,
    ! [M: num,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( suc @ N ) )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).

% bit_numeral_Bit1_Suc_iff
thf(fact_8735_real__sqrt__four,axiom,
    ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% real_sqrt_four
thf(fact_8736_signed__take__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% signed_take_bit_nonnegative_iff
thf(fact_8737_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
      = ( bit_se1146084159140164899it_int @ K @ N ) ) ).

% signed_take_bit_negative_iff
thf(fact_8738_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_8739_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( semiri4216267220026989637d_enat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_8740_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_8741_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_8742_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_8743_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( semiri4939895301339042750nteger @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_8744_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_8745_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_8746_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_8747_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_8748_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_8749_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_8750_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_8751_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_8752_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_8753_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_8754_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( ( semiri4216267220026989637d_enat @ Y )
        = ( power_8040749407984259932d_enat @ ( numera1916890842035813515d_enat @ X ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_8755_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( ( semiri681578069525770553at_rat @ Y )
        = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_8756_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ Y )
        = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_8757_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ Y )
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_8758_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ Y )
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_8759_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( ( semiri8010041392384452111omplex @ Y )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_8760_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( ( semiri4939895301339042750nteger @ Y )
        = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_8761_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( ( power_8040749407984259932d_enat @ ( numera1916890842035813515d_enat @ X ) @ N )
        = ( semiri4216267220026989637d_enat @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_8762_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
        = ( semiri681578069525770553at_rat @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_8763_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
        = ( semiri5074537144036343181t_real @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_8764_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = ( semiri1314217659103216013at_int @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_8765_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = ( semiri1316708129612266289at_nat @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_8766_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
        = ( semiri8010041392384452111omplex @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_8767_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N )
        = ( semiri4939895301339042750nteger @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_8768_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_8769_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_8770_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_8771_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_8772_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_8773_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_8774_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B ) @ W ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_8775_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_8776_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_8777_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_8778_real__of__nat__less__numeral__iff,axiom,
    ! [N: nat,W: num] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
      = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).

% real_of_nat_less_numeral_iff
thf(fact_8779_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,N: nat] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).

% numeral_less_real_of_nat_iff
thf(fact_8780_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M: nat] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).

% numeral_le_real_of_nat_iff
thf(fact_8781_bit__numeral__simps_I2_J,axiom,
    ! [W: num,N: num] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
      = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).

% bit_numeral_simps(2)
thf(fact_8782_bit__numeral__simps_I2_J,axiom,
    ! [W: num,N: num] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).

% bit_numeral_simps(2)
thf(fact_8783_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N ) ) ).

% bit_minus_numeral_Bit0_Suc_iff
thf(fact_8784_bit__numeral__simps_I3_J,axiom,
    ! [W: num,N: num] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
      = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).

% bit_numeral_simps(3)
thf(fact_8785_bit__numeral__simps_I3_J,axiom,
    ! [W: num,N: num] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).

% bit_numeral_simps(3)
thf(fact_8786_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N ) ) ) ).

% bit_minus_numeral_Bit1_Suc_iff
thf(fact_8787_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_8788_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_8789_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_8790_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_8791_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_8792_bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se9216721137139052372nteger @ A @ zero_zero_nat )
      = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_0
thf(fact_8793_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_8794_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_8795_real__sqrt__abs,axiom,
    ! [X: real] :
      ( ( sqrt @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( abs_abs_real @ X ) ) ).

% real_sqrt_abs
thf(fact_8796_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_8797_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,N: num] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N ) ) ) ).

% bit_minus_numeral_int(1)
thf(fact_8798_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,N: num] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ) ).

% bit_minus_numeral_int(2)
thf(fact_8799_real__sqrt__pow2__iff,axiom,
    ! [X: real] :
      ( ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% real_sqrt_pow2_iff
thf(fact_8800_real__sqrt__pow2,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X ) ) ).

% real_sqrt_pow2
thf(fact_8801_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] :
      ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_squared_eq
thf(fact_8802_real__sqrt__mult,axiom,
    ! [X: real,Y: real] :
      ( ( sqrt @ ( times_times_real @ X @ Y ) )
      = ( times_times_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_mult
thf(fact_8803_bit__and__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ N )
      = ( ( bit_se1146084159140164899it_int @ K @ N )
        & ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).

% bit_and_int_iff
thf(fact_8804_bit__or__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ N )
      = ( ( bit_se1146084159140164899it_int @ K @ N )
        | ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).

% bit_or_int_iff
thf(fact_8805_real__sqrt__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_gt_zero
thf(fact_8806_real__sqrt__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_ge_zero
thf(fact_8807_real__sqrt__eq__zero__cancel,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( sqrt @ X )
          = zero_zero_real )
       => ( X = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
thf(fact_8808_int__ops_I1_J,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% int_ops(1)
thf(fact_8809_int__ops_I3_J,axiom,
    ! [N: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% int_ops(3)
thf(fact_8810_int__cases,axiom,
    ! [Z2: int] :
      ( ! [N2: nat] :
          ( Z2
         != ( semiri1314217659103216013at_int @ N2 ) )
     => ~ ! [N2: nat] :
            ( Z2
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).

% int_cases
thf(fact_8811_int__of__nat__induct,axiom,
    ! [P: int > $o,Z2: int] :
      ( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
     => ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
       => ( P @ Z2 ) ) ) ).

% int_of_nat_induct
thf(fact_8812_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_8813_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_8814_nonneg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ~ ! [N2: nat] :
            ( K
           != ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% nonneg_int_cases
thf(fact_8815_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ? [N2: nat] :
          ( K
          = ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% zero_le_imp_eq_int
thf(fact_8816_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z2: int] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).

% zadd_int_left
thf(fact_8817_int__plus,axiom,
    ! [N: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% int_plus
thf(fact_8818_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_8819_int__ops_I7_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(7)
thf(fact_8820_int__ops_I2_J,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% int_ops(2)
thf(fact_8821_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_8822_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_8823_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).

% nat_less_as_int
thf(fact_8824_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).

% nat_leq_as_int
thf(fact_8825_real__div__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( divide_divide_real @ X @ ( sqrt @ X ) )
        = ( sqrt @ X ) ) ) ).

% real_div_sqrt
thf(fact_8826_sqrt__add__le__add__sqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ) ) ).

% sqrt_add_le_add_sqrt
thf(fact_8827_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ) ).

% le_real_sqrt_sumsq
thf(fact_8828_bit__not__int__iff_H,axiom,
    ! [K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% bit_not_int_iff'
thf(fact_8829_reals__Archimedean3,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ! [Y5: real] :
        ? [N2: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) ) ).

% reals_Archimedean3
thf(fact_8830_int__cases4,axiom,
    ! [M: int] :
      ( ! [N2: nat] :
          ( M
         != ( semiri1314217659103216013at_int @ N2 ) )
     => ~ ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( M
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% int_cases4
thf(fact_8831_real__of__nat__div4,axiom,
    ! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).

% real_of_nat_div4
thf(fact_8832_int__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).

% int_Suc
thf(fact_8833_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_8834_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W3: int,Z3: int] :
        ? [N3: nat] :
          ( Z3
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_8835_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
      = ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
thf(fact_8836_real__of__nat__div,axiom,
    ! [D: nat,N: nat] :
      ( ( dvd_dvd_nat @ D @ N )
     => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
        = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).

% real_of_nat_div
thf(fact_8837_nonpos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ~ ! [N2: nat] :
            ( K
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% nonpos_int_cases
thf(fact_8838_negative__zle__0,axiom,
    ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).

% negative_zle_0
thf(fact_8839_sqrt2__less__2,axiom,
    ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% sqrt2_less_2
thf(fact_8840_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M: nat,K: int] :
      ( ( ord_less_nat @ N @ M )
     => ( ( bit_se1146084159140164899it_int @ K @ N )
       => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).

% bit_imp_take_bit_positive
thf(fact_8841_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N )
      = ( ( ( ord_less_nat @ N @ M )
          & ( bit_se1146084159140164899it_int @ K @ N ) )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_8842_pos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ~ ! [N2: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N2 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% pos_int_cases
thf(fact_8843_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ? [N2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N2 )
          & ( K
            = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_8844_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N2: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N2 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
       => ~ ! [N2: nat] :
              ( ( K
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% int_cases3
thf(fact_8845_nat__less__real__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat,M4: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M4 ) ) ) ) ).

% nat_less_real_le
thf(fact_8846_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N3: nat,M4: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M4 ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_8847_zmult__zless__mono2__lemma,axiom,
    ! [I3: int,J: int,K: nat] :
      ( ( ord_less_int @ I3 @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I3 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).

% zmult_zless_mono2_lemma
thf(fact_8848_not__zle__0__negative,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).

% not_zle_0_negative
thf(fact_8849_negD,axiom,
    ! [X: int] :
      ( ( ord_less_int @ X @ zero_zero_int )
     => ? [N2: nat] :
          ( X
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).

% negD
thf(fact_8850_negative__zless__0,axiom,
    ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_8851_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_8852_real__of__nat__div__aux,axiom,
    ! [X: nat,D: nat] :
      ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ D ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).

% real_of_nat_div_aux
thf(fact_8853_signed__take__bit__eq__concat__bit,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N3: nat,K3: int] : ( bit_concat_bit @ N3 @ K3 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N3 ) ) ) ) ) ) ).

% signed_take_bit_eq_concat_bit
thf(fact_8854_real__less__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
     => ( ord_less_real @ X @ ( sqrt @ Y ) ) ) ).

% real_less_rsqrt
thf(fact_8855_real__le__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
     => ( ord_less_eq_real @ X @ ( sqrt @ Y ) ) ) ).

% real_le_rsqrt
thf(fact_8856_sqrt__le__D,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ Y )
     => ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sqrt_le_D
thf(fact_8857_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ N2 @ M2 )
             => ( ( bit_se1146084159140164899it_int @ K @ M2 )
                = ( bit_se1146084159140164899it_int @ K @ N2 ) ) )
         => ~ ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ one_one_nat ) )
                = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ) ) ).

% int_bit_bound
thf(fact_8858_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ! [M3: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ M3 )
             => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ X ) @ C ) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
thf(fact_8859_neg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ~ ! [N2: nat] :
            ( ( K
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% neg_int_cases
thf(fact_8860_zdiff__int__split,axiom,
    ! [P: int > $o,X: nat,Y: nat] :
      ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
      = ( ( ( ord_less_eq_nat @ Y @ X )
         => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
        & ( ( ord_less_nat @ X @ Y )
         => ( P @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_8861_real__of__nat__div2,axiom,
    ! [N: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) ) ).

% real_of_nat_div2
thf(fact_8862_real__of__nat__div3,axiom,
    ! [N: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) @ one_one_real ) ).

% real_of_nat_div3
thf(fact_8863_log__base__pow,axiom,
    ! [A: real,N: nat,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( log @ ( power_power_real @ A @ N ) @ X )
        = ( divide_divide_real @ ( log @ A @ X ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log_base_pow
thf(fact_8864_ln__realpow,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ln_ln_real @ ( power_power_real @ X @ N ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X ) ) ) ) ).

% ln_realpow
thf(fact_8865_log__nat__power,axiom,
    ! [X: real,B: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( log @ B @ ( power_power_real @ X @ N ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ).

% log_nat_power
thf(fact_8866_real__sqrt__unique,axiom,
    ! [Y: real,X: real] :
      ( ( ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( sqrt @ X )
          = Y ) ) ) ).

% real_sqrt_unique
thf(fact_8867_real__le__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sqrt @ X ) @ Y ) ) ) ) ).

% real_le_lsqrt
thf(fact_8868_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_8869_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X: real,Y: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = Y )
     => ( X = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
thf(fact_8870_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = X )
     => ( Y = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
thf(fact_8871_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_8872_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge2
thf(fact_8873_real__sqrt__sum__squares__ge1,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge1
thf(fact_8874_sqrt__ge__absD,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ Y ) )
     => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) ).

% sqrt_ge_absD
thf(fact_8875_bit__int__def,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [K3: int,N3: nat] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% bit_int_def
thf(fact_8876_log2__of__power__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( M
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( semiri5074537144036343181t_real @ N )
        = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% log2_of_power_eq
thf(fact_8877_linear__plus__1__le__power,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N ) ) ) ).

% linear_plus_1_le_power
thf(fact_8878_log__of__power__less,axiom,
    ! [M: nat,B: real,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_of_power_less
thf(fact_8879_Bernoulli__inequality,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).

% Bernoulli_inequality
thf(fact_8880_real__less__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sqrt @ X ) @ Y ) ) ) ) ).

% real_less_lsqrt
thf(fact_8881_sqrt__sum__squares__le__sum,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X @ Y ) ) ) ) ).

% sqrt_sum_squares_le_sum
thf(fact_8882_sqrt__even__pow2,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% sqrt_even_pow2
thf(fact_8883_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) ) ) ).

% sqrt_sum_squares_le_sum_abs
thf(fact_8884_real__sqrt__ge__abs2,axiom,
    ! [Y: real,X: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs2
thf(fact_8885_real__sqrt__ge__abs1,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs1
thf(fact_8886_ln__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ln_ln_real @ ( sqrt @ X ) )
        = ( divide_divide_real @ ( ln_ln_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% ln_sqrt
thf(fact_8887_arsinh__real__def,axiom,
    ( arsinh_real
    = ( ^ [X6: real] : ( ln_ln_real @ ( plus_plus_real @ X6 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arsinh_real_def
thf(fact_8888_log__of__power__le,axiom,
    ! [M: nat,B: real,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
     => ( ( 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 @ N ) ) ) ) ) ).

% log_of_power_le
thf(fact_8889_arsinh__real__aux,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% arsinh_real_aux
thf(fact_8890_real__sqrt__power__even,axiom,
    ! [N: nat,X: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( sqrt @ X ) @ N )
          = ( power_power_real @ X @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_power_even
thf(fact_8891_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_ge_zero
thf(fact_8892_arith__geo__mean__sqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X @ Y ) ) @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arith_geo_mean_sqrt
thf(fact_8893_less__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% less_log2_of_power
thf(fact_8894_le__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% le_log2_of_power
thf(fact_8895_set__bit__eq,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N3: nat,K3: int] :
          ( plus_plus_int @ K3
          @ ( times_times_int
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( bit_se1146084159140164899it_int @ K3 @ N3 ) )
            @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% set_bit_eq
thf(fact_8896_unset__bit__eq,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N3: nat,K3: int] : ( minus_minus_int @ K3 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N3 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% unset_bit_eq
thf(fact_8897_cos__x__y__le__one,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).

% cos_x_y_le_one
thf(fact_8898_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( 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 @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).

% real_sqrt_sum_squares_less
thf(fact_8899_arcosh__real__def,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( arcosh_real @ X )
        = ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arcosh_real_def
thf(fact_8900_take__bit__Suc__from__most,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K )
      = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N ) ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% take_bit_Suc_from_most
thf(fact_8901_log2__of__power__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log2_of_power_less
thf(fact_8902_Bernoulli__inequality__even,axiom,
    ! [N: nat,X: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).

% Bernoulli_inequality_even
thf(fact_8903_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X ) ) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
thf(fact_8904_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
thf(fact_8905_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( ( ord_less_real @ X @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_real @ Y @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ X )
         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
           => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).

% sqrt_sum_squares_half_less
thf(fact_8906_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( topolo6980174941875973593q_real
        @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% monoseq_arctan_series
thf(fact_8907_ln__series,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ( ln_ln_real @ X )
          = ( suminf_real
            @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N3 ) ) ) ) ) ) ) ).

% ln_series
thf(fact_8908_arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( arctan @ X )
        = ( suminf_real
          @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).

% arctan_series
thf(fact_8909_int__if,axiom,
    ! [P: $o,A: nat,B: nat] :
      ( ( P
       => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
          = ( semiri1314217659103216013at_int @ A ) ) )
      & ( ~ P
       => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
          = ( semiri1314217659103216013at_int @ B ) ) ) ) ).

% int_if
thf(fact_8910_nat__int__comparison_I1_J,axiom,
    ( ( ^ [Y3: nat,Z: nat] : Y3 = Z )
    = ( ^ [A3: nat,B3: nat] :
          ( ( semiri1314217659103216013at_int @ A3 )
          = ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).

% nat_int_comparison(1)
thf(fact_8911_bit__Suc__0__iff,axiom,
    ! [N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( N = zero_zero_nat ) ) ).

% bit_Suc_0_iff
thf(fact_8912_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).

% not_bit_Suc_0_Suc
thf(fact_8913_not__bit__Suc__0__numeral,axiom,
    ! [N: num] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).

% not_bit_Suc_0_numeral
thf(fact_8914_bit__nat__def,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [M4: nat,N3: nat] :
          ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% bit_nat_def
thf(fact_8915_monoseq__realpow,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( topolo6980174941875973593q_real @ ( power_power_real @ X ) ) ) ) ).

% monoseq_realpow
thf(fact_8916_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_8917_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_8918_pi__neq__zero,axiom,
    pi != zero_zero_real ).

% pi_neq_zero
thf(fact_8919_pi__gt__zero,axiom,
    ord_less_real @ zero_zero_real @ pi ).

% pi_gt_zero
thf(fact_8920_pi__not__less__zero,axiom,
    ~ ( ord_less_real @ pi @ zero_zero_real ) ).

% pi_not_less_zero
thf(fact_8921_pi__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ pi ).

% pi_ge_zero
thf(fact_8922_lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).

% lessThan_Suc
thf(fact_8923_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( ( set_ord_lessThan_nat @ N )
        = bot_bot_set_nat )
      = ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_8924_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_8925_pi__less__4,axiom,
    ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).

% pi_less_4
thf(fact_8926_pi__ge__two,axiom,
    ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).

% pi_ge_two
thf(fact_8927_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_8928_pi__half__neq__zero,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% pi_half_neq_zero
thf(fact_8929_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_8930_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_8931_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_8932_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_8933_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_8934_arctan__ubound,axiom,
    ! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arctan_ubound
thf(fact_8935_arctan__one,axiom,
    ( ( arctan @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% arctan_one
thf(fact_8936_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_8937_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_8938_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_8939_sum__split__even__odd,axiom,
    ! [F: nat > real,G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) @ ( F @ I ) @ ( G @ I ) )
        @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [I: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) )
          @ ( set_ord_lessThan_nat @ N ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) @ one_one_nat ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum_split_even_odd
thf(fact_8940_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_8941_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_8942_sin__cos__npi,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).

% sin_cos_npi
thf(fact_8943_sumr__cos__zero__one,axiom,
    ! [N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [M4: nat] : ( times_times_real @ ( cos_coeff @ M4 ) @ ( power_power_real @ zero_zero_real @ M4 ) )
        @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = one_one_real ) ).

% sumr_cos_zero_one
thf(fact_8944_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
          = ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
thf(fact_8945_summable__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( summable_real
        @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% summable_arctan_series
thf(fact_8946_sin__pi,axiom,
    ( ( sin_real @ pi )
    = zero_zero_real ) ).

% sin_pi
thf(fact_8947_cos__coeff__0,axiom,
    ( ( cos_coeff @ zero_zero_nat )
    = one_one_real ) ).

% cos_coeff_0
thf(fact_8948_sin__npi,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% sin_npi
thf(fact_8949_sin__npi2,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
      = zero_zero_real ) ).

% sin_npi2
thf(fact_8950_sin__npi__int,axiom,
    ! [N: int] :
      ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
      = zero_zero_real ) ).

% sin_npi_int
thf(fact_8951_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_8952_sin__two__pi,axiom,
    ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = zero_zero_real ) ).

% sin_two_pi
thf(fact_8953_sin__pi__half,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = one_one_real ) ).

% sin_pi_half
thf(fact_8954_sin__periodic,axiom,
    ! [X: real] :
      ( ( sin_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( sin_real @ X ) ) ).

% sin_periodic
thf(fact_8955_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_8956_sin__2npi,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
      = zero_zero_real ) ).

% sin_2npi
thf(fact_8957_sin__2pi__minus,axiom,
    ! [X: real] :
      ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
      = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).

% sin_2pi_minus
thf(fact_8958_sin__int__2pin,axiom,
    ! [N: int] :
      ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
      = zero_zero_real ) ).

% sin_int_2pin
thf(fact_8959_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_8960_sin__x__le__x,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( sin_real @ X ) @ X ) ) ).

% sin_x_le_x
thf(fact_8961_sin__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ pi )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_gt_zero
thf(fact_8962_sin__x__ge__neg__x,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ ( sin_real @ X ) ) ) ).

% sin_x_ge_neg_x
thf(fact_8963_sin__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_ge_zero
thf(fact_8964_sin__eq__0__pi,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
     => ( ( ord_less_real @ X @ pi )
       => ( ( ( sin_real @ X )
            = zero_zero_real )
         => ( X = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
thf(fact_8965_sin__zero__pi__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ pi )
     => ( ( ( sin_real @ X )
          = zero_zero_real )
        = ( X = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
thf(fact_8966_sin__zero__iff__int2,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [I: int] :
            ( X
            = ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) ) ) ).

% sin_zero_iff_int2
thf(fact_8967_sin__gt__zero__02,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_gt_zero_02
thf(fact_8968_sin__pi__divide__n__ge__0,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% sin_pi_divide_n_ge_0
thf(fact_8969_summable__power__series,axiom,
    ! [F: nat > real,Z2: real] :
      ( ! [I2: nat] : ( ord_less_eq_real @ ( F @ I2 ) @ one_one_real )
     => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ Z2 )
         => ( ( ord_less_real @ Z2 @ one_one_real )
           => ( summable_real
              @ ^ [I: nat] : ( times_times_real @ ( F @ I ) @ ( power_power_real @ Z2 @ I ) ) ) ) ) ) ) ).

% summable_power_series
thf(fact_8970_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_8971_sin__gt__zero2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_gt_zero2
thf(fact_8972_sin__lt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ pi @ X )
     => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).

% sin_lt_zero
thf(fact_8973_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_8974_sin__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ( sin_real @ X )
                = ( sin_real @ Y ) )
             => ( X = Y ) ) ) ) ) ) ).

% sin_inj_pi
thf(fact_8975_sin__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
              = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).

% sin_mono_le_eq
thf(fact_8976_sin__monotone__2pi__le,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).

% sin_monotone_2pi_le
thf(fact_8977_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_8978_sin__le__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ pi @ X )
     => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_eq_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).

% sin_le_zero
thf(fact_8979_sin__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).

% sin_less_zero
thf(fact_8980_sin__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
              = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).

% sin_mono_less_eq
thf(fact_8981_sin__monotone__2pi,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).

% sin_monotone_2pi
thf(fact_8982_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 )
            & ! [Y5: real] :
                ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
                  & ( ord_less_eq_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
                  & ( ( sin_real @ Y5 )
                    = Y ) )
               => ( Y5 = X5 ) ) ) ) ) ).

% sin_total
thf(fact_8983_sin__pi__divide__n__gt__0,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% sin_pi_divide_n_gt_0
thf(fact_8984_sin__arctan,axiom,
    ! [X: real] :
      ( ( sin_real @ ( arctan @ X ) )
      = ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_arctan
thf(fact_8985_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_8986_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [I: int] :
            ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I )
            & ( X
              = ( times_times_real @ ( ring_1_of_int_real @ I ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_iff_int
thf(fact_8987_sin__zero__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( sin_real @ X )
          = zero_zero_real )
       => ? [N2: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_lemma
thf(fact_8988_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [N3: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N3: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% sin_zero_iff
thf(fact_8989_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).

% ceiling_log2_div2
thf(fact_8990_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
     => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).

% ceiling_log_nat_eq_if
thf(fact_8991_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_8992_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ~ ! [T6: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T6 )
           => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( ( X
                  = ( cos_real @ T6 ) )
               => ( Y
                 != ( sin_real @ T6 ) ) ) ) ) ) ).

% sincos_total_2pi
thf(fact_8993_sin__tan,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( sin_real @ X )
        = ( divide_divide_real @ ( tan_real @ X ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_tan
thf(fact_8994_Maclaurin__exp__lt,axiom,
    ! [X: real,N: nat] :
      ( ( X != zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
            & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
            & ( ( exp_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M4: nat] : ( divide_divide_real @ ( power_power_real @ X @ M4 ) @ ( semiri2265585572941072030t_real @ M4 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_8995_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
          = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X )
            & ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).

% ceiling_log_eq_powr_iff
thf(fact_8996_powr__gt__zero,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X @ A ) )
      = ( X != zero_zero_real ) ) ).

% powr_gt_zero
thf(fact_8997_powr__nonneg__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_eq_real @ ( powr_real @ A @ X ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
thf(fact_8998_tan__pi,axiom,
    ( ( tan_real @ pi )
    = zero_zero_real ) ).

% tan_pi
thf(fact_8999_powr__eq__one__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( powr_real @ A @ X )
          = one_one_real )
        = ( X = zero_zero_real ) ) ) ).

% powr_eq_one_iff
thf(fact_9000_powr__one__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( ( powr_real @ X @ one_one_real )
        = X )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% powr_one_gt_zero_iff
thf(fact_9001_powr__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ one_one_real )
        = X ) ) ).

% powr_one
thf(fact_9002_numeral__powr__numeral__real,axiom,
    ! [M: num,N: num] :
      ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).

% numeral_powr_numeral_real
thf(fact_9003_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_9004_powr__log__cancel,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( powr_real @ A @ ( log @ A @ X ) )
            = X ) ) ) ) ).

% powr_log_cancel
thf(fact_9005_tan__npi,axiom,
    ! [N: nat] :
      ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% tan_npi
thf(fact_9006_tan__periodic__n,axiom,
    ! [X: real,N: num] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ N ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_n
thf(fact_9007_tan__periodic__nat,axiom,
    ! [X: real,N: nat] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_nat
thf(fact_9008_tan__periodic__int,axiom,
    ! [X: real,I3: int] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_int
thf(fact_9009_powr__numeral,axiom,
    ! [X: real,N: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( numeral_numeral_real @ N ) )
        = ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ).

% powr_numeral
thf(fact_9010_cos__pi__half,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = zero_zero_real ) ).

% cos_pi_half
thf(fact_9011_cos__two__pi,axiom,
    ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_real ) ).

% cos_two_pi
thf(fact_9012_cos__periodic,axiom,
    ! [X: real] :
      ( ( cos_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( cos_real @ X ) ) ).

% cos_periodic
thf(fact_9013_cos__2pi__minus,axiom,
    ! [X: real] :
      ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
      = ( cos_real @ X ) ) ).

% cos_2pi_minus
thf(fact_9014_tan__periodic,axiom,
    ! [X: real] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic
thf(fact_9015_cos__npi2,axiom,
    ! [N: nat] :
      ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).

% cos_npi2
thf(fact_9016_cos__npi,axiom,
    ! [N: nat] :
      ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).

% cos_npi
thf(fact_9017_cos__2npi,axiom,
    ! [N: nat] :
      ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
      = one_one_real ) ).

% cos_2npi
thf(fact_9018_cos__int__2pin,axiom,
    ! [N: int] :
      ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
      = one_one_real ) ).

% cos_int_2pin
thf(fact_9019_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_9020_square__powr__half,axiom,
    ! [X: real] :
      ( ( powr_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( abs_abs_real @ X ) ) ).

% square_powr_half
thf(fact_9021_cos__npi__int,axiom,
    ! [N: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
          = one_one_real ) )
      & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% cos_npi_int
thf(fact_9022_powr__powr,axiom,
    ! [X: real,A: real,B: real] :
      ( ( powr_real @ ( powr_real @ X @ A ) @ B )
      = ( powr_real @ X @ ( times_times_real @ A @ B ) ) ) ).

% powr_powr
thf(fact_9023_powr__non__neg,axiom,
    ! [A: real,X: real] :
      ~ ( ord_less_real @ ( powr_real @ A @ X ) @ zero_zero_real ) ).

% powr_non_neg
thf(fact_9024_powr__less__mono2__neg,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ X @ Y )
         => ( ord_less_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).

% powr_less_mono2_neg
thf(fact_9025_powr__mono2,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ Y )
         => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_mono2
thf(fact_9026_powr__ge__pzero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y ) ) ).

% powr_ge_pzero
thf(fact_9027_polar__Ex,axiom,
    ! [X: real,Y: real] :
    ? [R: real,A5: real] :
      ( ( X
        = ( times_times_real @ R @ ( cos_real @ A5 ) ) )
      & ( Y
        = ( times_times_real @ R @ ( sin_real @ A5 ) ) ) ) ).

% polar_Ex
thf(fact_9028_cos__arctan__not__zero,axiom,
    ! [X: real] :
      ( ( cos_real @ ( arctan @ X ) )
     != zero_zero_real ) ).

% cos_arctan_not_zero
thf(fact_9029_powr__mono2_H,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ Y )
         => ( ord_less_eq_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).

% powr_mono2'
thf(fact_9030_powr__less__mono2,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ X @ Y )
         => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_less_mono2
thf(fact_9031_powr__inj,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ( powr_real @ A @ X )
            = ( powr_real @ A @ Y ) )
          = ( X = Y ) ) ) ) ).

% powr_inj
thf(fact_9032_gr__one__powr,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y ) ) ) ) ).

% gr_one_powr
thf(fact_9033_ge__one__powr__ge__zero,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ) ).

% ge_one_powr_ge_zero
thf(fact_9034_powr__mono__both,axiom,
    ! [A: real,B: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ( ord_less_eq_real @ one_one_real @ X )
         => ( ( ord_less_eq_real @ X @ Y )
           => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ B ) ) ) ) ) ) ).

% powr_mono_both
thf(fact_9035_powr__le1,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ one_one_real )
         => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ one_one_real ) ) ) ) ).

% powr_le1
thf(fact_9036_powr__divide,axiom,
    ! [X: real,Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( powr_real @ ( divide_divide_real @ X @ Y ) @ A )
          = ( divide_divide_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_divide
thf(fact_9037_powr__mult,axiom,
    ! [X: real,Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( powr_real @ ( times_times_real @ X @ Y ) @ A )
          = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_mult
thf(fact_9038_divide__powr__uminus,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ A @ ( powr_real @ B @ C ) )
      = ( times_times_real @ A @ ( powr_real @ B @ ( uminus_uminus_real @ C ) ) ) ) ).

% divide_powr_uminus
thf(fact_9039_log__base__powr,axiom,
    ! [A: real,B: real,X: real] :
      ( ( A != zero_zero_real )
     => ( ( log @ ( powr_real @ A @ B ) @ X )
        = ( divide_divide_real @ ( log @ A @ X ) @ B ) ) ) ).

% log_base_powr
thf(fact_9040_ln__powr,axiom,
    ! [X: real,Y: real] :
      ( ( X != zero_zero_real )
     => ( ( ln_ln_real @ ( powr_real @ X @ Y ) )
        = ( times_times_real @ Y @ ( ln_ln_real @ X ) ) ) ) ).

% ln_powr
thf(fact_9041_log__powr,axiom,
    ! [X: real,B: real,Y: real] :
      ( ( X != zero_zero_real )
     => ( ( log @ B @ ( powr_real @ X @ Y ) )
        = ( times_times_real @ Y @ ( log @ B @ X ) ) ) ) ).

% log_powr
thf(fact_9042_cos__monotone__0__pi__le,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ pi )
         => ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).

% cos_monotone_0_pi_le
thf(fact_9043_cos__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
              = ( ord_less_eq_real @ Y @ X ) ) ) ) ) ) ).

% cos_mono_le_eq
thf(fact_9044_cos__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ( cos_real @ X )
                = ( cos_real @ Y ) )
             => ( X = Y ) ) ) ) ) ) ).

% cos_inj_pi
thf(fact_9045_cos__two__neq__zero,axiom,
    ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% cos_two_neq_zero
thf(fact_9046_powr__realpow,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N ) )
        = ( power_power_real @ X @ N ) ) ) ).

% powr_realpow
thf(fact_9047_less__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ Y @ ( log @ B @ X ) )
          = ( ord_less_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).

% less_log_iff
thf(fact_9048_log__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( log @ B @ X ) @ Y )
          = ( ord_less_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).

% log_less_iff
thf(fact_9049_less__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ X @ ( powr_real @ B @ Y ) )
          = ( ord_less_real @ ( log @ B @ X ) @ Y ) ) ) ) ).

% less_powr_iff
thf(fact_9050_powr__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( powr_real @ B @ Y ) @ X )
          = ( ord_less_real @ Y @ ( log @ B @ X ) ) ) ) ) ).

% powr_less_iff
thf(fact_9051_cos__monotone__0__pi,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ pi )
         => ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).

% cos_monotone_0_pi
thf(fact_9052_cos__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
              = ( ord_less_real @ Y @ X ) ) ) ) ) ) ).

% cos_mono_less_eq
thf(fact_9053_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ zero_zero_real )
         => ( ord_less_eq_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).

% cos_monotone_minus_pi_0'
thf(fact_9054_sin__zero__abs__cos__one,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
     => ( ( abs_abs_real @ ( cos_real @ X ) )
        = one_one_real ) ) ).

% sin_zero_abs_cos_one
thf(fact_9055_powr__neg__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( uminus_uminus_real @ one_one_real ) )
        = ( divide_divide_real @ one_one_real @ X ) ) ) ).

% powr_neg_one
thf(fact_9056_powr__mult__base,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( times_times_real @ X @ ( powr_real @ X @ Y ) )
        = ( powr_real @ X @ ( plus_plus_real @ one_one_real @ Y ) ) ) ) ).

% powr_mult_base
thf(fact_9057_cos__two__less__zero,axiom,
    ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_less_zero
thf(fact_9058_powr__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X )
          = ( ord_less_eq_real @ Y @ ( log @ B @ X ) ) ) ) ) ).

% powr_le_iff
thf(fact_9059_le__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) )
          = ( ord_less_eq_real @ ( log @ B @ X ) @ Y ) ) ) ) ).

% le_powr_iff
thf(fact_9060_log__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( log @ B @ X ) @ Y )
          = ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).

% log_le_iff
thf(fact_9061_le__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ Y @ ( log @ B @ X ) )
          = ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).

% le_log_iff
thf(fact_9062_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 )
      & ! [Y5: real] :
          ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
            & ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ Y5 )
              = zero_zero_real ) )
         => ( Y5 = X5 ) ) ) ).

% cos_is_zero
thf(fact_9063_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_9064_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ zero_zero_real )
         => ( ord_less_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).

% cos_monotone_minus_pi_0
thf(fact_9065_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 )
            & ! [Y5: real] :
                ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
                  & ( ord_less_eq_real @ Y5 @ pi )
                  & ( ( cos_real @ Y5 )
                    = Y ) )
               => ( Y5 = X5 ) ) ) ) ) ).

% cos_total
thf(fact_9066_square__fact__le__2__fact,axiom,
    ! [N: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% square_fact_le_2_fact
thf(fact_9067_cos__tan,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( cos_real @ X )
        = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_tan
thf(fact_9068_ln__powr__bound,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A ) @ A ) ) ) ) ).

% ln_powr_bound
thf(fact_9069_ln__powr__bound2,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X ) ) ) ) ).

% ln_powr_bound2
thf(fact_9070_tan__45,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = one_one_real ) ).

% tan_45
thf(fact_9071_add__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( plus_plus_real @ Y @ ( log @ B @ X ) )
            = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).

% add_log_eq_powr
thf(fact_9072_log__add__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( plus_plus_real @ ( log @ B @ X ) @ Y )
            = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ) ) ).

% log_add_eq_powr
thf(fact_9073_minus__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( minus_minus_real @ Y @ ( log @ B @ X ) )
            = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).

% minus_log_eq_powr
thf(fact_9074_tan__60,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).

% tan_60
thf(fact_9075_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_9076_sin__cos__le1,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) @ one_one_real ) ).

% sin_cos_le1
thf(fact_9077_tan__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).

% tan_gt_zero
thf(fact_9078_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_9079_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_9080_tan__mono__lt__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
              = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).

% tan_mono_lt_eq
thf(fact_9081_tan__monotone_H,axiom,
    ! [Y: real,X: 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 ) ) ) ) @ X )
         => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ Y @ X )
              = ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ) ) ).

% tan_monotone'
thf(fact_9082_tan__monotone,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ).

% tan_monotone
thf(fact_9083_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 )
      & ! [Y5: real] :
          ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
            & ( ord_less_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( tan_real @ Y5 )
              = Y ) )
         => ( Y5 = X5 ) ) ) ).

% tan_total
thf(fact_9084_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_9085_cos__double__less__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ).

% cos_double_less_one
thf(fact_9086_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_9087_cos__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).

% cos_gt_zero
thf(fact_9088_log__minus__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( minus_minus_real @ ( log @ B @ X ) @ Y )
            = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y ) ) ) ) ) ) ) ) ).

% log_minus_eq_powr
thf(fact_9089_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_9090_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_9091_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = one_one_real )
      = ( ? [X6: int] :
            ( X
            = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X6 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).

% cos_one_2pi_int
thf(fact_9092_Maclaurin__cos__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T6: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
      & ( ( cos_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M4: nat] : ( times_times_real @ ( cos_coeff @ M4 ) @ ( power_power_real @ X @ M4 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% Maclaurin_cos_expansion
thf(fact_9093_powr__half__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        = ( sqrt @ X ) ) ) ).

% powr_half_sqrt
thf(fact_9094_powr__neg__numeral,axiom,
    ! [X: real,N: num] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
        = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ) ).

% powr_neg_numeral
thf(fact_9095_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_9096_tan__pos__pi2__le,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).

% tan_pos_pi2_le
thf(fact_9097_tan__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ( ord_less_real @ ( tan_real @ X ) @ zero_zero_real ) ) ) ).

% tan_less_zero
thf(fact_9098_tan__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
              = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).

% tan_mono_le_eq
thf(fact_9099_tan__mono__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ).

% tan_mono_le
thf(fact_9100_tan__bound__pi2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
     => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X ) ) @ one_one_real ) ) ).

% tan_bound_pi2
thf(fact_9101_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_9102_cos__gt__zero__pi,axiom,
    ! [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 ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).

% cos_gt_zero_pi
thf(fact_9103_cos__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).

% cos_ge_zero
thf(fact_9104_arctan__unique,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ( tan_real @ X )
            = Y )
         => ( ( arctan @ Y )
            = X ) ) ) ) ).

% arctan_unique
thf(fact_9105_arctan__tan,axiom,
    ! [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 ) ) ) )
       => ( ( arctan @ ( tan_real @ X ) )
          = X ) ) ) ).

% arctan_tan
thf(fact_9106_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_9107_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = one_one_real )
      = ( ? [X6: nat] :
            ( X
            = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X6 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
        | ? [X6: nat] :
            ( X
            = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X6 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).

% cos_one_2pi
thf(fact_9108_Maclaurin__lemma,axiom,
    ! [H2: real,F: real > real,J: nat > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ? [B8: real] :
          ( ( F @ H2 )
          = ( plus_plus_real
            @ ( groups6591440286371151544t_real
              @ ^ [M4: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M4 ) @ ( semiri2265585572941072030t_real @ M4 ) ) @ ( power_power_real @ H2 @ M4 ) )
              @ ( set_ord_lessThan_nat @ N ) )
            @ ( times_times_real @ B8 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).

% Maclaurin_lemma
thf(fact_9109_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ T6 )
            & ( ord_less_real @ T6 @ X )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M4: nat] : ( times_times_real @ ( cos_coeff @ M4 ) @ ( power_power_real @ X @ M4 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_cos_expansion2
thf(fact_9110_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ? [T6: real] :
            ( ( ord_less_real @ X @ T6 )
            & ( ord_less_real @ T6 @ zero_zero_real )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M4: nat] : ( times_times_real @ ( cos_coeff @ M4 ) @ ( power_power_real @ X @ M4 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
thf(fact_9111_tan__total__pi4,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ? [Z4: real] :
          ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z4 )
          & ( ord_less_real @ Z4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
          & ( ( tan_real @ Z4 )
            = X ) ) ) ).

% tan_total_pi4
thf(fact_9112_cos__arctan,axiom,
    ! [X: real] :
      ( ( cos_real @ ( arctan @ X ) )
      = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_arctan
thf(fact_9113_Maclaurin__exp__le,axiom,
    ! [X: real,N: nat] :
    ? [T6: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
      & ( ( exp_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M4: nat] : ( divide_divide_real @ ( power_power_real @ X @ M4 ) @ ( semiri2265585572941072030t_real @ M4 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% Maclaurin_exp_le
thf(fact_9114_sincos__total__pi,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = one_one_real )
       => ? [T6: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T6 )
            & ( ord_less_eq_real @ T6 @ pi )
            & ( X
              = ( cos_real @ T6 ) )
            & ( Y
              = ( sin_real @ T6 ) ) ) ) ) ).

% sincos_total_pi
thf(fact_9115_sin__cos__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) )
     => ( ( sin_real @ X )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_cos_sqrt
thf(fact_9116_sin__expansion__lemma,axiom,
    ! [X: real,M: nat] :
      ( ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_expansion_lemma
thf(fact_9117_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = zero_zero_real )
      = ( ? [I: int] :
            ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I )
            & ( X
              = ( times_times_real @ ( ring_1_of_int_real @ I ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_iff_int
thf(fact_9118_cos__coeff__def,axiom,
    ( cos_coeff
    = ( ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ zero_zero_real ) ) ) ).

% cos_coeff_def
thf(fact_9119_cos__zero__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( cos_real @ X )
          = zero_zero_real )
       => ? [N2: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_lemma
thf(fact_9120_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = zero_zero_real )
      = ( ? [N3: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N3: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% cos_zero_iff
thf(fact_9121_cos__expansion__lemma,axiom,
    ! [X: real,M: nat] :
      ( ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_expansion_lemma
thf(fact_9122_sincos__total__pi__half,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
            = one_one_real )
         => ? [T6: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ T6 )
              & ( ord_less_eq_real @ T6 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( X
                = ( cos_real @ T6 ) )
              & ( Y
                = ( sin_real @ T6 ) ) ) ) ) ) ).

% sincos_total_pi_half
thf(fact_9123_sincos__total__2pi__le,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ? [T6: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ T6 )
          & ( ord_less_eq_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
          & ( X
            = ( cos_real @ T6 ) )
          & ( Y
            = ( sin_real @ T6 ) ) ) ) ).

% sincos_total_2pi_le
thf(fact_9124_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ T6 )
            & ( ord_less_real @ T6 @ X )
            & ( ( sin_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M4: nat] : ( times_times_real @ ( sin_coeff @ M4 ) @ ( power_power_real @ X @ M4 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_sin_expansion3
thf(fact_9125_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [T6: real] :
          ( ( ord_less_real @ zero_zero_real @ T6 )
          & ( ord_less_eq_real @ T6 @ X )
          & ( ( sin_real @ X )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M4: nat] : ( times_times_real @ ( sin_coeff @ M4 ) @ ( power_power_real @ X @ M4 ) )
                @ ( set_ord_lessThan_nat @ N ) )
              @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).

% Maclaurin_sin_expansion4
thf(fact_9126_Maclaurin__sin__expansion2,axiom,
    ! [X: real,N: nat] :
    ? [T6: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
      & ( ( sin_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M4: nat] : ( times_times_real @ ( sin_coeff @ M4 ) @ ( power_power_real @ X @ M4 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% Maclaurin_sin_expansion2
thf(fact_9127_Maclaurin__sin__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T6: real] :
      ( ( sin_real @ X )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [M4: nat] : ( times_times_real @ ( sin_coeff @ M4 ) @ ( power_power_real @ X @ M4 ) )
          @ ( set_ord_lessThan_nat @ N ) )
        @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ).

% Maclaurin_sin_expansion
thf(fact_9128_sin__coeff__0,axiom,
    ( ( sin_coeff @ zero_zero_nat )
    = zero_zero_real ) ).

% sin_coeff_0
thf(fact_9129_fact__less__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% fact_less_mono_nat
thf(fact_9130_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_Suc_0_nat
thf(fact_9131_fact__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ ( suc @ M ) )
     => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
        = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).

% fact_diff_Suc
thf(fact_9132_fact__div__fact__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( ( ord_less_eq_nat @ R2 @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R2 ) ) ) @ ( power_power_nat @ N @ R2 ) ) ) ).

% fact_div_fact_le_pow
thf(fact_9133_sin__coeff__Suc,axiom,
    ! [N: nat] :
      ( ( sin_coeff @ ( suc @ N ) )
      = ( divide_divide_real @ ( cos_coeff @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).

% sin_coeff_Suc
thf(fact_9134_cos__coeff__Suc,axiom,
    ! [N: nat] :
      ( ( cos_coeff @ ( suc @ N ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).

% cos_coeff_Suc
thf(fact_9135_sin__coeff__def,axiom,
    ( sin_coeff
    = ( ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) ) ) ) ).

% sin_coeff_def
thf(fact_9136_sin__paired,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
      @ ( sin_real @ X ) ) ).

% sin_paired
thf(fact_9137_cos__arcsin,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_arcsin
thf(fact_9138_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_9139_arcsin__0,axiom,
    ( ( arcsin @ zero_zero_real )
    = zero_zero_real ) ).

% arcsin_0
thf(fact_9140_arccos__1,axiom,
    ( ( arccos @ one_one_real )
    = zero_zero_real ) ).

% arccos_1
thf(fact_9141_arccos__0,axiom,
    ( ( arccos @ zero_zero_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arccos_0
thf(fact_9142_arcsin__1,axiom,
    ( ( arcsin @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arcsin_1
thf(fact_9143_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_9144_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_9145_arccos__cos,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( arccos @ ( cos_real @ X ) )
          = X ) ) ) ).

% arccos_cos
thf(fact_9146_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_9147_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_9148_sin__arccos__nonzero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ( sin_real @ ( arccos @ X ) )
         != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
thf(fact_9149_arccos__cos2,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X )
       => ( ( arccos @ ( cos_real @ X ) )
          = ( uminus_uminus_real @ X ) ) ) ) ).

% arccos_cos2
thf(fact_9150_cos__arcsin__nonzero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X ) )
         != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
thf(fact_9151_power__half__series,axiom,
    ( sums_real
    @ ^ [N3: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N3 ) )
    @ one_one_real ) ).

% power_half_series
thf(fact_9152_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_9153_sums__if_H,axiom,
    ! [G: nat > real,X: real] :
      ( ( sums_real @ G @ X )
     => ( sums_real
        @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        @ X ) ) ).

% sums_if'
thf(fact_9154_sums__if,axiom,
    ! [G: nat > real,X: real,F: nat > real,Y: real] :
      ( ( sums_real @ G @ X )
     => ( ( sums_real @ F @ Y )
       => ( sums_real
          @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( F @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          @ ( plus_plus_real @ X @ Y ) ) ) ) ).

% sums_if
thf(fact_9155_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_9156_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_9157_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_9158_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_9159_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_9160_arcsin__sin,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( arcsin @ ( sin_real @ X ) )
          = X ) ) ) ).

% arcsin_sin
thf(fact_9161_cos__paired,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( power_power_real @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      @ ( cos_real @ X ) ) ).

% cos_paired
thf(fact_9162_le__arcsin__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ Y @ ( arcsin @ X ) )
              = ( ord_less_eq_real @ ( sin_real @ Y ) @ X ) ) ) ) ) ) ).

% le_arcsin_iff
thf(fact_9163_arcsin__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( arcsin @ X ) @ Y )
              = ( ord_less_eq_real @ X @ ( sin_real @ Y ) ) ) ) ) ) ) ).

% arcsin_le_iff
thf(fact_9164_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_9165_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_9166_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_9167_sin__arccos,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( sin_real @ ( arccos @ X ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_arccos
thf(fact_9168_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N ) )
          = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
thf(fact_9169_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_9170_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_9171_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_9172_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_9173_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_gt
thf(fact_9174_floor__eq,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = N ) ) ) ).

% floor_eq
thf(fact_9175_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_ge
thf(fact_9176_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).

% real_of_int_floor_gt_diff_one
thf(fact_9177_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).

% real_of_int_floor_ge_diff_one
thf(fact_9178_floor__eq2,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = N ) ) ) ).

% floor_eq2
thf(fact_9179_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_9180_floor__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X ) )
            = K )
          = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X )
            & ( ord_less_real @ X @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).

% floor_log_eq_powr_iff
thf(fact_9181_floor__log2__div2,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).

% floor_log2_div2
thf(fact_9182_floor__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
     => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% floor_log_nat_eq_if
thf(fact_9183_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri1408675320244567234ct_nat @ M )
        = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
          @ ( groups708209901874060359at_nat
            @ ^ [X6: nat] : X6
            @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).

% fact_eq_fact_times
thf(fact_9184_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
        = ( groups708209901874060359at_nat
          @ ^ [X6: nat] : X6
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).

% fact_div_fact
thf(fact_9185_central__binomial__lower__bound,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ) ) ).

% central_binomial_lower_bound
thf(fact_9186_Maclaurin__sin__bound,axiom,
    ! [X: real,N: nat] :
      ( ord_less_eq_real
      @ ( abs_abs_real
        @ ( minus_minus_real @ ( sin_real @ X )
          @ ( groups6591440286371151544t_real
            @ ^ [M4: nat] : ( times_times_real @ ( sin_coeff @ M4 ) @ ( power_power_real @ X @ M4 ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) )
      @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N ) ) ) ).

% Maclaurin_sin_bound
thf(fact_9187_complex__unimodular__polar,axiom,
    ! [Z2: complex] :
      ( ( ( real_V1022390504157884413omplex @ Z2 )
        = one_one_real )
     => ~ ! [T6: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T6 )
           => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( Z2
               != ( complex2 @ ( cos_real @ T6 ) @ ( sin_real @ T6 ) ) ) ) ) ) ).

% complex_unimodular_polar
thf(fact_9188_binomial__Suc__n,axiom,
    ! [N: nat] :
      ( ( binomial @ ( suc @ N ) @ N )
      = ( suc @ N ) ) ).

% binomial_Suc_n
thf(fact_9189_binomial__1,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
      = N ) ).

% binomial_1
thf(fact_9190_binomial__0__Suc,axiom,
    ! [K: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_9191_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( binomial @ N @ K )
        = zero_zero_nat )
      = ( ord_less_nat @ N @ K ) ) ).

% binomial_eq_0_iff
thf(fact_9192_binomial__Suc__Suc,axiom,
    ! [N: nat,K: nat] :
      ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
      = ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).

% binomial_Suc_Suc
thf(fact_9193_binomial__n__0,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_9194_zero__less__binomial__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
      = ( ord_less_eq_nat @ K @ N ) ) ).

% zero_less_binomial_iff
thf(fact_9195_complex__eq__cancel__iff2,axiom,
    ! [X: real,Y: real,Xa2: real] :
      ( ( ( complex2 @ X @ Y )
        = ( real_V4546457046886955230omplex @ Xa2 ) )
      = ( ( X = Xa2 )
        & ( Y = zero_zero_real ) ) ) ).

% complex_eq_cancel_iff2
thf(fact_9196_complex__of__real__code,axiom,
    ( real_V4546457046886955230omplex
    = ( ^ [X6: real] : ( complex2 @ X6 @ zero_zero_real ) ) ) ).

% complex_of_real_code
thf(fact_9197_complex__of__real__def,axiom,
    ( real_V4546457046886955230omplex
    = ( ^ [R5: real] : ( complex2 @ R5 @ zero_zero_real ) ) ) ).

% complex_of_real_def
thf(fact_9198_Complex__mult__complex__of__real,axiom,
    ! [X: real,Y: real,R2: real] :
      ( ( times_times_complex @ ( complex2 @ X @ Y ) @ ( real_V4546457046886955230omplex @ R2 ) )
      = ( complex2 @ ( times_times_real @ X @ R2 ) @ ( times_times_real @ Y @ R2 ) ) ) ).

% Complex_mult_complex_of_real
thf(fact_9199_complex__of__real__mult__Complex,axiom,
    ! [R2: real,X: real,Y: real] :
      ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( complex2 @ X @ Y ) )
      = ( complex2 @ ( times_times_real @ R2 @ X ) @ ( times_times_real @ R2 @ Y ) ) ) ).

% complex_of_real_mult_Complex
thf(fact_9200_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( binomial @ N @ K )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_9201_Suc__times__binomial,axiom,
    ! [K: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
      = ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).

% Suc_times_binomial
thf(fact_9202_Suc__times__binomial__eq,axiom,
    ! [N: nat,K: nat] :
      ( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
      = ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).

% Suc_times_binomial_eq
thf(fact_9203_choose__mult__lemma,axiom,
    ! [M: nat,R2: nat,K: nat] :
      ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
      = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R2 ) @ M ) ) ) ).

% choose_mult_lemma
thf(fact_9204_binomial__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( ( ord_less_eq_nat @ R2 @ N )
     => ( ord_less_eq_nat @ ( binomial @ N @ R2 ) @ ( power_power_nat @ N @ R2 ) ) ) ).

% binomial_le_pow
thf(fact_9205_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_9206_divide__real__def,axiom,
    ( divide_divide_real
    = ( ^ [X6: real,Y6: real] : ( times_times_real @ X6 @ ( inverse_inverse_real @ Y6 ) ) ) ) ).

% divide_real_def
thf(fact_9207_Complex__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( ( complex2 @ A @ B )
        = zero_zero_complex )
      = ( ( A = zero_zero_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_0
thf(fact_9208_zero__complex_Ocode,axiom,
    ( zero_zero_complex
    = ( complex2 @ zero_zero_real @ zero_zero_real ) ) ).

% zero_complex.code
thf(fact_9209_zero__less__binomial,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).

% zero_less_binomial
thf(fact_9210_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_9211_binomial__Suc__Suc__eq__times,axiom,
    ! [N: nat,K: nat] :
      ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).

% binomial_Suc_Suc_eq_times
thf(fact_9212_choose__mult,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
          = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).

% choose_mult
thf(fact_9213_binomial__absorb__comp,axiom,
    ! [N: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
      = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).

% binomial_absorb_comp
thf(fact_9214_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_9215_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_9216_complex__mult,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( times_times_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
      = ( complex2 @ ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).

% complex_mult
thf(fact_9217_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_9218_one__complex_Ocode,axiom,
    ( one_one_complex
    = ( complex2 @ one_one_real @ zero_zero_real ) ) ).

% one_complex.code
thf(fact_9219_binomial__absorption,axiom,
    ! [K: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
      = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).

% binomial_absorption
thf(fact_9220_forall__pos__mono__1,axiom,
    ! [P: real > $o,E2: real] :
      ( ! [D3: real,E: real] :
          ( ( ord_less_real @ D3 @ E )
         => ( ( P @ D3 )
           => ( P @ E ) ) )
     => ( ! [N2: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( P @ E2 ) ) ) ) ).

% forall_pos_mono_1
thf(fact_9221_binomial__fact__lemma,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
        = ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% binomial_fact_lemma
thf(fact_9222_forall__pos__mono,axiom,
    ! [P: real > $o,E2: real] :
      ( ! [D3: real,E: real] :
          ( ( ord_less_real @ D3 @ E )
         => ( ( P @ D3 )
           => ( P @ E ) ) )
     => ( ! [N2: nat] :
            ( ( N2 != zero_zero_nat )
           => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( P @ E2 ) ) ) ) ).

% forall_pos_mono
thf(fact_9223_real__arch__inverse,axiom,
    ! [E2: real] :
      ( ( ord_less_real @ zero_zero_real @ E2 )
      = ( ? [N3: nat] :
            ( ( N3 != zero_zero_nat )
            & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) )
            & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ E2 ) ) ) ) ).

% real_arch_inverse
thf(fact_9224_sqrt__divide__self__eq,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( divide_divide_real @ ( sqrt @ X ) @ X )
        = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ) ).

% sqrt_divide_self_eq
thf(fact_9225_ln__inverse,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ln_ln_real @ ( inverse_inverse_real @ X ) )
        = ( uminus_uminus_real @ ( ln_ln_real @ X ) ) ) ) ).

% ln_inverse
thf(fact_9226_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_9227_prod__int__plus__eq,axiom,
    ! [I3: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I3 @ ( plus_plus_nat @ I3 @ J ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X6: int] : X6
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I3 ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I3 @ J ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_9228_binomial__maximum_H,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).

% binomial_maximum'
thf(fact_9229_binomial__mono,axiom,
    ! [K: nat,K5: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ K5 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K5 ) @ N )
       => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K5 ) ) ) ) ).

% binomial_mono
thf(fact_9230_binomial__antimono,axiom,
    ! [K: nat,K5: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ K5 )
     => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
       => ( ( ord_less_eq_nat @ K5 @ N )
         => ( ord_less_eq_nat @ ( binomial @ N @ K5 ) @ ( binomial @ N @ K ) ) ) ) ) ).

% binomial_antimono
thf(fact_9231_binomial__maximum,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% binomial_maximum
thf(fact_9232_binomial__le__pow2,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% binomial_le_pow2
thf(fact_9233_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( binomial @ N @ K )
          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).

% choose_reduce_nat
thf(fact_9234_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
        = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% times_binomial_minus1_eq
thf(fact_9235_binomial__altdef__nat,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( binomial @ N @ K )
        = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).

% binomial_altdef_nat
thf(fact_9236_log__inverse,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( log @ A @ ( inverse_inverse_real @ X ) )
            = ( uminus_uminus_real @ ( log @ A @ X ) ) ) ) ) ) ).

% log_inverse
thf(fact_9237_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).

% binomial_less_binomial_Suc
thf(fact_9238_binomial__strict__antimono,axiom,
    ! [K: nat,K5: nat,N: nat] :
      ( ( ord_less_nat @ K @ K5 )
     => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
       => ( ( ord_less_eq_nat @ K5 @ N )
         => ( ord_less_nat @ ( binomial @ N @ K5 ) @ ( binomial @ N @ K ) ) ) ) ) ).

% binomial_strict_antimono
thf(fact_9239_binomial__strict__mono,axiom,
    ! [K: nat,K5: nat,N: nat] :
      ( ( ord_less_nat @ K @ K5 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K5 ) @ N )
       => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K5 ) ) ) ) ).

% binomial_strict_mono
thf(fact_9240_central__binomial__odd,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( binomial @ N @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% central_binomial_odd
thf(fact_9241_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( binomial @ N @ ( suc @ K ) )
        = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).

% binomial_addition_formula
thf(fact_9242_complex__norm,axiom,
    ! [X: real,Y: real] :
      ( ( real_V1022390504157884413omplex @ ( complex2 @ X @ Y ) )
      = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_norm
thf(fact_9243_exp__plus__inverse__exp,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) ) ).

% exp_plus_inverse_exp
thf(fact_9244_choose__two,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% choose_two
thf(fact_9245_plus__inverse__ge__2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) ) ) ).

% plus_inverse_ge_2
thf(fact_9246_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( inverse_inverse_real @ X ) ) ) ).

% real_inv_sqrt_pow2
thf(fact_9247_tan__cot,axiom,
    ! [X: real] :
      ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
      = ( inverse_inverse_real @ ( tan_real @ X ) ) ) ).

% tan_cot
thf(fact_9248_real__le__x__sinh,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ X @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_x_sinh
thf(fact_9249_real__le__abs__sinh,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_abs_sinh
thf(fact_9250_binomial__code,axiom,
    ( binomial
    = ( ^ [N3: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N3 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N3 @ ( minus_minus_nat @ N3 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N3 @ K3 ) @ one_one_nat ) @ N3 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).

% binomial_code
thf(fact_9251_atMost__0,axiom,
    ( ( set_ord_atMost_nat @ zero_zero_nat )
    = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).

% atMost_0
thf(fact_9252_divide__complex__def,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [X6: complex,Y6: complex] : ( times_times_complex @ X6 @ ( invers8013647133539491842omplex @ Y6 ) ) ) ) ).

% divide_complex_def
thf(fact_9253_complex__exp__exists,axiom,
    ! [Z2: complex] :
    ? [A5: complex,R: real] :
      ( Z2
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ ( exp_complex @ A5 ) ) ) ).

% complex_exp_exists
thf(fact_9254_atMost__atLeast0,axiom,
    ( set_ord_atMost_nat
    = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).

% atMost_atLeast0
thf(fact_9255_lessThan__Suc__atMost,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( set_ord_atMost_nat @ K ) ) ).

% lessThan_Suc_atMost
thf(fact_9256_atMost__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K ) )
      = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).

% atMost_Suc
thf(fact_9257_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_9258_sum__choose__upper,axiom,
    ! [M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ K3 @ M )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).

% sum_choose_upper
thf(fact_9259_sum__choose__lower,axiom,
    ! [R2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K3 ) @ K3 )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N ) ) @ N ) ) ).

% sum_choose_lower
thf(fact_9260_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J2: nat] : ( binomial @ ( plus_plus_nat @ N @ J2 ) @ N )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).

% choose_rising_sum(1)
thf(fact_9261_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J2: nat] : ( binomial @ ( plus_plus_nat @ N @ J2 ) @ N )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ M ) ) ).

% choose_rising_sum(2)
thf(fact_9262_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3542108847815614940at_nat
          @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
          @ ( set_ord_atMost_nat @ M ) )
        = ( binomial @ ( suc @ N ) @ M ) ) ) ).

% sum_choose_diagonal
thf(fact_9263_vandermonde,axiom,
    ! [M: nat,N: nat,R2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N @ ( minus_minus_nat @ R2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ R2 ) )
      = ( binomial @ ( plus_plus_nat @ M @ N ) @ R2 ) ) ).

% vandermonde
thf(fact_9264_choose__row__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ N ) @ ( set_ord_atMost_nat @ N ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% choose_row_sum
thf(fact_9265_binomial,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% binomial
thf(fact_9266_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_atMost_eq_remove0
thf(fact_9267_polynomial__product__nat,axiom,
    ! [M: nat,A: nat > nat,N: nat,B: nat > nat,X: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ M @ I2 )
         => ( ( A @ I2 )
            = zero_zero_nat ) )
     => ( ! [J3: nat] :
            ( ( ord_less_nat @ N @ J3 )
           => ( ( B @ J3 )
              = zero_zero_nat ) )
       => ( ( times_times_nat
            @ ( groups3542108847815614940at_nat
              @ ^ [I: nat] : ( times_times_nat @ ( A @ I ) @ ( power_power_nat @ X @ I ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups3542108847815614940at_nat
              @ ^ [J2: nat] : ( times_times_nat @ ( B @ J2 ) @ ( power_power_nat @ X @ J2 ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups3542108847815614940at_nat
            @ ^ [R5: nat] :
                ( times_times_nat
                @ ( groups3542108847815614940at_nat
                  @ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_nat @ X @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_9268_choose__square__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( power_power_nat @ ( binomial @ N @ K3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).

% choose_square_sum
thf(fact_9269_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_9270_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_9271_choose__linear__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( times_times_nat @ I @ ( binomial @ N @ I ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% choose_linear_sum
thf(fact_9272_of__nat__id,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N3: nat] : N3 ) ) ).

% of_nat_id
thf(fact_9273_real__scaleR__def,axiom,
    real_V1485227260804924795R_real = times_times_real ).

% real_scaleR_def
thf(fact_9274_complex__scaleR,axiom,
    ! [R2: real,A: real,B: real] :
      ( ( real_V2046097035970521341omplex @ R2 @ ( complex2 @ A @ B ) )
      = ( complex2 @ ( times_times_real @ R2 @ A ) @ ( times_times_real @ R2 @ B ) ) ) ).

% complex_scaleR
thf(fact_9275_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_9276_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_9277_sinh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( ( sinh_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% sinh_real_zero_iff
thf(fact_9278_sinh__real__neg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( sinh_real @ X ) @ zero_zero_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% sinh_real_neg_iff
thf(fact_9279_sinh__real__pos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% sinh_real_pos_iff
thf(fact_9280_sinh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% sinh_real_nonpos_iff
thf(fact_9281_sinh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% sinh_real_nonneg_iff
thf(fact_9282_complex__i__mult__minus,axiom,
    ! [X: complex] :
      ( ( times_times_complex @ imaginary_unit @ ( times_times_complex @ imaginary_unit @ X ) )
      = ( uminus1482373934393186551omplex @ X ) ) ).

% complex_i_mult_minus
thf(fact_9283_divide__i,axiom,
    ! [X: complex] :
      ( ( divide1717551699836669952omplex @ X @ imaginary_unit )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X ) ) ).

% divide_i
thf(fact_9284_i__squared,axiom,
    ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% i_squared
thf(fact_9285_divide__numeral__i,axiom,
    ! [Z2: complex,N: num] :
      ( ( divide1717551699836669952omplex @ Z2 @ ( times_times_complex @ ( numera6690914467698888265omplex @ N ) @ imaginary_unit ) )
      = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z2 ) ) @ ( numera6690914467698888265omplex @ N ) ) ) ).

% divide_numeral_i
thf(fact_9286_power2__i,axiom,
    ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power2_i
thf(fact_9287_exp__pi__i_H,axiom,
    ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% exp_pi_i'
thf(fact_9288_exp__pi__i,axiom,
    ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% exp_pi_i
thf(fact_9289_i__even__power,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) ) ).

% i_even_power
thf(fact_9290_cosh__real__nonzero,axiom,
    ! [X: real] :
      ( ( cosh_real @ X )
     != zero_zero_real ) ).

% cosh_real_nonzero
thf(fact_9291_complex__i__not__zero,axiom,
    imaginary_unit != zero_zero_complex ).

% complex_i_not_zero
thf(fact_9292_cosh__real__pos,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X ) ) ).

% cosh_real_pos
thf(fact_9293_cosh__real__nonpos__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_eq_real @ Y @ X ) ) ) ) ).

% cosh_real_nonpos_le_iff
thf(fact_9294_cosh__real__nonneg__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_eq_real @ X @ Y ) ) ) ) ).

% cosh_real_nonneg_le_iff
thf(fact_9295_cosh__real__nonneg,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X ) ) ).

% cosh_real_nonneg
thf(fact_9296_i__times__eq__iff,axiom,
    ! [W: complex,Z2: complex] :
      ( ( ( times_times_complex @ imaginary_unit @ W )
        = Z2 )
      = ( W
        = ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z2 ) ) ) ) ).

% i_times_eq_iff
thf(fact_9297_cosh__real__strict__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) ) ) ).

% cosh_real_strict_mono
thf(fact_9298_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_real @ X @ Y ) ) ) ) ).

% cosh_real_nonneg_less_iff
thf(fact_9299_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_real @ Y @ X ) ) ) ) ).

% cosh_real_nonpos_less_iff
thf(fact_9300_arcosh__cosh__real,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( arcosh_real @ ( cosh_real @ X ) )
        = X ) ) ).

% arcosh_cosh_real
thf(fact_9301_imaginary__unit_Ocode,axiom,
    ( imaginary_unit
    = ( complex2 @ zero_zero_real @ one_one_real ) ) ).

% imaginary_unit.code
thf(fact_9302_Complex__eq__i,axiom,
    ! [X: real,Y: real] :
      ( ( ( complex2 @ X @ Y )
        = imaginary_unit )
      = ( ( X = zero_zero_real )
        & ( Y = one_one_real ) ) ) ).

% Complex_eq_i
thf(fact_9303_Complex__mult__i,axiom,
    ! [A: real,B: real] :
      ( ( times_times_complex @ ( complex2 @ A @ B ) @ imaginary_unit )
      = ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).

% Complex_mult_i
thf(fact_9304_i__mult__Complex,axiom,
    ! [A: real,B: real] :
      ( ( times_times_complex @ imaginary_unit @ ( complex2 @ A @ B ) )
      = ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).

% i_mult_Complex
thf(fact_9305_Complex__eq,axiom,
    ( complex2
    = ( ^ [A3: real,B3: real] : ( plus_plus_complex @ ( real_V4546457046886955230omplex @ A3 ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B3 ) ) ) ) ) ).

% Complex_eq
thf(fact_9306_i__complex__of__real,axiom,
    ! [R2: real] :
      ( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R2 ) )
      = ( complex2 @ zero_zero_real @ R2 ) ) ).

% i_complex_of_real
thf(fact_9307_complex__of__real__i,axiom,
    ! [R2: real] :
      ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ imaginary_unit )
      = ( complex2 @ zero_zero_real @ R2 ) ) ).

% complex_of_real_i
thf(fact_9308_complex__split__polar,axiom,
    ! [Z2: complex] :
    ? [R: real,A5: real] :
      ( Z2
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A5 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A5 ) ) ) ) ) ) ).

% complex_split_polar
thf(fact_9309_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_9310_cmod__complex__polar,axiom,
    ! [R2: real,A: real] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) ) )
      = ( abs_abs_real @ R2 ) ) ).

% cmod_complex_polar
thf(fact_9311_cosh__ln__real,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( cosh_real @ ( ln_ln_real @ X ) )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% cosh_ln_real
thf(fact_9312_sinh__ln__real,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( sinh_real @ ( ln_ln_real @ X ) )
        = ( divide_divide_real @ ( minus_minus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% sinh_ln_real
thf(fact_9313_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_9314_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_9315_Arg__ii,axiom,
    ( ( arg @ imaginary_unit )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% Arg_ii
thf(fact_9316_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_9317_csqrt__eq__0,axiom,
    ! [Z2: complex] :
      ( ( ( csqrt @ Z2 )
        = zero_zero_complex )
      = ( Z2 = zero_zero_complex ) ) ).

% csqrt_eq_0
thf(fact_9318_csqrt__0,axiom,
    ( ( csqrt @ zero_zero_complex )
    = zero_zero_complex ) ).

% csqrt_0
thf(fact_9319_cis__zero,axiom,
    ( ( cis @ zero_zero_real )
    = one_one_complex ) ).

% cis_zero
thf(fact_9320_power2__csqrt,axiom,
    ! [Z2: complex] :
      ( ( power_power_complex @ ( csqrt @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = Z2 ) ).

% power2_csqrt
thf(fact_9321_cis__pi__half,axiom,
    ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = imaginary_unit ) ).

% cis_pi_half
thf(fact_9322_cis__2pi,axiom,
    ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_complex ) ).

% cis_2pi
thf(fact_9323_cis__neq__zero,axiom,
    ! [A: real] :
      ( ( cis @ A )
     != zero_zero_complex ) ).

% cis_neq_zero
thf(fact_9324_cis__mult,axiom,
    ! [A: real,B: real] :
      ( ( times_times_complex @ ( cis @ A ) @ ( cis @ B ) )
      = ( cis @ ( plus_plus_real @ A @ B ) ) ) ).

% cis_mult
thf(fact_9325_Arg__zero,axiom,
    ( ( arg @ zero_zero_complex )
    = zero_zero_real ) ).

% Arg_zero
thf(fact_9326_DeMoivre,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_complex @ ( cis @ A ) @ N )
      = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).

% DeMoivre
thf(fact_9327_cis__conv__exp,axiom,
    ( cis
    = ( ^ [B3: real] : ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B3 ) ) ) ) ) ).

% cis_conv_exp
thf(fact_9328_of__real__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( real_V4546457046886955230omplex @ ( sqrt @ X ) )
        = ( csqrt @ ( real_V4546457046886955230omplex @ X ) ) ) ) ).

% of_real_sqrt
thf(fact_9329_bij__betw__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( bij_betw_nat_complex
        @ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        @ ( set_ord_lessThan_nat @ N )
        @ ( collect_complex
          @ ^ [Z3: complex] :
              ( ( power_power_complex @ Z3 @ N )
              = one_one_complex ) ) ) ) ).

% bij_betw_roots_unity
thf(fact_9330_cot__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ( ord_less_real @ ( cot_real @ X ) @ zero_zero_real ) ) ) ).

% cot_less_zero
thf(fact_9331_cot__periodic,axiom,
    ! [X: real] :
      ( ( cot_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( cot_real @ X ) ) ).

% cot_periodic
thf(fact_9332_arctan__def,axiom,
    ( arctan
    = ( ^ [Y6: real] :
          ( the_real
          @ ^ [X6: real] :
              ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X6 )
              & ( ord_less_real @ X6 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( tan_real @ X6 )
                = Y6 ) ) ) ) ) ).

% arctan_def
thf(fact_9333_cot__pi,axiom,
    ( ( cot_real @ pi )
    = zero_zero_real ) ).

% cot_pi
thf(fact_9334_cot__npi,axiom,
    ! [N: nat] :
      ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% cot_npi
thf(fact_9335_ln__neg__is__const,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ln_ln_real @ X )
        = ( the_real
          @ ^ [X6: real] : $false ) ) ) ).

% ln_neg_is_const
thf(fact_9336_arccos__def,axiom,
    ( arccos
    = ( ^ [Y6: real] :
          ( the_real
          @ ^ [X6: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ X6 )
              & ( ord_less_eq_real @ X6 @ pi )
              & ( ( cos_real @ X6 )
                = Y6 ) ) ) ) ) ).

% arccos_def
thf(fact_9337_pi__half,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
    = ( the_real
      @ ^ [X6: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X6 )
          & ( ord_less_eq_real @ X6 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
          & ( ( cos_real @ X6 )
            = zero_zero_real ) ) ) ) ).

% pi_half
thf(fact_9338_pi__def,axiom,
    ( pi
    = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
      @ ( the_real
        @ ^ [X6: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X6 )
            & ( ord_less_eq_real @ X6 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ X6 )
              = zero_zero_real ) ) ) ) ) ).

% pi_def
thf(fact_9339_cot__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cot_real @ X ) ) ) ) ).

% cot_gt_zero
thf(fact_9340_tan__cot_H,axiom,
    ! [X: real] :
      ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
      = ( cot_real @ X ) ) ).

% tan_cot'
thf(fact_9341_arcsin__def,axiom,
    ( arcsin
    = ( ^ [Y6: real] :
          ( the_real
          @ ^ [X6: real] :
              ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X6 )
              & ( ord_less_eq_real @ X6 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( sin_real @ X6 )
                = Y6 ) ) ) ) ) ).

% arcsin_def
thf(fact_9342_bij__betw__nth__root__unity,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( power_power_complex @ Z3 @ N )
                = one_one_complex ) )
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( power_power_complex @ Z3 @ N )
                = C ) ) ) ) ) ).

% bij_betw_nth_root_unity
thf(fact_9343_modulo__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
          = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L )
                @ ( minus_minus_int
                  @ ( semiri1314217659103216013at_int
                    @ ( times_times_nat @ N
                      @ ( zero_n2687167440665602831ol_nat
                        @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
                  @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).

% modulo_int_unfold
thf(fact_9344_powr__int,axiom,
    ! [X: real,I3: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ I3 )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ I3 ) )
            = ( power_power_real @ X @ ( nat2 @ I3 ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ I3 )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ I3 ) )
            = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I3 ) ) ) ) ) ) ) ) ).

% powr_int
thf(fact_9345_divide__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
          = zero_zero_int ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( uminus_uminus_int
                @ ( semiri1314217659103216013at_int
                  @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
                    @ ( zero_n2687167440665602831ol_nat
                      @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).

% divide_int_unfold
thf(fact_9346_real__root__zero,axiom,
    ! [N: nat] :
      ( ( root @ N @ zero_zero_real )
      = zero_zero_real ) ).

% real_root_zero
thf(fact_9347_nat__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_numeral
thf(fact_9348_real__root__Suc__0,axiom,
    ! [X: real] :
      ( ( root @ ( suc @ zero_zero_nat ) @ X )
      = X ) ).

% real_root_Suc_0
thf(fact_9349_real__root__eq__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X )
          = ( root @ N @ Y ) )
        = ( X = Y ) ) ) ).

% real_root_eq_iff
thf(fact_9350_root__0,axiom,
    ! [X: real] :
      ( ( root @ zero_zero_nat @ X )
      = zero_zero_real ) ).

% root_0
thf(fact_9351_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_9352_real__root__eq__0__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X )
          = zero_zero_real )
        = ( X = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
thf(fact_9353_real__root__less__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
        = ( ord_less_real @ X @ Y ) ) ) ).

% real_root_less_iff
thf(fact_9354_nat__0__iff,axiom,
    ! [I3: int] :
      ( ( ( nat2 @ I3 )
        = zero_zero_nat )
      = ( ord_less_eq_int @ I3 @ zero_zero_int ) ) ).

% nat_0_iff
thf(fact_9355_nat__le__0,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ Z2 @ zero_zero_int )
     => ( ( nat2 @ Z2 )
        = zero_zero_nat ) ) ).

% nat_le_0
thf(fact_9356_real__root__le__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% real_root_le_iff
thf(fact_9357_zless__nat__conj,axiom,
    ! [W: int,Z2: int] :
      ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
      = ( ( ord_less_int @ zero_zero_int @ Z2 )
        & ( ord_less_int @ W @ Z2 ) ) ) ).

% zless_nat_conj
thf(fact_9358_real__root__eq__1__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X )
          = one_one_real )
        = ( X = one_one_real ) ) ) ).

% real_root_eq_1_iff
thf(fact_9359_real__root__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ one_one_real )
        = one_one_real ) ) ).

% real_root_one
thf(fact_9360_nat__neg__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = zero_zero_nat ) ).

% nat_neg_numeral
thf(fact_9361_nat__zminus__int,axiom,
    ! [N: nat] :
      ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
      = zero_zero_nat ) ).

% nat_zminus_int
thf(fact_9362_int__nat__eq,axiom,
    ! [Z2: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
          = Z2 ) )
      & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
          = zero_zero_int ) ) ) ).

% int_nat_eq
thf(fact_9363_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R2: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R2 ) ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
thf(fact_9364_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( R2 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
thf(fact_9365_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R2 ) ) @ K )
      = ( ( dvd_dvd_int @ L @ K )
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
thf(fact_9366_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L ) @ K )
      = ( ( dvd_dvd_int @ L @ K )
        & ( ( R2 = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
thf(fact_9367_real__root__lt__0__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X ) @ zero_zero_real )
        = ( ord_less_real @ X @ zero_zero_real ) ) ) ).

% real_root_lt_0_iff
thf(fact_9368_real__root__gt__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y ) )
        = ( ord_less_real @ zero_zero_real @ Y ) ) ) ).

% real_root_gt_0_iff
thf(fact_9369_real__root__le__0__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X ) @ zero_zero_real )
        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).

% real_root_le_0_iff
thf(fact_9370_real__root__ge__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y ) )
        = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).

% real_root_ge_0_iff
thf(fact_9371_zero__less__nat__eq,axiom,
    ! [Z2: int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
      = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).

% zero_less_nat_eq
thf(fact_9372_real__root__lt__1__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X ) @ one_one_real )
        = ( ord_less_real @ X @ one_one_real ) ) ) ).

% real_root_lt_1_iff
thf(fact_9373_real__root__gt__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y ) )
        = ( ord_less_real @ one_one_real @ Y ) ) ) ).

% real_root_gt_1_iff
thf(fact_9374_real__root__ge__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y ) )
        = ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).

% real_root_ge_1_iff
thf(fact_9375_real__root__le__1__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X ) @ one_one_real )
        = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).

% real_root_le_1_iff
thf(fact_9376_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_9377_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( nat2 @ Y )
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% nat_eq_numeral_power_cancel_iff
thf(fact_9378_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = ( nat2 @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
thf(fact_9379_nat__ceiling__le__eq,axiom,
    ! [X: real,A: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
      = ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).

% nat_ceiling_le_eq
thf(fact_9380_one__less__nat__eq,axiom,
    ! [Z2: int] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
      = ( ord_less_int @ one_one_int @ Z2 ) ) ).

% one_less_nat_eq
thf(fact_9381_real__root__pow__pos2,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N @ X ) @ N )
          = X ) ) ) ).

% real_root_pow_pos2
thf(fact_9382_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_9383_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% nat_less_numeral_power_cancel_iff
thf(fact_9384_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_less_nat_cancel_iff
thf(fact_9385_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_le_nat_cancel_iff
thf(fact_9386_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% nat_le_numeral_power_cancel_iff
thf(fact_9387_real__root__mult__exp,axiom,
    ! [M: nat,N: nat,X: real] :
      ( ( root @ ( times_times_nat @ M @ N ) @ X )
      = ( root @ M @ ( root @ N @ X ) ) ) ).

% real_root_mult_exp
thf(fact_9388_real__root__mult,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( root @ N @ ( times_times_real @ X @ Y ) )
      = ( times_times_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ).

% real_root_mult
thf(fact_9389_real__root__pos__pos__le,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ).

% real_root_pos_pos_le
thf(fact_9390_nat__zero__as__int,axiom,
    ( zero_zero_nat
    = ( nat2 @ zero_zero_int ) ) ).

% nat_zero_as_int
thf(fact_9391_nat__numeral__as__int,axiom,
    ( numeral_numeral_nat
    = ( ^ [I: num] : ( nat2 @ ( numeral_numeral_int @ I ) ) ) ) ).

% nat_numeral_as_int
thf(fact_9392_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N2: nat,L3: int] :
          ( K
         != ( times_times_int @ ( sgn_sgn_int @ L3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% int_sgnE
thf(fact_9393_eq__nat__nat__iff,axiom,
    ! [Z2: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
       => ( ( ( nat2 @ Z2 )
            = ( nat2 @ Z6 ) )
          = ( Z2 = Z6 ) ) ) ) ).

% eq_nat_nat_iff
thf(fact_9394_all__nat,axiom,
    ( ( ^ [P3: nat > $o] :
        ! [X7: nat] : ( P3 @ X7 ) )
    = ( ^ [P4: nat > $o] :
        ! [X6: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X6 )
         => ( P4 @ ( nat2 @ X6 ) ) ) ) ) ).

% all_nat
thf(fact_9395_ex__nat,axiom,
    ( ( ^ [P3: nat > $o] :
        ? [X7: nat] : ( P3 @ X7 ) )
    = ( ^ [P4: nat > $o] :
        ? [X6: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X6 )
          & ( P4 @ ( nat2 @ X6 ) ) ) ) ) ).

% ex_nat
thf(fact_9396_nat__one__as__int,axiom,
    ( one_one_nat
    = ( nat2 @ one_one_int ) ) ).

% nat_one_as_int
thf(fact_9397_unset__bit__nat__def,axiom,
    ( bit_se4205575877204974255it_nat
    = ( ^ [M4: nat,N3: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M4 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% unset_bit_nat_def
thf(fact_9398_nat__mask__eq,axiom,
    ! [N: nat] :
      ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N ) )
      = ( bit_se2002935070580805687sk_nat @ N ) ) ).

% nat_mask_eq
thf(fact_9399_real__root__less__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ X @ Y )
       => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).

% real_root_less_mono
thf(fact_9400_real__root__le__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).

% real_root_le_mono
thf(fact_9401_real__root__power,axiom,
    ! [N: nat,X: real,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( power_power_real @ X @ K ) )
        = ( power_power_real @ ( root @ N @ X ) @ K ) ) ) ).

% real_root_power
thf(fact_9402_real__root__abs,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( abs_abs_real @ X ) )
        = ( abs_abs_real @ ( root @ N @ X ) ) ) ) ).

% real_root_abs
thf(fact_9403_nat__mono__iff,axiom,
    ! [Z2: int,W: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
        = ( ord_less_int @ W @ Z2 ) ) ) ).

% nat_mono_iff
thf(fact_9404_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z2: int] :
      ( ( ord_less_nat @ M @ ( nat2 @ Z2 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z2 ) ) ).

% zless_nat_eq_int_zless
thf(fact_9405_int__eq__iff,axiom,
    ! [M: nat,Z2: int] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = Z2 )
      = ( ( M
          = ( nat2 @ Z2 ) )
        & ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ) ).

% int_eq_iff
thf(fact_9406_nat__0__le,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
        = Z2 ) ) ).

% nat_0_le
thf(fact_9407_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_9408_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_9409_int__minus,axiom,
    ! [N: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
      = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).

% int_minus
thf(fact_9410_nat__abs__mult__distrib,axiom,
    ! [W: int,Z2: int] :
      ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z2 ) ) )
      = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z2 ) ) ) ) ).

% nat_abs_mult_distrib
thf(fact_9411_and__nat__def,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M4: nat,N3: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% and_nat_def
thf(fact_9412_nat__plus__as__int,axiom,
    ( plus_plus_nat
    = ( ^ [A3: nat,B3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).

% nat_plus_as_int
thf(fact_9413_or__nat__def,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M4: nat,N3: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% or_nat_def
thf(fact_9414_nat__times__as__int,axiom,
    ( times_times_nat
    = ( ^ [A3: nat,B3: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).

% nat_times_as_int
thf(fact_9415_real__nat__ceiling__ge,axiom,
    ! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).

% real_nat_ceiling_ge
thf(fact_9416_nat__minus__as__int,axiom,
    ( minus_minus_nat
    = ( ^ [A3: nat,B3: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).

% nat_minus_as_int
thf(fact_9417_nat__div__as__int,axiom,
    ( divide_divide_nat
    = ( ^ [A3: nat,B3: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).

% nat_div_as_int
thf(fact_9418_nat__mod__as__int,axiom,
    ( modulo_modulo_nat
    = ( ^ [A3: nat,B3: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).

% nat_mod_as_int
thf(fact_9419_real__root__gt__zero,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).

% real_root_gt_zero
thf(fact_9420_real__root__strict__decreasing,axiom,
    ! [N: nat,N5: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ( ord_less_real @ one_one_real @ X )
         => ( ord_less_real @ ( root @ N5 @ X ) @ ( root @ N @ X ) ) ) ) ) ).

% real_root_strict_decreasing
thf(fact_9421_sqrt__def,axiom,
    ( sqrt
    = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% sqrt_def
thf(fact_9422_root__abs__power,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y @ N ) ) )
        = ( abs_abs_real @ Y ) ) ) ).

% root_abs_power
thf(fact_9423_zsgn__def,axiom,
    ( sgn_sgn_int
    = ( ^ [I: int] : ( if_int @ ( I = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zsgn_def
thf(fact_9424_nat__less__eq__zless,axiom,
    ! [W: int,Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ W )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
        = ( ord_less_int @ W @ Z2 ) ) ) ).

% nat_less_eq_zless
thf(fact_9425_nat__le__eq__zle,axiom,
    ! [W: int,Z2: int] :
      ( ( ( ord_less_int @ zero_zero_int @ W )
        | ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
     => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
        = ( ord_less_eq_int @ W @ Z2 ) ) ) ).

% nat_le_eq_zle
thf(fact_9426_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_9427_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_9428_split__nat,axiom,
    ! [P: nat > $o,I3: int] :
      ( ( P @ ( nat2 @ I3 ) )
      = ( ! [N3: nat] :
            ( ( I3
              = ( semiri1314217659103216013at_int @ N3 ) )
           => ( P @ N3 ) )
        & ( ( ord_less_int @ I3 @ zero_zero_int )
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_nat
thf(fact_9429_le__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
        = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).

% le_nat_iff
thf(fact_9430_nat__add__distrib,axiom,
    ! [Z2: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
       => ( ( nat2 @ ( plus_plus_int @ Z2 @ Z6 ) )
          = ( plus_plus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_9431_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_9432_nat__mult__distrib,axiom,
    ! [Z2: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( nat2 @ ( times_times_int @ Z2 @ Z6 ) )
        = ( times_times_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) ) ) ) ).

% nat_mult_distrib
thf(fact_9433_Suc__as__int,axiom,
    ( suc
    = ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_9434_nat__diff__distrib,axiom,
    ! [Z6: int,Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
     => ( ( ord_less_eq_int @ Z6 @ Z2 )
       => ( ( nat2 @ ( minus_minus_int @ Z2 @ Z6 ) )
          = ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) ) ) ) ) ).

% nat_diff_distrib
thf(fact_9435_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
          = ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).

% nat_diff_distrib'
thf(fact_9436_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_9437_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).

% nat_div_distrib
thf(fact_9438_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).

% nat_div_distrib'
thf(fact_9439_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_9440_nat__power__eq,axiom,
    ! [Z2: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( nat2 @ ( power_power_int @ Z2 @ N ) )
        = ( power_power_nat @ ( nat2 @ Z2 ) @ N ) ) ) ).

% nat_power_eq
thf(fact_9441_nat__floor__neg,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
        = zero_zero_nat ) ) ).

% nat_floor_neg
thf(fact_9442_nat__mod__distrib,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( nat2 @ ( modulo_modulo_int @ X @ Y ) )
          = ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).

% nat_mod_distrib
thf(fact_9443_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_9444_floor__eq3,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N ) ) ) ).

% floor_eq3
thf(fact_9445_le__nat__floor,axiom,
    ! [X: nat,A: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
     => ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).

% le_nat_floor
thf(fact_9446_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_9447_nat__take__bit__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
        = ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).

% nat_take_bit_eq
thf(fact_9448_take__bit__nat__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
        = ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).

% take_bit_nat_eq
thf(fact_9449_bit__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% bit_nat_iff
thf(fact_9450_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_9451_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_9452_real__root__pos__pos,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).

% real_root_pos_pos
thf(fact_9453_real__root__strict__increasing,axiom,
    ! [N: nat,N5: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ X @ one_one_real )
           => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N5 @ X ) ) ) ) ) ) ).

% real_root_strict_increasing
thf(fact_9454_real__root__decreasing,axiom,
    ! [N: nat,N5: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ( ord_less_eq_real @ one_one_real @ X )
         => ( ord_less_eq_real @ ( root @ N5 @ X ) @ ( root @ N @ X ) ) ) ) ) ).

% real_root_decreasing
thf(fact_9455_real__root__pow__pos,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N @ X ) @ N )
          = X ) ) ) ).

% real_root_pow_pos
thf(fact_9456_real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( root @ N @ ( power_power_real @ X @ N ) )
          = X ) ) ) ).

% real_root_power_cancel
thf(fact_9457_real__root__pos__unique,axiom,
    ! [N: nat,Y: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( power_power_real @ Y @ N )
            = X )
         => ( ( root @ N @ X )
            = Y ) ) ) ) ).

% real_root_pos_unique
thf(fact_9458_odd__real__root__pow,axiom,
    ! [N: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( root @ N @ X ) @ N )
        = X ) ) ).

% odd_real_root_pow
thf(fact_9459_odd__real__root__unique,axiom,
    ! [N: nat,Y: real,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ( power_power_real @ Y @ N )
          = X )
       => ( ( root @ N @ X )
          = Y ) ) ) ).

% odd_real_root_unique
thf(fact_9460_odd__real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( root @ N @ ( power_power_real @ X @ N ) )
        = X ) ) ).

% odd_real_root_power_cancel
thf(fact_9461_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_9462_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( suc @ ( nat2 @ Z2 ) )
        = ( nat2 @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ) ).

% Suc_nat_eq_nat_zadd1
thf(fact_9463_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_9464_nat__mult__distrib__neg,axiom,
    ! [Z2: int,Z6: int] :
      ( ( ord_less_eq_int @ Z2 @ zero_zero_int )
     => ( ( nat2 @ ( times_times_int @ Z2 @ Z6 ) )
        = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z2 ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).

% nat_mult_distrib_neg
thf(fact_9465_floor__eq4,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N ) ) ) ).

% floor_eq4
thf(fact_9466_diff__nat__eq__if,axiom,
    ! [Z6: int,Z2: int] :
      ( ( ( ord_less_int @ Z6 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) )
          = ( nat2 @ Z2 ) ) )
      & ( ~ ( ord_less_int @ Z6 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) )
          = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z2 @ Z6 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z2 @ Z6 ) ) ) ) ) ) ).

% diff_nat_eq_if
thf(fact_9467_real__root__increasing,axiom,
    ! [N: nat,N5: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ( ord_less_eq_real @ zero_zero_real @ X )
         => ( ( ord_less_eq_real @ X @ one_one_real )
           => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N5 @ X ) ) ) ) ) ) ).

% real_root_increasing
thf(fact_9468_nat__dvd__iff,axiom,
    ! [Z2: int,M: nat] :
      ( ( dvd_dvd_nat @ ( nat2 @ Z2 ) @ M )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
         => ( dvd_dvd_int @ Z2 @ ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
thf(fact_9469_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q2: int] :
      ( ( ( sgn_sgn_int @ R2 )
        = ( sgn_sgn_int @ L ) )
     => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
       => ( ( K
            = ( plus_plus_int @ ( times_times_int @ Q2 @ L ) @ R2 ) )
         => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q2 @ R2 ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_9470_ln__root,axiom,
    ! [N: nat,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( ln_ln_real @ ( root @ N @ B ) )
          = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% ln_root
thf(fact_9471_log__root,axiom,
    ! [N: nat,A: real,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( log @ B @ ( root @ N @ A ) )
          = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_root
thf(fact_9472_log__base__root,axiom,
    ! [N: nat,B: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( log @ ( root @ N @ B ) @ X )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ) ).

% log_base_root
thf(fact_9473_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A12: int,A23: int,A32: product_prod_int_int] :
          ( ? [K3: int] :
              ( ( A12 = K3 )
              & ( A23 = zero_zero_int )
              & ( A32
                = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
          | ? [L2: int,K3: int,Q4: int] :
              ( ( A12 = K3 )
              & ( A23 = L2 )
              & ( A32
                = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
              & ( L2 != zero_zero_int )
              & ( K3
                = ( times_times_int @ Q4 @ L2 ) ) )
          | ? [R5: int,L2: int,K3: int,Q4: int] :
              ( ( A12 = K3 )
              & ( A23 = L2 )
              & ( A32
                = ( product_Pair_int_int @ Q4 @ 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 @ Q4 @ L2 ) @ R5 ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_9474_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A33: product_prod_int_int] :
      ( ( eucl_rel_int @ A1 @ A22 @ A33 )
     => ( ( ( A22 = zero_zero_int )
         => ( A33
           != ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
       => ( ! [Q3: int] :
              ( ( A33
                = ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
             => ( ( A22 != zero_zero_int )
               => ( A1
                 != ( times_times_int @ Q3 @ A22 ) ) ) )
         => ~ ! [R: int,Q3: int] :
                ( ( A33
                  = ( product_Pair_int_int @ Q3 @ R ) )
               => ( ( ( sgn_sgn_int @ R )
                    = ( sgn_sgn_int @ A22 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R ) @ ( abs_abs_int @ A22 ) )
                   => ( A1
                     != ( plus_plus_int @ ( times_times_int @ Q3 @ A22 ) @ R ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_9475_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_9476_root__powr__inverse,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( root @ N @ X )
          = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).

% root_powr_inverse
thf(fact_9477_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_9478_powr__real__of__int,axiom,
    ! [X: real,N: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
            = ( power_power_real @ X @ ( nat2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
            = ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).

% powr_real_of_int
thf(fact_9479_arctan__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( ( arctan @ ( divide_divide_real @ one_one_real @ X ) )
        = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X ) ) ) ) ).

% arctan_inverse
thf(fact_9480_cis__multiple__2pi,axiom,
    ! [N: real] :
      ( ( member_real @ N @ ring_1_Ints_real )
     => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
        = one_one_complex ) ) ).

% cis_multiple_2pi
thf(fact_9481_sgn__le__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sgn_sgn_real @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% sgn_le_0_iff
thf(fact_9482_zero__le__sgn__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% zero_le_sgn_iff
thf(fact_9483_sgn__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( sgn_sgn_real @ ( root @ N @ X ) )
        = ( sgn_sgn_real @ X ) ) ) ).

% sgn_root
thf(fact_9484_cis__Arg,axiom,
    ! [Z2: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( cis @ ( arg @ Z2 ) )
        = ( sgn_sgn_complex @ Z2 ) ) ) ).

% cis_Arg
thf(fact_9485_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_9486_sin__times__pi__eq__0,axiom,
    ! [X: real] :
      ( ( ( sin_real @ ( times_times_real @ X @ pi ) )
        = zero_zero_real )
      = ( member_real @ X @ ring_1_Ints_real ) ) ).

% sin_times_pi_eq_0
thf(fact_9487_sgn__power__injE,axiom,
    ! [A: real,N: nat,X: real,B: real] :
      ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
        = X )
     => ( ( X
          = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( A = B ) ) ) ) ).

% sgn_power_injE
thf(fact_9488_sgn__power__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X ) ) @ N ) )
        = X ) ) ).

% sgn_power_root
thf(fact_9489_root__sgn__power,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) ) )
        = Y ) ) ).

% root_sgn_power
thf(fact_9490_split__root,axiom,
    ! [P: real > $o,N: nat,X: real] :
      ( ( P @ ( root @ N @ X ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ zero_zero_real ) )
        & ( ( ord_less_nat @ zero_zero_nat @ N )
         => ! [Y6: real] :
              ( ( ( times_times_real @ ( sgn_sgn_real @ Y6 ) @ ( power_power_real @ ( abs_abs_real @ Y6 ) @ N ) )
                = X )
             => ( P @ Y6 ) ) ) ) ) ).

% split_root
thf(fact_9491_floor__real__def,axiom,
    ( archim6058952711729229775r_real
    = ( ^ [X6: real] :
          ( the_int
          @ ^ [Z3: int] :
              ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X6 )
              & ( ord_less_real @ X6 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).

% floor_real_def
thf(fact_9492_sin__integer__2pi,axiom,
    ! [N: real] :
      ( ( member_real @ N @ ring_1_Ints_real )
     => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
        = zero_zero_real ) ) ).

% sin_integer_2pi
thf(fact_9493_cos__integer__2pi,axiom,
    ! [N: real] :
      ( ( member_real @ N @ ring_1_Ints_real )
     => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
        = one_one_real ) ) ).

% cos_integer_2pi
thf(fact_9494_Arg__correct,axiom,
    ! [Z2: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( ( sgn_sgn_complex @ Z2 )
          = ( cis @ ( arg @ Z2 ) ) )
        & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z2 ) )
        & ( ord_less_eq_real @ ( arg @ Z2 ) @ pi ) ) ) ).

% Arg_correct
thf(fact_9495_Arg__def,axiom,
    ( arg
    = ( ^ [Z3: complex] :
          ( if_real @ ( Z3 = zero_zero_complex ) @ zero_zero_real
          @ ( fChoice_real
            @ ^ [A3: real] :
                ( ( ( sgn_sgn_complex @ Z3 )
                  = ( cis @ A3 ) )
                & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A3 )
                & ( ord_less_eq_real @ A3 @ pi ) ) ) ) ) ) ).

% Arg_def
thf(fact_9496_xor__Suc__0__eq,axiom,
    ! [N: nat] :
      ( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% xor_Suc_0_eq
thf(fact_9497_Suc__0__xor__eq,axiom,
    ! [N: nat] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% Suc_0_xor_eq
thf(fact_9498_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_9499_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_9500_xor__nat__numerals_I3_J,axiom,
    ! [X: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% xor_nat_numerals(3)
thf(fact_9501_xor__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).

% xor_nat_numerals(4)
thf(fact_9502_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M4: nat,N3: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M4 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_9503_xor__nat__rec,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M4: nat,N3: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
             != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_9504_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_9505_push__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% push_bit_nonnegative_int_iff
thf(fact_9506_push__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% push_bit_negative_int_iff
thf(fact_9507_concat__bit__of__zero__1,axiom,
    ! [N: nat,L: int] :
      ( ( bit_concat_bit @ N @ zero_zero_int @ L )
      = ( bit_se545348938243370406it_int @ N @ L ) ) ).

% concat_bit_of_zero_1
thf(fact_9508_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_9509_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_9510_push__bit__of__Suc__0,axiom,
    ! [N: nat] :
      ( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% push_bit_of_Suc_0
thf(fact_9511_flip__bit__int__def,axiom,
    ( bit_se2159334234014336723it_int
    = ( ^ [N3: nat,K3: int] : ( bit_se6526347334894502574or_int @ K3 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).

% flip_bit_int_def
thf(fact_9512_bit__xor__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ N )
      = ( ( bit_se1146084159140164899it_int @ K @ N )
       != ( bit_se1146084159140164899it_int @ L @ N ) ) ) ).

% bit_xor_int_iff
thf(fact_9513_push__bit__nat__eq,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se547839408752420682it_nat @ N @ ( nat2 @ K ) )
      = ( nat2 @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).

% push_bit_nat_eq
thf(fact_9514_XOR__lower,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X @ Y ) ) ) ) ).

% XOR_lower
thf(fact_9515_set__bit__nat__def,axiom,
    ( bit_se7882103937844011126it_nat
    = ( ^ [M4: nat,N3: nat] : ( bit_se1412395901928357646or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M4 @ one_one_nat ) ) ) ) ).

% set_bit_nat_def
thf(fact_9516_flip__bit__nat__def,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [M4: nat,N3: nat] : ( bit_se6528837805403552850or_nat @ N3 @ ( bit_se547839408752420682it_nat @ M4 @ one_one_nat ) ) ) ) ).

% flip_bit_nat_def
thf(fact_9517_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
      = ( ( ord_less_eq_nat @ M @ N )
        & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).

% bit_push_bit_iff_int
thf(fact_9518_xor__nat__def,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M4: nat,N3: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% xor_nat_def
thf(fact_9519_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q2 ) @ N )
      = ( ( ord_less_eq_nat @ M @ N )
        & ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N @ M ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_9520_concat__bit__eq,axiom,
    ( bit_concat_bit
    = ( ^ [N3: nat,K3: int,L2: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N3 @ K3 ) @ ( bit_se545348938243370406it_int @ N3 @ L2 ) ) ) ) ).

% concat_bit_eq
thf(fact_9521_concat__bit__def,axiom,
    ( bit_concat_bit
    = ( ^ [N3: nat,K3: int,L2: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N3 @ K3 ) @ ( bit_se545348938243370406it_int @ N3 @ L2 ) ) ) ) ).

% concat_bit_def
thf(fact_9522_set__bit__int__def,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N3: nat,K3: int] : ( bit_se1409905431419307370or_int @ K3 @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ).

% set_bit_int_def
thf(fact_9523_push__bit__nat__def,axiom,
    ( bit_se547839408752420682it_nat
    = ( ^ [N3: nat,M4: nat] : ( times_times_nat @ M4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% push_bit_nat_def
thf(fact_9524_push__bit__int__def,axiom,
    ( bit_se545348938243370406it_int
    = ( ^ [N3: nat,K3: int] : ( times_times_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% push_bit_int_def
thf(fact_9525_push__bit__minus__one,axiom,
    ! [N: nat] :
      ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% push_bit_minus_one
thf(fact_9526_XOR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
         => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% XOR_upper
thf(fact_9527_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_9528_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_9529_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X6: nat] : X6
        @ ( set_or4665077453230672383an_nat @ M @ N ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Ico_nat
thf(fact_9530_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_9531_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_9532_atLeastLessThan__singleton,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
      = ( insert_nat @ M @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_9533_or__minus__minus__numerals,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).

% or_minus_minus_numerals
thf(fact_9534_and__minus__minus__numerals,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).

% and_minus_minus_numerals
thf(fact_9535_bit__not__int__iff,axiom,
    ! [K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% bit_not_int_iff
thf(fact_9536_ex__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M4: nat] :
            ( ( ord_less_nat @ M4 @ N )
            & ( P @ M4 ) ) )
      = ( ? [X6: nat] :
            ( ( member_nat @ X6 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
            & ( P @ X6 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_9537_all__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M4: nat] :
            ( ( ord_less_nat @ M4 @ N )
           => ( P @ M4 ) ) )
      = ( ! [X6: nat] :
            ( ( member_nat @ X6 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
           => ( P @ X6 ) ) ) ) ).

% all_nat_less_eq
thf(fact_9538_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
      = ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% atLeastLessThanSuc_atLeastAtMost
thf(fact_9539_lessThan__atLeast0,axiom,
    ( set_ord_lessThan_nat
    = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).

% lessThan_atLeast0
thf(fact_9540_atLeastLessThan0,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_9541_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_9542_not__int__def,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K3: int] : ( minus_minus_int @ ( uminus_uminus_int @ K3 ) @ one_one_int ) ) ) ).

% not_int_def
thf(fact_9543_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_9544_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_9545_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).

% atLeast0_lessThan_Suc
thf(fact_9546_unset__bit__int__def,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N3: nat,K3: int] : ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N3 @ one_one_int ) ) ) ) ) ).

% unset_bit_int_def
thf(fact_9547_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_9548_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N5: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( finite_finite_nat @ N5 ) ) ).

% subset_eq_atLeast0_lessThan_finite
thf(fact_9549_atLeastLessThan__add__Un,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ( set_or4665077453230672383an_nat @ I3 @ ( plus_plus_nat @ J @ K ) )
        = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I3 @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% atLeastLessThan_add_Un
thf(fact_9550_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_9551_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_9552_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_9553_and__not__numerals_I2_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = one_one_int ) ).

% and_not_numerals(2)
thf(fact_9554_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_9555_or__not__numerals_I2_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).

% or_not_numerals(2)
thf(fact_9556_bit__minus__int__iff,axiom,
    ! [K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N )
      = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N ) ) ).

% bit_minus_int_iff
thf(fact_9557_int__numeral__or__not__num__neg,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).

% int_numeral_or_not_num_neg
thf(fact_9558_int__numeral__not__or__num__neg,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).

% int_numeral_not_or_num_neg
thf(fact_9559_numeral__or__not__num__eq,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) )
      = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% numeral_or_not_num_eq
thf(fact_9560_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
          = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
thf(fact_9561_prod__Suc__fact,axiom,
    ! [N: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
      = ( semiri1408675320244567234ct_nat @ N ) ) ).

% prod_Suc_fact
thf(fact_9562_prod__Suc__Suc__fact,axiom,
    ! [N: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( semiri1408675320244567234ct_nat @ N ) ) ).

% prod_Suc_Suc_fact
thf(fact_9563_and__not__numerals_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% and_not_numerals(5)
thf(fact_9564_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_9565_or__not__numerals_I3_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).

% or_not_numerals(3)
thf(fact_9566_and__not__numerals_I3_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = zero_zero_int ) ).

% and_not_numerals(3)
thf(fact_9567_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_9568_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_9569_and__not__numerals_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% and_not_numerals(9)
thf(fact_9570_and__not__numerals_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% and_not_numerals(6)
thf(fact_9571_or__not__numerals_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% or_not_numerals(6)
thf(fact_9572_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_lessThan_eq_remove0
thf(fact_9573_or__not__numerals_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).

% or_not_numerals(5)
thf(fact_9574_and__not__numerals_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).

% and_not_numerals(8)
thf(fact_9575_or__not__numerals_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).

% or_not_numerals(9)
thf(fact_9576_or__not__numerals_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).

% or_not_numerals(8)
thf(fact_9577_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_9578_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_9579_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A: nat > nat,B: nat > nat] :
      ( ! [I2: nat,J3: nat] :
          ( ( ord_less_eq_nat @ I2 @ J3 )
         => ( ( ord_less_nat @ J3 @ N )
           => ( ord_less_eq_nat @ ( A @ I2 ) @ ( A @ J3 ) ) ) )
     => ( ! [I2: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I2 @ J3 )
           => ( ( ord_less_nat @ J3 @ N )
             => ( ord_less_eq_nat @ ( B @ J3 ) @ ( B @ I2 ) ) ) )
       => ( ord_less_eq_nat
          @ ( times_times_nat @ N
            @ ( groups3542108847815614940at_nat
              @ ^ [I: nat] : ( times_times_nat @ ( A @ I ) @ ( B @ I ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
          @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).

% Chebyshev_sum_upper_nat
thf(fact_9580_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).

% finite_atLeastZeroLessThan_int
thf(fact_9581_Cauchy__iff2,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X4: nat > real] :
        ! [J2: nat] :
        ? [M8: nat] :
        ! [M4: nat] :
          ( ( ord_less_eq_nat @ M8 @ M4 )
         => ! [N3: nat] :
              ( ( ord_less_eq_nat @ M8 @ N3 )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X4 @ M4 ) @ ( X4 @ N3 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J2 ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_9582_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_9583_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_9584_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X22: $o] :
      ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X22 ) )
      = zero_zero_nat ) ).

% VEBT.size_gen(2)
thf(fact_9585_csqrt_Osimps_I1_J,axiom,
    ! [Z2: complex] :
      ( ( re @ ( csqrt @ Z2 ) )
      = ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% csqrt.simps(1)
thf(fact_9586_cos__Arg__i__mult__zero,axiom,
    ! [Y: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( ( re @ Y )
          = zero_zero_real )
       => ( ( cos_real @ ( arg @ Y ) )
          = zero_zero_real ) ) ) ).

% cos_Arg_i_mult_zero
thf(fact_9587_imaginary__unit_Osimps_I1_J,axiom,
    ( ( re @ imaginary_unit )
    = zero_zero_real ) ).

% imaginary_unit.simps(1)
thf(fact_9588_zero__complex_Osimps_I1_J,axiom,
    ( ( re @ zero_zero_complex )
    = zero_zero_real ) ).

% zero_complex.simps(1)
thf(fact_9589_scaleR__complex_Osimps_I1_J,axiom,
    ! [R2: real,X: complex] :
      ( ( re @ ( real_V2046097035970521341omplex @ R2 @ X ) )
      = ( times_times_real @ R2 @ ( re @ X ) ) ) ).

% scaleR_complex.simps(1)
thf(fact_9590_Re__csqrt,axiom,
    ! [Z2: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z2 ) ) ) ).

% Re_csqrt
thf(fact_9591_cmod__plus__Re__le__0__iff,axiom,
    ! [Z2: complex] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ zero_zero_real )
      = ( ( re @ Z2 )
        = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ).

% cmod_plus_Re_le_0_iff
thf(fact_9592_cos__n__Re__cis__pow__n,axiom,
    ! [N: nat,A: real] :
      ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
      = ( re @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).

% cos_n_Re_cis_pow_n
thf(fact_9593_csqrt_Ocode,axiom,
    ( csqrt
    = ( ^ [Z3: complex] :
          ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( re @ Z3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          @ ( times_times_real
            @ ( if_real
              @ ( ( im @ Z3 )
                = zero_zero_real )
              @ one_one_real
              @ ( sgn_sgn_real @ ( im @ Z3 ) ) )
            @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( re @ Z3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% csqrt.code
thf(fact_9594_csqrt_Osimps_I2_J,axiom,
    ! [Z2: complex] :
      ( ( im @ ( csqrt @ Z2 ) )
      = ( times_times_real
        @ ( if_real
          @ ( ( im @ Z2 )
            = zero_zero_real )
          @ one_one_real
          @ ( sgn_sgn_real @ ( im @ Z2 ) ) )
        @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% csqrt.simps(2)
thf(fact_9595_csqrt__of__real__nonpos,axiom,
    ! [X: complex] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( ord_less_eq_real @ ( re @ X ) @ zero_zero_real )
       => ( ( csqrt @ X )
          = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X ) ) ) ) ) ) ) ) ).

% csqrt_of_real_nonpos
thf(fact_9596_complex__Im__fact,axiom,
    ! [N: nat] :
      ( ( im @ ( semiri5044797733671781792omplex @ N ) )
      = zero_zero_real ) ).

% complex_Im_fact
thf(fact_9597_complex__Im__of__int,axiom,
    ! [Z2: int] :
      ( ( im @ ( ring_17405671764205052669omplex @ Z2 ) )
      = zero_zero_real ) ).

% complex_Im_of_int
thf(fact_9598_complex__Im__of__nat,axiom,
    ! [N: nat] :
      ( ( im @ ( semiri8010041392384452111omplex @ N ) )
      = zero_zero_real ) ).

% complex_Im_of_nat
thf(fact_9599_Im__complex__of__real,axiom,
    ! [Z2: real] :
      ( ( im @ ( real_V4546457046886955230omplex @ Z2 ) )
      = zero_zero_real ) ).

% Im_complex_of_real
thf(fact_9600_Im__power__real,axiom,
    ! [X: complex,N: nat] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( im @ ( power_power_complex @ X @ N ) )
        = zero_zero_real ) ) ).

% Im_power_real
thf(fact_9601_complex__Im__numeral,axiom,
    ! [V: num] :
      ( ( im @ ( numera6690914467698888265omplex @ V ) )
      = zero_zero_real ) ).

% complex_Im_numeral
thf(fact_9602_Im__i__times,axiom,
    ! [Z2: complex] :
      ( ( im @ ( times_times_complex @ imaginary_unit @ Z2 ) )
      = ( re @ Z2 ) ) ).

% Im_i_times
thf(fact_9603_Re__power__real,axiom,
    ! [X: complex,N: nat] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( re @ ( power_power_complex @ X @ N ) )
        = ( power_power_real @ ( re @ X ) @ N ) ) ) ).

% Re_power_real
thf(fact_9604_Re__i__times,axiom,
    ! [Z2: complex] :
      ( ( re @ ( times_times_complex @ imaginary_unit @ Z2 ) )
      = ( uminus_uminus_real @ ( im @ Z2 ) ) ) ).

% Re_i_times
thf(fact_9605_csqrt__of__real__nonneg,axiom,
    ! [X: complex] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) )
       => ( ( csqrt @ X )
          = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X ) ) ) ) ) ) ).

% csqrt_of_real_nonneg
thf(fact_9606_csqrt__minus,axiom,
    ! [X: complex] :
      ( ( ( ord_less_real @ ( im @ X ) @ zero_zero_real )
        | ( ( ( im @ X )
            = zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) ) ) )
     => ( ( csqrt @ ( uminus1482373934393186551omplex @ X ) )
        = ( times_times_complex @ imaginary_unit @ ( csqrt @ X ) ) ) ) ).

% csqrt_minus
thf(fact_9607_zero__complex_Osimps_I2_J,axiom,
    ( ( im @ zero_zero_complex )
    = zero_zero_real ) ).

% zero_complex.simps(2)
thf(fact_9608_one__complex_Osimps_I2_J,axiom,
    ( ( im @ one_one_complex )
    = zero_zero_real ) ).

% one_complex.simps(2)
thf(fact_9609_scaleR__complex_Osimps_I2_J,axiom,
    ! [R2: real,X: complex] :
      ( ( im @ ( real_V2046097035970521341omplex @ R2 @ X ) )
      = ( times_times_real @ R2 @ ( im @ X ) ) ) ).

% scaleR_complex.simps(2)
thf(fact_9610_complex__is__Int__iff,axiom,
    ! [Z2: complex] :
      ( ( member_complex @ Z2 @ ring_1_Ints_complex )
      = ( ( ( im @ Z2 )
          = zero_zero_real )
        & ? [I: int] :
            ( ( re @ Z2 )
            = ( ring_1_of_int_real @ I ) ) ) ) ).

% complex_is_Int_iff
thf(fact_9611_times__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( im @ ( times_times_complex @ X @ Y ) )
      = ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) ) ) ).

% times_complex.simps(2)
thf(fact_9612_cmod__eq__Re,axiom,
    ! [Z2: complex] :
      ( ( ( im @ Z2 )
        = zero_zero_real )
     => ( ( real_V1022390504157884413omplex @ Z2 )
        = ( abs_abs_real @ ( re @ Z2 ) ) ) ) ).

% cmod_eq_Re
thf(fact_9613_cmod__eq__Im,axiom,
    ! [Z2: complex] :
      ( ( ( re @ Z2 )
        = zero_zero_real )
     => ( ( real_V1022390504157884413omplex @ Z2 )
        = ( abs_abs_real @ ( im @ Z2 ) ) ) ) ).

% cmod_eq_Im
thf(fact_9614_Im__eq__0,axiom,
    ! [Z2: complex] :
      ( ( ( abs_abs_real @ ( re @ Z2 ) )
        = ( real_V1022390504157884413omplex @ Z2 ) )
     => ( ( im @ Z2 )
        = zero_zero_real ) ) ).

% Im_eq_0
thf(fact_9615_times__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( re @ ( times_times_complex @ X @ Y ) )
      = ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) ) ).

% times_complex.simps(1)
thf(fact_9616_scaleR__complex_Ocode,axiom,
    ( real_V2046097035970521341omplex
    = ( ^ [R5: real,X6: complex] : ( complex2 @ ( times_times_real @ R5 @ ( re @ X6 ) ) @ ( times_times_real @ R5 @ ( im @ X6 ) ) ) ) ) ).

% scaleR_complex.code
thf(fact_9617_csqrt__principal,axiom,
    ! [Z2: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z2 ) ) )
      | ( ( ( re @ ( csqrt @ Z2 ) )
          = zero_zero_real )
        & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z2 ) ) ) ) ) ).

% csqrt_principal
thf(fact_9618_sin__n__Im__cis__pow__n,axiom,
    ! [N: nat,A: real] :
      ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
      = ( im @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).

% sin_n_Im_cis_pow_n
thf(fact_9619_Re__exp,axiom,
    ! [Z2: complex] :
      ( ( re @ ( exp_complex @ Z2 ) )
      = ( times_times_real @ ( exp_real @ ( re @ Z2 ) ) @ ( cos_real @ ( im @ Z2 ) ) ) ) ).

% Re_exp
thf(fact_9620_Im__exp,axiom,
    ! [Z2: complex] :
      ( ( im @ ( exp_complex @ Z2 ) )
      = ( times_times_real @ ( exp_real @ ( re @ Z2 ) ) @ ( sin_real @ ( im @ Z2 ) ) ) ) ).

% Im_exp
thf(fact_9621_complex__eq,axiom,
    ! [A: complex] :
      ( A
      = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( re @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( im @ A ) ) ) ) ) ).

% complex_eq
thf(fact_9622_times__complex_Ocode,axiom,
    ( times_times_complex
    = ( ^ [X6: complex,Y6: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X6 ) @ ( re @ Y6 ) ) @ ( times_times_real @ ( im @ X6 ) @ ( im @ Y6 ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X6 ) @ ( im @ Y6 ) ) @ ( times_times_real @ ( im @ X6 ) @ ( re @ Y6 ) ) ) ) ) ) ).

% times_complex.code
thf(fact_9623_exp__eq__polar,axiom,
    ( exp_complex
    = ( ^ [Z3: complex] : ( times_times_complex @ ( real_V4546457046886955230omplex @ ( exp_real @ ( re @ Z3 ) ) ) @ ( cis @ ( im @ Z3 ) ) ) ) ) ).

% exp_eq_polar
thf(fact_9624_cmod__power2,axiom,
    ! [Z2: complex] :
      ( ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cmod_power2
thf(fact_9625_Im__power2,axiom,
    ! [X: complex] :
      ( ( im @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X ) ) @ ( im @ X ) ) ) ).

% Im_power2
thf(fact_9626_Re__power2,axiom,
    ! [X: complex] :
      ( ( re @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( minus_minus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% Re_power2
thf(fact_9627_complex__eq__0,axiom,
    ! [Z2: complex] :
      ( ( Z2 = zero_zero_complex )
      = ( ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_real ) ) ).

% complex_eq_0
thf(fact_9628_norm__complex__def,axiom,
    ( real_V1022390504157884413omplex
    = ( ^ [Z3: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% norm_complex_def
thf(fact_9629_inverse__complex_Osimps_I1_J,axiom,
    ! [X: complex] :
      ( ( re @ ( invers8013647133539491842omplex @ X ) )
      = ( 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 ) ) ) ) ) ) ).

% inverse_complex.simps(1)
thf(fact_9630_complex__neq__0,axiom,
    ! [Z2: complex] :
      ( ( Z2 != zero_zero_complex )
      = ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_neq_0
thf(fact_9631_Re__divide,axiom,
    ! [X: complex,Y: complex] :
      ( ( re @ ( divide1717551699836669952omplex @ X @ Y ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( 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_9632_csqrt__unique,axiom,
    ! [W: complex,Z2: complex] :
      ( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = Z2 )
     => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
          | ( ( ( re @ W )
              = zero_zero_real )
            & ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
       => ( ( csqrt @ Z2 )
          = W ) ) ) ).

% csqrt_unique
thf(fact_9633_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_9634_inverse__complex_Osimps_I2_J,axiom,
    ! [X: complex] :
      ( ( im @ ( invers8013647133539491842omplex @ X ) )
      = ( 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.simps(2)
thf(fact_9635_Im__divide,axiom,
    ! [X: complex,Y: complex] :
      ( ( im @ ( divide1717551699836669952omplex @ X @ Y ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( re @ X ) @ ( 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_9636_complex__abs__le__norm,axiom,
    ! [Z2: complex] : ( ord_less_eq_real @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z2 ) ) @ ( abs_abs_real @ ( im @ Z2 ) ) ) @ ( times_times_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ).

% complex_abs_le_norm
thf(fact_9637_complex__unit__circle,axiom,
    ! [Z2: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( plus_plus_real @ ( power_power_real @ ( divide_divide_real @ ( re @ Z2 ) @ ( real_V1022390504157884413omplex @ Z2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( divide_divide_real @ ( im @ Z2 ) @ ( real_V1022390504157884413omplex @ Z2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real ) ) ).

% complex_unit_circle
thf(fact_9638_inverse__complex_Ocode,axiom,
    ( invers8013647133539491842omplex
    = ( ^ [X6: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X6 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X6 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% inverse_complex.code
thf(fact_9639_Complex__divide,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [X6: complex,Y6: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X6 ) @ ( re @ Y6 ) ) @ ( times_times_real @ ( im @ X6 ) @ ( im @ Y6 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X6 ) @ ( re @ Y6 ) ) @ ( times_times_real @ ( re @ X6 ) @ ( im @ Y6 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y6 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% Complex_divide
thf(fact_9640_Im__Reals__divide,axiom,
    ! [R2: complex,Z2: complex] :
      ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
     => ( ( im @ ( divide1717551699836669952omplex @ R2 @ Z2 ) )
        = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R2 ) ) @ ( im @ Z2 ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Im_Reals_divide
thf(fact_9641_Re__Reals__divide,axiom,
    ! [R2: complex,Z2: complex] :
      ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
     => ( ( re @ ( divide1717551699836669952omplex @ R2 @ Z2 ) )
        = ( divide_divide_real @ ( times_times_real @ ( re @ R2 ) @ ( re @ Z2 ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Re_Reals_divide
thf(fact_9642_real__eq__imaginary__iff,axiom,
    ! [Y: complex,X: complex] :
      ( ( member_complex @ Y @ real_V2521375963428798218omplex )
     => ( ( member_complex @ X @ real_V2521375963428798218omplex )
       => ( ( X
            = ( times_times_complex @ imaginary_unit @ Y ) )
          = ( ( X = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% real_eq_imaginary_iff
thf(fact_9643_imaginary__eq__real__iff,axiom,
    ! [Y: complex,X: complex] :
      ( ( member_complex @ Y @ real_V2521375963428798218omplex )
     => ( ( member_complex @ X @ real_V2521375963428798218omplex )
       => ( ( ( times_times_complex @ imaginary_unit @ Y )
            = X )
          = ( ( X = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% imaginary_eq_real_iff
thf(fact_9644_complex__is__Real__iff,axiom,
    ! [Z2: complex] :
      ( ( member_complex @ Z2 @ real_V2521375963428798218omplex )
      = ( ( im @ Z2 )
        = zero_zero_real ) ) ).

% complex_is_Real_iff
thf(fact_9645_Complex__in__Reals,axiom,
    ! [X: real] : ( member_complex @ ( complex2 @ X @ zero_zero_real ) @ real_V2521375963428798218omplex ) ).

% Complex_in_Reals
thf(fact_9646_complex__mult__cnj,axiom,
    ! [Z2: complex] :
      ( ( times_times_complex @ Z2 @ ( cnj @ Z2 ) )
      = ( real_V4546457046886955230omplex @ ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_mult_cnj
thf(fact_9647_divmod__step__integer__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L2: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_9648_complex__cnj__mult,axiom,
    ! [X: complex,Y: complex] :
      ( ( cnj @ ( times_times_complex @ X @ Y ) )
      = ( times_times_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).

% complex_cnj_mult
thf(fact_9649_complex__cnj__zero,axiom,
    ( ( cnj @ zero_zero_complex )
    = zero_zero_complex ) ).

% complex_cnj_zero
thf(fact_9650_complex__cnj__zero__iff,axiom,
    ! [Z2: complex] :
      ( ( ( cnj @ Z2 )
        = zero_zero_complex )
      = ( Z2 = zero_zero_complex ) ) ).

% complex_cnj_zero_iff
thf(fact_9651_complex__In__mult__cnj__zero,axiom,
    ! [Z2: complex] :
      ( ( im @ ( times_times_complex @ Z2 @ ( cnj @ Z2 ) ) )
      = zero_zero_real ) ).

% complex_In_mult_cnj_zero
thf(fact_9652_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_9653_times__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(1)
thf(fact_9654_times__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(2)
thf(fact_9655_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L )
      = ( uminus1351360451143612070nteger @ L ) ) ).

% minus_integer_code(2)
thf(fact_9656_less__eq__integer__code_I1_J,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).

% less_eq_integer_code(1)
thf(fact_9657_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% minus_integer_code(1)
thf(fact_9658_divmod__integer_H__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M4: num,N3: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M4 ) @ ( numera6620942414471956472nteger @ N3 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M4 ) @ ( numera6620942414471956472nteger @ N3 ) ) ) ) ) ).

% divmod_integer'_def
thf(fact_9659_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% plus_integer_code(1)
thf(fact_9660_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L )
      = L ) ).

% plus_integer_code(2)
thf(fact_9661_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
thf(fact_9662_zero__integer_Orsp,axiom,
    zero_zero_int = zero_zero_int ).

% zero_integer.rsp
thf(fact_9663_Re__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ( re @ ( divide1717551699836669952omplex @ A @ B ) )
        = zero_zero_real )
      = ( ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) )
        = zero_zero_real ) ) ).

% Re_complex_div_eq_0
thf(fact_9664_Im__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ( im @ ( divide1717551699836669952omplex @ A @ B ) )
        = zero_zero_real )
      = ( ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) )
        = zero_zero_real ) ) ).

% Im_complex_div_eq_0
thf(fact_9665_complex__mod__sqrt__Re__mult__cnj,axiom,
    ( real_V1022390504157884413omplex
    = ( ^ [Z3: complex] : ( sqrt @ ( re @ ( times_times_complex @ Z3 @ ( cnj @ Z3 ) ) ) ) ) ) ).

% complex_mod_sqrt_Re_mult_cnj
thf(fact_9666_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_9667_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_9668_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_9669_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_9670_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_9671_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_9672_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_9673_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_9674_complex__mod__mult__cnj,axiom,
    ! [Z2: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z2 @ ( cnj @ Z2 ) ) )
      = ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% complex_mod_mult_cnj
thf(fact_9675_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_9676_complex__norm__square,axiom,
    ! [Z2: complex] :
      ( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( times_times_complex @ Z2 @ ( cnj @ Z2 ) ) ) ).

% complex_norm_square
thf(fact_9677_complex__add__cnj,axiom,
    ! [Z2: complex] :
      ( ( plus_plus_complex @ Z2 @ ( cnj @ Z2 ) )
      = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z2 ) ) ) ) ).

% complex_add_cnj
thf(fact_9678_complex__diff__cnj,axiom,
    ! [Z2: complex] :
      ( ( minus_minus_complex @ Z2 @ ( cnj @ Z2 ) )
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z2 ) ) ) @ imaginary_unit ) ) ).

% complex_diff_cnj
thf(fact_9679_complex__div__cnj,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A3: complex,B3: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ ( cnj @ B3 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% complex_div_cnj
thf(fact_9680_cnj__add__mult__eq__Re,axiom,
    ! [Z2: complex,W: complex] :
      ( ( plus_plus_complex @ ( times_times_complex @ Z2 @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z2 ) @ W ) )
      = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z2 @ ( cnj @ W ) ) ) ) ) ) ).

% cnj_add_mult_eq_Re
thf(fact_9681_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_9682_card__Collect__less__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) )
      = N ) ).

% card_Collect_less_nat
thf(fact_9683_card__atMost,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
      = ( suc @ U ) ) ).

% card_atMost
thf(fact_9684_card__Collect__le__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I: nat] : ( ord_less_eq_nat @ I @ N ) ) )
      = ( suc @ N ) ) ).

% card_Collect_le_nat
thf(fact_9685_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_9686_uminus__integer__code_I1_J,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% uminus_integer_code(1)
thf(fact_9687_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_9688_modulo__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ( modulo364778990260209775nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( modulo_modulo_int @ Xa2 @ X ) ) ) ).

% modulo_integer.abs_eq
thf(fact_9689_zero__integer__def,axiom,
    ( zero_z3403309356797280102nteger
    = ( code_integer_of_int @ zero_zero_int ) ) ).

% zero_integer_def
thf(fact_9690_less__integer__code_I1_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).

% less_integer_code(1)
thf(fact_9691_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_9692_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_9693_times__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( times_times_int @ Xa2 @ X ) ) ) ).

% times_integer.abs_eq
thf(fact_9694_card__less__Suc2,axiom,
    ! [M7: set_nat,I3: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ ( suc @ K3 ) @ M7 )
                & ( ord_less_nat @ K3 @ I3 ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I3 ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_9695_card__less__Suc,axiom,
    ! [M7: set_nat,I3: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K3: nat] :
                  ( ( member_nat @ ( suc @ K3 ) @ M7 )
                  & ( ord_less_nat @ K3 @ I3 ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I3 ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_9696_card__less,axiom,
    ! [M7: set_nat,I3: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I3 ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_9697_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
      = ( nat2 @ U ) ) ).

% card_atLeastZeroLessThan_int
thf(fact_9698_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_9699_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).

% less_eq_nat.simps(2)
thf(fact_9700_max__Suc2,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_max_nat @ M @ ( suc @ N ) )
      = ( case_nat_nat @ ( suc @ N )
        @ ^ [M5: nat] : ( suc @ ( ord_max_nat @ M5 @ N ) )
        @ M ) ) ).

% max_Suc2
thf(fact_9701_max__Suc1,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_max_nat @ ( suc @ N ) @ M )
      = ( case_nat_nat @ ( suc @ N )
        @ ^ [M5: nat] : ( suc @ ( ord_max_nat @ N @ M5 ) )
        @ M ) ) ).

% max_Suc1
thf(fact_9702_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N5: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( ord_less_eq_nat @ ( finite_card_nat @ N5 ) @ N ) ) ).

% subset_eq_atLeast0_lessThan_card
thf(fact_9703_card__sum__le__nat__sum,axiom,
    ! [S3: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X6: nat] : X6
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S3 ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X6: nat] : X6
        @ S3 ) ) ).

% card_sum_le_nat_sum
thf(fact_9704_card__nth__roots,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( finite_card_complex
            @ ( collect_complex
              @ ^ [Z3: complex] :
                  ( ( power_power_complex @ Z3 @ N )
                  = C ) ) )
          = N ) ) ) ).

% card_nth_roots
thf(fact_9705_card__roots__unity__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( finite_card_complex
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( power_power_complex @ Z3 @ N )
                = one_one_complex ) ) )
        = N ) ) ).

% card_roots_unity_eq
thf(fact_9706_diff__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [K3: nat] : K3
        @ ( minus_minus_nat @ M @ N ) ) ) ).

% diff_Suc
thf(fact_9707_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_9708_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,S6: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S6 ) ) @ ( S6 = one_one_Code_integer ) )
            @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_cut_integer_code
thf(fact_9709_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_9710_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] :
      ( ( code_divmod_abs @ zero_z3403309356797280102nteger @ J )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ) ).

% divmod_abs_code(6)
thf(fact_9711_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] :
      ( ( code_divmod_abs @ J @ zero_z3403309356797280102nteger )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ J ) ) ) ).

% divmod_abs_code(5)
thf(fact_9712_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_9713_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,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S6 ) ) )
                @ ( 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,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S6 ) ) )
                    @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ).

% divmod_integer_code
thf(fact_9714_pred__def,axiom,
    ( pred
    = ( case_nat_nat @ zero_zero_nat
      @ ^ [X24: nat] : X24 ) ) ).

% pred_def
thf(fact_9715_bezw__0,axiom,
    ! [X: nat] :
      ( ( bezw @ X @ zero_zero_nat )
      = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).

% bezw_0
thf(fact_9716_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_9717_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_9718_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X @ Xa2 )
        = Y )
     => ( ( ( ord_less_eq_nat @ Xa2 @ X )
         => ( Y
            = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa2 @ X )
         => ( Y
            = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_9719_drop__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% drop_bit_nonnegative_int_iff
thf(fact_9720_drop__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% drop_bit_negative_int_iff
thf(fact_9721_drop__bit__minus__one,axiom,
    ! [N: nat] :
      ( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% drop_bit_minus_one
thf(fact_9722_snd__divmod__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( product_snd_nat_nat @ ( divmod_nat @ M @ N ) )
      = ( modulo_modulo_nat @ M @ N ) ) ).

% snd_divmod_nat
thf(fact_9723_drop__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% drop_bit_Suc_minus_bit0
thf(fact_9724_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_9725_drop__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_Suc_minus_bit1
thf(fact_9726_drop__bit__push__bit__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N @ K ) )
      = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N @ M ) @ K ) ) ) ).

% drop_bit_push_bit_int
thf(fact_9727_drop__bit__int__def,axiom,
    ( bit_se8568078237143864401it_int
    = ( ^ [N3: nat,K3: int] : ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% drop_bit_int_def
thf(fact_9728_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K3: nat,M4: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M4 @ K3 ) @ ( product_Pair_nat_nat @ M4 @ ( minus_minus_nat @ K3 @ M4 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M4 @ ( suc @ K3 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_9729_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_9730_vebt__maxt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( ( B5
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B5
                   => ( ( A5
                       => ( Y
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A5
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A5 @ B5 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y
                      = ( some_nat @ Ma2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_maxt.pelims
thf(fact_9731_vebt__mint_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( ( A5
                   => ( Y
                      = ( some_nat @ zero_zero_nat ) ) )
                  & ( ~ A5
                   => ( ( B5
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B5
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A5 @ B5 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y
                      = ( some_nat @ Mi2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_mint.pelims
thf(fact_9732_snd__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( produc6174133586879617921nteger @ ( code_divmod_integer @ K @ L ) )
      = ( modulo364778990260209775nteger @ K @ L ) ) ).

% snd_divmod_integer
thf(fact_9733_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_9734_drop__bit__of__Suc__0,axiom,
    ! [N: nat] :
      ( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% drop_bit_of_Suc_0
thf(fact_9735_fst__divmod__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N ) )
      = ( divide_divide_nat @ M @ N ) ) ).

% fst_divmod_nat
thf(fact_9736_drop__bit__nat__eq,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se8570568707652914677it_nat @ N @ ( nat2 @ K ) )
      = ( nat2 @ ( bit_se8568078237143864401it_int @ N @ K ) ) ) ).

% drop_bit_nat_eq
thf(fact_9737_drop__bit__nat__def,axiom,
    ( bit_se8570568707652914677it_nat
    = ( ^ [N3: nat,M4: nat] : ( divide_divide_nat @ M4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% drop_bit_nat_def
thf(fact_9738_minus__one__mod__numeral,axiom,
    ! [N: num] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
      = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).

% minus_one_mod_numeral
thf(fact_9739_one__mod__minus__numeral,axiom,
    ! [N: num] :
      ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).

% one_mod_minus_numeral
thf(fact_9740_numeral__mod__minus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ) ).

% numeral_mod_minus_numeral
thf(fact_9741_minus__numeral__mod__numeral,axiom,
    ! [M: num,N: num] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).

% minus_numeral_mod_numeral
thf(fact_9742_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_9743_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Y )
     => ( ( bezw @ X @ Y )
        = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ).

% bezw_non_0
thf(fact_9744_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X6: nat,Y6: nat] : ( if_Pro3027730157355071871nt_int @ ( Y6 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y6 @ ( modulo_modulo_nat @ X6 @ Y6 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y6 @ ( modulo_modulo_nat @ X6 @ Y6 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y6 @ ( modulo_modulo_nat @ X6 @ Y6 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X6 @ Y6 ) ) ) ) ) ) ) ) ).

% bezw.simps
thf(fact_9745_bezw_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( ( bezw @ X @ Xa2 )
        = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y
            = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y
            = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) ) ) ).

% bezw.elims
thf(fact_9746_bezw_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( ( bezw @ X @ Xa2 )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y
                  = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y
                  = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).

% bezw.pelims
thf(fact_9747_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: $o] :
      ( ( ( vEBT_VEBT_minNull @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ( ( X
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( Y
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ $true @ Uv2 ) )
               => ( ~ Y
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
           => ( ! [Uu2: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ Uu2 @ $true ) )
                 => ( ~ Y
                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
                   => ( Y
                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                     => ( ~ Y
                       => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
thf(fact_9748_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ! [Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ $true @ Uv2 ) )
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
         => ( ! [Uu2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
thf(fact_9749_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ( ( X
              = ( vEBT_Leaf @ $false @ $false ) )
           => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
thf(fact_9750_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X @ Xa2 )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X )
               => ( Y
                  = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_nat @ Xa2 @ X )
               => ( Y
                  = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).

% prod_decode_aux.pelims
thf(fact_9751_divmod__integer__eq__cases,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L2: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
            @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
              @ ( if_Pro6119634080678213985nteger
                @ ( ( sgn_sgn_Code_integer @ K3 )
                  = ( sgn_sgn_Code_integer @ L2 ) )
                @ ( code_divmod_abs @ K3 @ L2 )
                @ ( produc6916734918728496179nteger
                  @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S6 ) ) )
                  @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ).

% divmod_integer_eq_cases
thf(fact_9752_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N @ K2 )
         => ( P @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
           => ( ! [I4: nat] :
                  ( ( ord_less_nat @ K2 @ I4 )
                 => ( P @ I4 ) )
             => ( P @ K2 ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_9753_xor__minus__numerals_I2_J,axiom,
    ! [K: int,N: num] :
      ( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).

% xor_minus_numerals(2)
thf(fact_9754_xor__minus__numerals_I1_J,axiom,
    ! [N: num,K: int] :
      ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K ) ) ) ).

% xor_minus_numerals(1)
thf(fact_9755_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_9756_sub__BitM__One__eq,axiom,
    ! [N: num] :
      ( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).

% sub_BitM_One_eq
thf(fact_9757_infinite__nat__iff__unbounded,axiom,
    ! [S3: set_nat] :
      ( ( ~ ( finite_finite_nat @ S3 ) )
      = ( ! [M4: nat] :
          ? [N3: nat] :
            ( ( ord_less_nat @ M4 @ N3 )
            & ( member_nat @ N3 @ S3 ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_9758_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set_nat] :
      ( ! [M3: nat] :
          ( ( ord_less_nat @ K @ M3 )
         => ? [N7: nat] :
              ( ( ord_less_nat @ M3 @ N7 )
              & ( member_nat @ N7 @ S3 ) ) )
     => ~ ( finite_finite_nat @ S3 ) ) ).

% unbounded_k_infinite
thf(fact_9759_finite__enumerate,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ? [R: nat > nat] :
          ( ( strict1292158309912662752at_nat @ R @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S3 ) ) )
          & ! [N7: nat] :
              ( ( ord_less_nat @ N7 @ ( finite_card_nat @ S3 ) )
             => ( member_nat @ ( R @ N7 ) @ S3 ) ) ) ) ).

% finite_enumerate
thf(fact_9760_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_9761_Suc__funpow,axiom,
    ! [N: nat] :
      ( ( compow_nat_nat @ N @ suc )
      = ( plus_plus_nat @ N ) ) ).

% Suc_funpow
thf(fact_9762_of__nat__of__integer,axiom,
    ! [K: code_integer] :
      ( ( semiri4939895301339042750nteger @ ( code_nat_of_integer @ K ) )
      = ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ K ) ) ).

% of_nat_of_integer
thf(fact_9763_nat__of__integer__code__post_I1_J,axiom,
    ( ( code_nat_of_integer @ zero_z3403309356797280102nteger )
    = zero_zero_nat ) ).

% nat_of_integer_code_post(1)
thf(fact_9764_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_9765_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,J2: code_integer] : ( if_nat @ ( J2 = 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_9766_max__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
    @ ^ [X6: nat,Y6: nat] : ( ord_less_eq_nat @ Y6 @ X6 )
    @ ^ [X6: nat,Y6: nat] : ( ord_less_nat @ Y6 @ X6 ) ) ).

% max_nat.semilattice_neutr_order_axioms
thf(fact_9767_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,J2: code_integer] : ( if_int @ ( J2 = 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_9768_zero__integer_Orep__eq,axiom,
    ( ( code_int_of_integer @ zero_z3403309356797280102nteger )
    = zero_zero_int ) ).

% zero_integer.rep_eq
thf(fact_9769_times__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X @ Xa2 ) )
      = ( times_times_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).

% times_integer.rep_eq
thf(fact_9770_modulo__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( modulo364778990260209775nteger @ X @ Xa2 ) )
      = ( modulo_modulo_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).

% modulo_integer.rep_eq
thf(fact_9771_times__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X6: nat,Y6: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X6 @ U2 ) @ ( times_times_nat @ Y6 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X6 @ V4 ) @ ( times_times_nat @ Y6 @ U2 ) ) ) )
          @ Xa2
          @ X ) ) ) ).

% times_int.abs_eq
thf(fact_9772_zero__int__def,axiom,
    ( zero_zero_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).

% zero_int_def
thf(fact_9773_int__def,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N3: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N3 @ zero_zero_nat ) ) ) ) ).

% int_def
thf(fact_9774_one__int__def,axiom,
    ( one_one_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).

% one_int_def
thf(fact_9775_less__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X6: nat,Y6: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X6 @ V4 ) @ ( plus_plus_nat @ U2 @ Y6 ) ) )
        @ Xa2
        @ X ) ) ).

% less_int.abs_eq
thf(fact_9776_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X6: nat,Y6: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X6 @ V4 ) @ ( plus_plus_nat @ U2 @ Y6 ) ) )
        @ Xa2
        @ X ) ) ).

% less_eq_int.abs_eq
thf(fact_9777_plus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X6: nat,Y6: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X6 @ U2 ) @ ( plus_plus_nat @ Y6 @ V4 ) ) )
          @ Xa2
          @ X ) ) ) ).

% plus_int.abs_eq
thf(fact_9778_minus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X6: nat,Y6: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X6 @ V4 ) @ ( plus_plus_nat @ Y6 @ U2 ) ) )
          @ Xa2
          @ X ) ) ) ).

% minus_int.abs_eq
thf(fact_9779_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( num_of_nat @ ( suc @ N ) )
          = ( inc @ ( num_of_nat @ N ) ) ) )
      & ( ~ ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( num_of_nat @ ( suc @ N ) )
          = one ) ) ) ).

% num_of_nat.simps(2)
thf(fact_9780_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] :
      ( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
      = Q2 ) ).

% num_of_nat_numeral_eq
thf(fact_9781_num__of__nat_Osimps_I1_J,axiom,
    ( ( num_of_nat @ zero_zero_nat )
    = one ) ).

% num_of_nat.simps(1)
thf(fact_9782_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
        = N ) ) ).

% numeral_num_of_nat
thf(fact_9783_num__of__nat__One,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ one_one_nat )
     => ( ( num_of_nat @ N )
        = one ) ) ).

% num_of_nat_One
thf(fact_9784_num__of__nat__double,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
        = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).

% num_of_nat_double
thf(fact_9785_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
          = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).

% num_of_nat_plus_distrib
thf(fact_9786_less__eq__int_Orep__eq,axiom,
    ( ord_less_eq_int
    = ( ^ [X6: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y6: nat,Z3: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y6 @ V4 ) @ ( plus_plus_nat @ U2 @ Z3 ) ) )
          @ ( rep_Integ @ X6 )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_eq_int.rep_eq
thf(fact_9787_less__int_Orep__eq,axiom,
    ( ord_less_int
    = ( ^ [X6: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y6: nat,Z3: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y6 @ V4 ) @ ( plus_plus_nat @ U2 @ Z3 ) ) )
          @ ( rep_Integ @ X6 )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_int.rep_eq
thf(fact_9788_pow_Osimps_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( pow @ X @ ( bit1 @ Y ) )
      = ( times_times_num @ ( sqr @ ( pow @ X @ Y ) ) @ X ) ) ).

% pow.simps(3)
thf(fact_9789_Gcd__remove0__nat,axiom,
    ! [M7: set_nat] :
      ( ( finite_finite_nat @ M7 )
     => ( ( gcd_Gcd_nat @ M7 )
        = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M7 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).

% Gcd_remove0_nat
thf(fact_9790_sqr_Osimps_I2_J,axiom,
    ! [N: num] :
      ( ( sqr @ ( bit0 @ N ) )
      = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).

% sqr.simps(2)
thf(fact_9791_sqr_Osimps_I1_J,axiom,
    ( ( sqr @ one )
    = one ) ).

% sqr.simps(1)
thf(fact_9792_sqr__conv__mult,axiom,
    ( sqr
    = ( ^ [X6: num] : ( times_times_num @ X6 @ X6 ) ) ) ).

% sqr_conv_mult
thf(fact_9793_pow_Osimps_I2_J,axiom,
    ! [X: num,Y: num] :
      ( ( pow @ X @ ( bit0 @ Y ) )
      = ( sqr @ ( pow @ X @ Y ) ) ) ).

% pow.simps(2)
thf(fact_9794_sqr_Osimps_I3_J,axiom,
    ! [N: num] :
      ( ( sqr @ ( bit1 @ N ) )
      = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).

% sqr.simps(3)
thf(fact_9795_Gcd__int__greater__eq__0,axiom,
    ! [K6: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K6 ) ) ).

% Gcd_int_greater_eq_0
thf(fact_9796_integer__of__num__triv_I2_J,axiom,
    ( ( code_integer_of_num @ ( bit0 @ one ) )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% integer_of_num_triv(2)
thf(fact_9797_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X: nat] :
      ( ( ( ord_less_nat @ C @ Y )
       => ( ( image_nat_nat
            @ ^ [I: nat] : ( minus_minus_nat @ I @ C )
            @ ( set_or4665077453230672383an_nat @ X @ Y ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
      & ( ~ ( ord_less_nat @ C @ Y )
       => ( ( ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I: nat] : ( minus_minus_nat @ I @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I: nat] : ( minus_minus_nat @ I @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_9798_bij__betw__Suc,axiom,
    ! [M7: set_nat,N5: set_nat] :
      ( ( bij_betw_nat_nat @ suc @ M7 @ N5 )
      = ( ( image_nat_nat @ suc @ M7 )
        = N5 ) ) ).

% bij_betw_Suc
thf(fact_9799_image__Suc__atLeastAtMost,axiom,
    ! [I3: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I3 @ J ) )
      = ( set_or1269000886237332187st_nat @ ( suc @ I3 ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastAtMost
thf(fact_9800_image__Suc__atLeastLessThan,axiom,
    ! [I3: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I3 @ J ) )
      = ( set_or4665077453230672383an_nat @ ( suc @ I3 ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastLessThan
thf(fact_9801_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).

% zero_notin_Suc_image
thf(fact_9802_image__Suc__lessThan,axiom,
    ! [N: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).

% image_Suc_lessThan
thf(fact_9803_image__Suc__atMost,axiom,
    ! [N: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).

% image_Suc_atMost
thf(fact_9804_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).

% atLeast0_atMost_Suc_eq_insert_0
thf(fact_9805_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_9806_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% lessThan_Suc_eq_insert_0
thf(fact_9807_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% atMost_Suc_eq_insert_0
thf(fact_9808_integer__of__num__triv_I1_J,axiom,
    ( ( code_integer_of_num @ one )
    = one_one_Code_integer ) ).

% integer_of_num_triv(1)
thf(fact_9809_integer__of__num_I2_J,axiom,
    ! [N: num] :
      ( ( code_integer_of_num @ ( bit0 @ N ) )
      = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).

% integer_of_num(2)
thf(fact_9810_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] :
      ( ( image_int_int
        @ ^ [X6: int] : ( plus_plus_int @ X6 @ L )
        @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
      = ( set_or4662586982721622107an_int @ L @ U ) ) ).

% image_add_int_atLeastLessThan
thf(fact_9811_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_9812_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( case_option_int_num @ zero_zero_int
        @ ^ [Q4: 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 @ Q4 ) ) )
        @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).

% take_bit_numeral_minus_numeral_int
thf(fact_9813_rat__inverse__code,axiom,
    ! [P2: rat] :
      ( ( quotient_of @ ( inverse_inverse_rat @ P2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( A3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A3 ) @ B3 ) @ ( abs_abs_int @ A3 ) ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_inverse_code
thf(fact_9814_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_9815_take__bit__num__simps_I2_J,axiom,
    ! [N: nat] :
      ( ( bit_take_bit_num @ ( suc @ N ) @ one )
      = ( some_num @ one ) ) ).

% take_bit_num_simps(2)
thf(fact_9816_take__bit__num__simps_I5_J,axiom,
    ! [R2: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ one )
      = ( some_num @ one ) ) ).

% take_bit_num_simps(5)
thf(fact_9817_rat__zero__code,axiom,
    ( ( quotient_of @ zero_zero_rat )
    = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% rat_zero_code
thf(fact_9818_divide__rat__def,axiom,
    ( divide_divide_rat
    = ( ^ [Q4: rat,R5: rat] : ( times_times_rat @ Q4 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).

% divide_rat_def
thf(fact_9819_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [N: nat] :
      ( ( bit_take_bit_num @ N @ one )
      = ( case_nat_option_num @ none_num
        @ ^ [N3: nat] : ( some_num @ one )
        @ N ) ) ).

% Code_Abstract_Nat.take_bit_num_code(1)
thf(fact_9820_quotient__of__denom__pos,axiom,
    ! [R2: rat,P2: int,Q2: int] :
      ( ( ( quotient_of @ R2 )
        = ( product_Pair_int_int @ P2 @ Q2 ) )
     => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).

% quotient_of_denom_pos
thf(fact_9821_quotient__of__denom__pos_H,axiom,
    ! [R2: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R2 ) ) ) ).

% quotient_of_denom_pos'
thf(fact_9822_take__bit__num__def,axiom,
    ( bit_take_bit_num
    = ( ^ [N3: nat,M4: num] :
          ( if_option_num
          @ ( ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M4 ) )
            = zero_zero_nat )
          @ none_num
          @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M4 ) ) ) ) ) ) ) ).

% take_bit_num_def
thf(fact_9823_and__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).

% and_minus_numerals(3)
thf(fact_9824_and__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).

% and_minus_numerals(7)
thf(fact_9825_and__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).

% and_minus_numerals(4)
thf(fact_9826_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).

% take_bit_num_simps(4)
thf(fact_9827_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
        @ ( bit_take_bit_num @ N @ M ) ) ) ).

% take_bit_num_simps(3)
thf(fact_9828_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit1 @ M ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ) ).

% take_bit_num_simps(7)
thf(fact_9829_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
        @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ).

% take_bit_num_simps(6)
thf(fact_9830_and__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).

% and_minus_numerals(8)
thf(fact_9831_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_9832_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_9833_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( case_o6005452278849405969um_num @ ( some_num @ one )
        @ ^ [N8: num] : ( some_num @ ( bit1 @ N8 ) )
        @ ( bit_and_not_num @ M @ N ) ) ) ).

% and_not_num.simps(8)
thf(fact_9834_and__not__num_Osimps_I1_J,axiom,
    ( ( bit_and_not_num @ one @ one )
    = none_num ) ).

% and_not_num.simps(1)
thf(fact_9835_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ R2 )
     => ~ ! [S2: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ S2 )
           => ! [T6: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ T6 )
               => ( R2
                 != ( plus_plus_rat @ S2 @ T6 ) ) ) ) ) ).

% obtain_pos_sum
thf(fact_9836_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_9837_and__not__num_Osimps_I2_J,axiom,
    ! [N: num] :
      ( ( bit_and_not_num @ one @ ( bit0 @ N ) )
      = ( some_num @ one ) ) ).

% and_not_num.simps(2)
thf(fact_9838_and__not__num_Osimps_I3_J,axiom,
    ! [N: num] :
      ( ( bit_and_not_num @ one @ ( bit1 @ N ) )
      = none_num ) ).

% and_not_num.simps(3)
thf(fact_9839_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
      = ( case_nat_option_num @ none_num
        @ ^ [N3: nat] :
            ( case_o6005452278849405969um_num @ none_num
            @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
            @ ( bit_take_bit_num @ N3 @ M ) )
        @ N ) ) ).

% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_9840_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_9841_and__not__num__eq__Some__iff,axiom,
    ! [M: num,N: num,Q2: num] :
      ( ( ( bit_and_not_num @ M @ N )
        = ( some_num @ Q2 ) )
      = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
        = ( numeral_numeral_int @ Q2 ) ) ) ).

% and_not_num_eq_Some_iff
thf(fact_9842_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
      = ( case_nat_option_num @ none_num
        @ ^ [N3: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N3 @ M ) ) )
        @ N ) ) ).

% Code_Abstract_Nat.take_bit_num_code(3)
thf(fact_9843_and__not__num__eq__None__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( bit_and_not_num @ M @ N )
        = none_num )
      = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
        = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
thf(fact_9844_int__numeral__not__and__num,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N @ M ) ) ) ).

% int_numeral_not_and_num
thf(fact_9845_int__numeral__and__not__num,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N ) ) ) ).

% int_numeral_and_not_num
thf(fact_9846_rat__divide__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( ( quotient_of @ ( divide_divide_rat @ P2 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,C4: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B3: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ C4 @ B3 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_divide_code
thf(fact_9847_rat__times__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( ( quotient_of @ ( times_times_rat @ P2 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,C4: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B3: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ B3 ) @ ( times_times_int @ C4 @ D2 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_times_code
thf(fact_9848_rat__minus__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( ( quotient_of @ ( minus_minus_rat @ P2 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,C4: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B3: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ B3 @ C4 ) ) @ ( times_times_int @ C4 @ D2 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_minus_code
thf(fact_9849_normalize__denom__zero,axiom,
    ! [P2: int] :
      ( ( normalize @ ( product_Pair_int_int @ P2 @ zero_zero_int ) )
      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% normalize_denom_zero
thf(fact_9850_normalize__negative,axiom,
    ! [Q2: int,P2: int] :
      ( ( ord_less_int @ Q2 @ zero_zero_int )
     => ( ( normalize @ ( product_Pair_int_int @ P2 @ Q2 ) )
        = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P2 ) @ ( uminus_uminus_int @ Q2 ) ) ) ) ) ).

% normalize_negative
thf(fact_9851_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P2: int,Q2: int] :
      ( ( ( normalize @ R2 )
        = ( product_Pair_int_int @ P2 @ Q2 ) )
     => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).

% normalize_denom_pos
thf(fact_9852_normalize__crossproduct,axiom,
    ! [Q2: int,S: int,P2: int,R2: int] :
      ( ( Q2 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( ( normalize @ ( product_Pair_int_int @ P2 @ Q2 ) )
            = ( normalize @ ( product_Pair_int_int @ R2 @ S ) ) )
         => ( ( times_times_int @ P2 @ S )
            = ( times_times_int @ R2 @ Q2 ) ) ) ) ) ).

% normalize_crossproduct
thf(fact_9853_rat__plus__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( ( quotient_of @ ( plus_plus_rat @ P2 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,C4: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B3: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ B3 @ C4 ) ) @ ( times_times_int @ C4 @ D2 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_plus_code
thf(fact_9854_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_9855_and__not__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( ( bit_and_not_num @ X @ Xa2 )
        = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y != none_num ) ) )
       => ( ( ( X = one )
           => ( ? [N2: num] :
                  ( Xa2
                  = ( bit0 @ N2 ) )
             => ( Y
               != ( some_num @ one ) ) ) )
         => ( ( ( X = one )
             => ( ? [N2: num] :
                    ( Xa2
                    = ( bit1 @ N2 ) )
               => ( Y != none_num ) ) )
           => ( ! [M3: num] :
                  ( ( X
                    = ( bit0 @ M3 ) )
                 => ( ( Xa2 = one )
                   => ( Y
                     != ( some_num @ ( bit0 @ M3 ) ) ) ) )
             => ( ! [M3: num] :
                    ( ( X
                      = ( bit0 @ M3 ) )
                   => ! [N2: num] :
                        ( ( Xa2
                          = ( bit0 @ N2 ) )
                       => ( Y
                         != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M3 @ N2 ) ) ) ) )
               => ( ! [M3: num] :
                      ( ( X
                        = ( bit0 @ M3 ) )
                     => ! [N2: num] :
                          ( ( Xa2
                            = ( bit1 @ N2 ) )
                         => ( Y
                           != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M3 @ N2 ) ) ) ) )
                 => ( ! [M3: num] :
                        ( ( X
                          = ( bit1 @ M3 ) )
                       => ( ( Xa2 = one )
                         => ( Y
                           != ( some_num @ ( bit0 @ M3 ) ) ) ) )
                   => ( ! [M3: num] :
                          ( ( X
                            = ( bit1 @ M3 ) )
                         => ! [N2: num] :
                              ( ( Xa2
                                = ( bit0 @ N2 ) )
                             => ( Y
                               != ( case_o6005452278849405969um_num @ ( some_num @ one )
                                  @ ^ [N8: num] : ( some_num @ ( bit1 @ N8 ) )
                                  @ ( bit_and_not_num @ M3 @ N2 ) ) ) ) )
                     => ~ ! [M3: num] :
                            ( ( X
                              = ( bit1 @ M3 ) )
                           => ! [N2: num] :
                                ( ( Xa2
                                  = ( bit1 @ N2 ) )
                               => ( Y
                                 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M3 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
thf(fact_9856_Bit__Operations_Otake__bit__num__code,axiom,
    ( bit_take_bit_num
    = ( ^ [N3: nat,M4: num] :
          ( produc478579273971653890on_num
          @ ^ [A3: nat,X6: num] :
              ( case_nat_option_num @ none_num
              @ ^ [O: nat] :
                  ( case_num_option_num @ ( some_num @ one )
                  @ ^ [P5: num] :
                      ( case_o6005452278849405969um_num @ none_num
                      @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
                      @ ( bit_take_bit_num @ O @ P5 ) )
                  @ ^ [P5: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P5 ) ) )
                  @ X6 )
              @ A3 )
          @ ( product_Pair_nat_num @ N3 @ M4 ) ) ) ) ).

% Bit_Operations.take_bit_num_code
thf(fact_9857_gcd__pos__int,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
      = ( ( M != zero_zero_int )
        | ( N != zero_zero_int ) ) ) ).

% gcd_pos_int
thf(fact_9858_gcd__0__left__int,axiom,
    ! [X: int] :
      ( ( gcd_gcd_int @ zero_zero_int @ X )
      = ( abs_abs_int @ X ) ) ).

% gcd_0_left_int
thf(fact_9859_gcd__0__int,axiom,
    ! [X: int] :
      ( ( gcd_gcd_int @ X @ zero_zero_int )
      = ( abs_abs_int @ X ) ) ).

% gcd_0_int
thf(fact_9860_gcd__mult__distrib__int,axiom,
    ! [K: int,M: int,N: int] :
      ( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N ) )
      = ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) ) ) ).

% gcd_mult_distrib_int
thf(fact_9861_bezout__int,axiom,
    ! [X: int,Y: int] :
    ? [U3: int,V2: int] :
      ( ( plus_plus_int @ ( times_times_int @ U3 @ X ) @ ( times_times_int @ V2 @ Y ) )
      = ( gcd_gcd_int @ X @ Y ) ) ).

% bezout_int
thf(fact_9862_gcd__red__int,axiom,
    ( gcd_gcd_int
    = ( ^ [X6: int,Y6: int] : ( gcd_gcd_int @ Y6 @ ( modulo_modulo_int @ X6 @ Y6 ) ) ) ) ).

% gcd_red_int
thf(fact_9863_gcd__ge__0__int,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X @ Y ) ) ).

% gcd_ge_0_int
thf(fact_9864_and__not__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).

% and_not_num.simps(5)
thf(fact_9865_and__not__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).

% and_not_num.simps(6)
thf(fact_9866_and__not__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).

% and_not_num.simps(9)
thf(fact_9867_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_9868_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_9869_gcd__cases__int,axiom,
    ! [X: int,Y: int,P: int > $o] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( P @ ( gcd_gcd_int @ X @ Y ) ) ) )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
         => ( ( ord_less_eq_int @ Y @ zero_zero_int )
           => ( P @ ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y ) ) ) ) )
       => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
           => ( ( ord_less_eq_int @ zero_zero_int @ Y )
             => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y ) ) ) )
         => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
             => ( ( ord_less_eq_int @ Y @ zero_zero_int )
               => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y ) ) ) ) )
           => ( P @ ( gcd_gcd_int @ X @ Y ) ) ) ) ) ) ).

% gcd_cases_int
thf(fact_9870_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_9871_gcd__non__0__int,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ Y )
     => ( ( gcd_gcd_int @ X @ Y )
        = ( gcd_gcd_int @ Y @ ( modulo_modulo_int @ X @ Y ) ) ) ) ).

% gcd_non_0_int
thf(fact_9872_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_9873_and__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( ( bit_un7362597486090784418nd_num @ X @ Xa2 )
        = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y
             != ( some_num @ one ) ) ) )
       => ( ( ( X = one )
           => ( ? [N2: num] :
                  ( Xa2
                  = ( bit0 @ N2 ) )
             => ( Y != none_num ) ) )
         => ( ( ( X = one )
             => ( ? [N2: num] :
                    ( Xa2
                    = ( bit1 @ N2 ) )
               => ( Y
                 != ( some_num @ one ) ) ) )
           => ( ( ? [M3: num] :
                    ( X
                    = ( bit0 @ M3 ) )
               => ( ( Xa2 = one )
                 => ( Y != none_num ) ) )
             => ( ! [M3: num] :
                    ( ( X
                      = ( bit0 @ M3 ) )
                   => ! [N2: num] :
                        ( ( Xa2
                          = ( bit0 @ N2 ) )
                       => ( Y
                         != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N2 ) ) ) ) )
               => ( ! [M3: num] :
                      ( ( X
                        = ( bit0 @ M3 ) )
                     => ! [N2: num] :
                          ( ( Xa2
                            = ( bit1 @ N2 ) )
                         => ( Y
                           != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N2 ) ) ) ) )
                 => ( ( ? [M3: num] :
                          ( X
                          = ( bit1 @ M3 ) )
                     => ( ( Xa2 = one )
                       => ( Y
                         != ( some_num @ one ) ) ) )
                   => ( ! [M3: num] :
                          ( ( X
                            = ( bit1 @ M3 ) )
                         => ! [N2: num] :
                              ( ( Xa2
                                = ( bit0 @ N2 ) )
                             => ( Y
                               != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N2 ) ) ) ) )
                     => ~ ! [M3: num] :
                            ( ( X
                              = ( bit1 @ M3 ) )
                           => ! [N2: num] :
                                ( ( Xa2
                                  = ( bit1 @ N2 ) )
                               => ( Y
                                 != ( case_o6005452278849405969um_num @ ( some_num @ one )
                                    @ ^ [N8: num] : ( some_num @ ( bit1 @ N8 ) )
                                    @ ( bit_un7362597486090784418nd_num @ M3 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
thf(fact_9874_xor__num_Oelims,axiom,
    ! [X: num,Xa2: num,Y: option_num] :
      ( ( ( bit_un2480387367778600638or_num @ X @ Xa2 )
        = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y != none_num ) ) )
       => ( ( ( X = one )
           => ! [N2: num] :
                ( ( Xa2
                  = ( bit0 @ N2 ) )
               => ( Y
                 != ( some_num @ ( bit1 @ N2 ) ) ) ) )
         => ( ( ( X = one )
             => ! [N2: num] :
                  ( ( Xa2
                    = ( bit1 @ N2 ) )
                 => ( Y
                   != ( some_num @ ( bit0 @ N2 ) ) ) ) )
           => ( ! [M3: num] :
                  ( ( X
                    = ( bit0 @ M3 ) )
                 => ( ( Xa2 = one )
                   => ( Y
                     != ( some_num @ ( bit1 @ M3 ) ) ) ) )
             => ( ! [M3: num] :
                    ( ( X
                      = ( bit0 @ M3 ) )
                   => ! [N2: num] :
                        ( ( Xa2
                          = ( bit0 @ N2 ) )
                       => ( Y
                         != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N2 ) ) ) ) )
               => ( ! [M3: num] :
                      ( ( X
                        = ( bit0 @ M3 ) )
                     => ! [N2: num] :
                          ( ( Xa2
                            = ( bit1 @ N2 ) )
                         => ( Y
                           != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N2 ) ) ) ) ) )
                 => ( ! [M3: num] :
                        ( ( X
                          = ( bit1 @ M3 ) )
                       => ( ( Xa2 = one )
                         => ( Y
                           != ( some_num @ ( bit0 @ M3 ) ) ) ) )
                   => ( ! [M3: num] :
                          ( ( X
                            = ( bit1 @ M3 ) )
                         => ! [N2: num] :
                              ( ( Xa2
                                = ( bit0 @ N2 ) )
                             => ( Y
                               != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N2 ) ) ) ) ) )
                     => ~ ! [M3: num] :
                            ( ( X
                              = ( bit1 @ M3 ) )
                           => ! [N2: num] :
                                ( ( Xa2
                                  = ( bit1 @ N2 ) )
                               => ( Y
                                 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
thf(fact_9875_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( gcd_gcd_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.eq_neutr_iff
thf(fact_9876_gcd__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( gcd_gcd_nat @ zero_zero_nat @ A )
      = A ) ).

% gcd_nat.left_neutral
thf(fact_9877_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat
        = ( gcd_gcd_nat @ A @ B ) )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.neutr_eq_iff
thf(fact_9878_gcd__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( gcd_gcd_nat @ A @ zero_zero_nat )
      = A ) ).

% gcd_nat.right_neutral
thf(fact_9879_gcd__0__nat,axiom,
    ! [X: nat] :
      ( ( gcd_gcd_nat @ X @ zero_zero_nat )
      = X ) ).

% gcd_0_nat
thf(fact_9880_gcd__0__left__nat,axiom,
    ! [X: nat] :
      ( ( gcd_gcd_nat @ zero_zero_nat @ X )
      = X ) ).

% gcd_0_left_nat
thf(fact_9881_gcd__Suc__0,axiom,
    ! [M: nat] :
      ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
      = ( suc @ zero_zero_nat ) ) ).

% gcd_Suc_0
thf(fact_9882_gcd__pos__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
      = ( ( M != zero_zero_nat )
        | ( N != zero_zero_nat ) ) ) ).

% gcd_pos_nat
thf(fact_9883_gcd__red__nat,axiom,
    ( gcd_gcd_nat
    = ( ^ [X6: nat,Y6: nat] : ( gcd_gcd_nat @ Y6 @ ( modulo_modulo_nat @ X6 @ Y6 ) ) ) ) ).

% gcd_red_nat
thf(fact_9884_gcd__non__0__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero_nat )
     => ( ( gcd_gcd_nat @ X @ Y )
        = ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ).

% gcd_non_0_nat
thf(fact_9885_gcd__nat_Osimps,axiom,
    ( gcd_gcd_nat
    = ( ^ [X6: nat,Y6: nat] : ( if_nat @ ( Y6 = zero_zero_nat ) @ X6 @ ( gcd_gcd_nat @ Y6 @ ( modulo_modulo_nat @ X6 @ Y6 ) ) ) ) ) ).

% gcd_nat.simps
thf(fact_9886_gcd__nat_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( ( gcd_gcd_nat @ X @ Xa2 )
        = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y = X ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y
            = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) ) ) ) ).

% gcd_nat.elims
thf(fact_9887_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_9888_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_9889_gcd__mult__distrib__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N ) )
      = ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% gcd_mult_distrib_nat
thf(fact_9890_Gcd__in,axiom,
    ! [A2: set_nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( member_nat @ A5 @ A2 )
         => ( ( member_nat @ B5 @ A2 )
           => ( member_nat @ ( gcd_gcd_nat @ A5 @ B5 ) @ A2 ) ) )
     => ( ( A2 != bot_bot_set_nat )
       => ( member_nat @ ( gcd_Gcd_nat @ A2 ) @ A2 ) ) ) ).

% Gcd_in
thf(fact_9891_and__num_Osimps_I1_J,axiom,
    ( ( bit_un7362597486090784418nd_num @ one @ one )
    = ( some_num @ one ) ) ).

% and_num.simps(1)
thf(fact_9892_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X5: nat,Y4: nat] :
          ( ( times_times_nat @ A @ X5 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_nat
thf(fact_9893_xor__num_Osimps_I1_J,axiom,
    ( ( bit_un2480387367778600638or_num @ one @ one )
    = none_num ) ).

% xor_num.simps(1)
thf(fact_9894_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X5: nat,Y4: nat] :
      ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y4 ) @ ( times_times_nat @ A @ X5 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ A @ X5 ) @ ( times_times_nat @ B @ Y4 ) )
          = ( gcd_gcd_nat @ A @ B ) ) )
      | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y4 ) @ ( times_times_nat @ B @ X5 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ B @ X5 ) @ ( times_times_nat @ A @ Y4 ) )
          = ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_gcd_nat'
thf(fact_9895_xor__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).

% xor_num.simps(5)
thf(fact_9896_and__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).

% and_num.simps(5)
thf(fact_9897_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_9898_and__num_Osimps_I7_J,axiom,
    ! [M: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
      = ( some_num @ one ) ) ).

% and_num.simps(7)
thf(fact_9899_and__num_Osimps_I3_J,axiom,
    ! [N: num] :
      ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N ) )
      = ( some_num @ one ) ) ).

% and_num.simps(3)
thf(fact_9900_and__num_Osimps_I2_J,axiom,
    ! [N: num] :
      ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N ) )
      = none_num ) ).

% and_num.simps(2)
thf(fact_9901_and__num_Osimps_I4_J,axiom,
    ! [M: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
      = none_num ) ).

% and_num.simps(4)
thf(fact_9902_and__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).

% and_num.simps(8)
thf(fact_9903_and__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).

% and_num.simps(6)
thf(fact_9904_xor__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).

% xor_num.simps(9)
thf(fact_9905_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
    @ ^ [M4: nat,N3: nat] :
        ( ( dvd_dvd_nat @ M4 @ N3 )
        & ( M4 != N3 ) ) ) ).

% gcd_nat.semilattice_neutr_order_axioms
thf(fact_9906_xor__num_Osimps_I2_J,axiom,
    ! [N: num] :
      ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N ) )
      = ( some_num @ ( bit1 @ N ) ) ) ).

% xor_num.simps(2)
thf(fact_9907_xor__num_Osimps_I3_J,axiom,
    ! [N: num] :
      ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N ) )
      = ( some_num @ ( bit0 @ N ) ) ) ).

% xor_num.simps(3)
thf(fact_9908_xor__num_Osimps_I4_J,axiom,
    ! [M: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
      = ( some_num @ ( bit1 @ M ) ) ) ).

% xor_num.simps(4)
thf(fact_9909_xor__num_Osimps_I7_J,axiom,
    ! [M: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
      = ( some_num @ ( bit0 @ M ) ) ) ).

% xor_num.simps(7)
thf(fact_9910_and__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( case_o6005452278849405969um_num @ ( some_num @ one )
        @ ^ [N8: num] : ( some_num @ ( bit1 @ N8 ) )
        @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).

% and_num.simps(9)
thf(fact_9911_bezw__aux,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X @ Y ) )
      = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ) ).

% bezw_aux
thf(fact_9912_xor__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).

% xor_num.simps(8)
thf(fact_9913_xor__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).

% xor_num.simps(6)
thf(fact_9914_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( ( gcd_gcd_nat @ X @ Xa2 )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y = X ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y
                  = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).

% gcd_nat.pelims
thf(fact_9915_xor__num__dict,axiom,
    bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).

% xor_num_dict
thf(fact_9916_and__num__dict,axiom,
    bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).

% and_num_dict
thf(fact_9917_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,J2: code_integer] : ( if_num @ ( J2 = 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_9918_Sup__nat__empty,axiom,
    ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Sup_nat_empty
thf(fact_9919_Inf__real__def,axiom,
    ( comple4887499456419720421f_real
    = ( ^ [X4: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X4 ) ) ) ) ) ).

% Inf_real_def
thf(fact_9920_Inf__nat__def1,axiom,
    ! [K6: set_nat] :
      ( ( K6 != bot_bot_set_nat )
     => ( member_nat @ ( complete_Inf_Inf_nat @ K6 ) @ K6 ) ) ).

% Inf_nat_def1
thf(fact_9921_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_9922_binomial__def,axiom,
    ( binomial
    = ( ^ [N3: nat,K3: nat] :
          ( finite_card_set_nat
          @ ( collect_set_nat
            @ ^ [K7: set_nat] :
                ( ( member_set_nat @ K7 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) )
                & ( ( finite_card_nat @ K7 )
                  = K3 ) ) ) ) ) ) ).

% binomial_def
thf(fact_9923_range__mod,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( image_nat_nat
          @ ^ [M4: nat] : ( modulo_modulo_nat @ M4 @ N )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).

% range_mod
thf(fact_9924_suminf__eq__SUP__real,axiom,
    ! [X9: nat > real] :
      ( ( summable_real @ X9 )
     => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X9 @ I2 ) )
       => ( ( suminf_real @ X9 )
          = ( comple1385675409528146559p_real
            @ ( image_nat_real
              @ ^ [I: nat] : ( groups6591440286371151544t_real @ X9 @ ( set_ord_lessThan_nat @ I ) )
              @ top_top_set_nat ) ) ) ) ) ).

% suminf_eq_SUP_real
thf(fact_9925_card__UNIV__bool,axiom,
    ( ( finite_card_o @ top_top_set_o )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% card_UNIV_bool
thf(fact_9926_range__mult,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
          = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
      & ( ( A != zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
          = top_top_set_real ) ) ) ).

% range_mult
thf(fact_9927_root__def,axiom,
    ( root
    = ( ^ [N3: nat,X6: real] :
          ( if_real @ ( N3 = zero_zero_nat ) @ zero_zero_real
          @ ( the_in5290026491893676941l_real @ top_top_set_real
            @ ^ [Y6: real] : ( times_times_real @ ( sgn_sgn_real @ Y6 ) @ ( power_power_real @ ( abs_abs_real @ Y6 ) @ N3 ) )
            @ X6 ) ) ) ) ).

% root_def
thf(fact_9928_UNIV__bool,axiom,
    ( top_top_set_o
    = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% UNIV_bool
thf(fact_9929_rat__less__eq__code,axiom,
    ( ord_less_eq_rat
    = ( ^ [P5: rat,Q4: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A3: int,C4: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B3: int,D2: int] : ( ord_less_eq_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ C4 @ B3 ) )
              @ ( quotient_of @ Q4 ) )
          @ ( quotient_of @ P5 ) ) ) ) ).

% rat_less_eq_code
thf(fact_9930_rat__less__code,axiom,
    ( ord_less_rat
    = ( ^ [P5: rat,Q4: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A3: int,C4: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B3: int,D2: int] : ( ord_less_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ C4 @ B3 ) )
              @ ( quotient_of @ Q4 ) )
          @ ( quotient_of @ P5 ) ) ) ) ).

% rat_less_code
thf(fact_9931_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_9932_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_9933_char_Osize_I2_J,axiom,
    ! [X15: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
      ( ( size_size_char @ ( char2 @ X15 @ X2 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
      = zero_zero_nat ) ).

% char.size(2)
thf(fact_9934_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_9935_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_9936_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_9937_char_Osize__gen,axiom,
    ! [X15: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
      ( ( size_char @ ( char2 @ X15 @ X2 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
      = zero_zero_nat ) ).

% char.size_gen
thf(fact_9938_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_9939_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_9940_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_9941_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I: int,J2: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J2 @ I ) @ Js @ ( upto_aux @ I @ ( minus_minus_int @ J2 @ one_one_int ) @ ( cons_int @ J2 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_9942_upto_Opsimps,axiom,
    ! [I3: int,J: int] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J ) )
     => ( ( ( ord_less_eq_int @ I3 @ J )
         => ( ( upto @ I3 @ J )
            = ( cons_int @ I3 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J ) ) ) )
        & ( ~ ( ord_less_eq_int @ I3 @ J )
         => ( ( upto @ I3 @ J )
            = nil_int ) ) ) ) ).

% upto.psimps
thf(fact_9943_upto_Opelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( ( upto @ X @ Xa2 )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_int @ X @ Xa2 )
               => ( Y
                  = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_int @ X @ Xa2 )
               => ( Y = nil_int ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).

% upto.pelims
thf(fact_9944_upto__empty,axiom,
    ! [J: int,I3: int] :
      ( ( ord_less_int @ J @ I3 )
     => ( ( upto @ I3 @ J )
        = nil_int ) ) ).

% upto_empty
thf(fact_9945_upto__Nil2,axiom,
    ! [I3: int,J: int] :
      ( ( nil_int
        = ( upto @ I3 @ J ) )
      = ( ord_less_int @ J @ I3 ) ) ).

% upto_Nil2
thf(fact_9946_upto__Nil,axiom,
    ! [I3: int,J: int] :
      ( ( ( upto @ I3 @ J )
        = nil_int )
      = ( ord_less_int @ J @ I3 ) ) ).

% upto_Nil
thf(fact_9947_upto__single,axiom,
    ! [I3: int] :
      ( ( upto @ I3 @ I3 )
      = ( cons_int @ I3 @ nil_int ) ) ).

% upto_single
thf(fact_9948_nth__upto,axiom,
    ! [I3: int,K: nat,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I3 @ ( semiri1314217659103216013at_int @ K ) ) @ J )
     => ( ( nth_int @ ( upto @ I3 @ J ) @ K )
        = ( plus_plus_int @ I3 @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).

% nth_upto
thf(fact_9949_length__upto,axiom,
    ! [I3: int,J: int] :
      ( ( size_size_list_int @ ( upto @ I3 @ J ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I3 ) @ one_one_int ) ) ) ).

% length_upto
thf(fact_9950_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(1)
thf(fact_9951_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(4)
thf(fact_9952_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(3)
thf(fact_9953_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(2)
thf(fact_9954_upto__aux__def,axiom,
    ( upto_aux
    = ( ^ [I: int,J2: int] : ( append_int @ ( upto @ I @ J2 ) ) ) ) ).

% upto_aux_def
thf(fact_9955_upto__code,axiom,
    ( upto
    = ( ^ [I: int,J2: int] : ( upto_aux @ I @ J2 @ nil_int ) ) ) ).

% upto_code
thf(fact_9956_atLeastAtMost__upto,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I: int,J2: int] : ( set_int2 @ ( upto @ I @ J2 ) ) ) ) ).

% atLeastAtMost_upto
thf(fact_9957_distinct__upto,axiom,
    ! [I3: int,J: int] : ( distinct_int @ ( upto @ I3 @ J ) ) ).

% distinct_upto
thf(fact_9958_upto__split2,axiom,
    ! [I3: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I3 @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I3 @ K )
          = ( append_int @ ( upto @ I3 @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).

% upto_split2
thf(fact_9959_upto__split1,axiom,
    ! [I3: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I3 @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I3 @ K )
          = ( append_int @ ( upto @ I3 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).

% upto_split1
thf(fact_9960_atLeastLessThan__upto,axiom,
    ( set_or4662586982721622107an_int
    = ( ^ [I: int,J2: int] : ( set_int2 @ ( upto @ I @ ( minus_minus_int @ J2 @ one_one_int ) ) ) ) ) ).

% atLeastLessThan_upto
thf(fact_9961_upto__rec1,axiom,
    ! [I3: int,J: int] :
      ( ( ord_less_eq_int @ I3 @ J )
     => ( ( upto @ I3 @ J )
        = ( cons_int @ I3 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J ) ) ) ) ).

% upto_rec1
thf(fact_9962_upto_Osimps,axiom,
    ( upto
    = ( ^ [I: int,J2: int] : ( if_list_int @ ( ord_less_eq_int @ I @ J2 ) @ ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J2 ) ) @ nil_int ) ) ) ).

% upto.simps
thf(fact_9963_upto_Oelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( ( upto @ X @ Xa2 )
        = Y )
     => ( ( ( ord_less_eq_int @ X @ Xa2 )
         => ( Y
            = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_int @ X @ Xa2 )
         => ( Y = nil_int ) ) ) ) ).

% upto.elims
thf(fact_9964_upto__rec2,axiom,
    ! [I3: int,J: int] :
      ( ( ord_less_eq_int @ I3 @ J )
     => ( ( upto @ I3 @ J )
        = ( append_int @ ( upto @ I3 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).

% upto_rec2
thf(fact_9965_upto__split3,axiom,
    ! [I3: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I3 @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I3 @ K )
          = ( append_int @ ( upto @ I3 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).

% upto_split3
thf(fact_9966_DERIV__even__real__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( ord_less_real @ X @ zero_zero_real )
         => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_even_real_root
thf(fact_9967_DERIV__real__root__generic,axiom,
    ! [N: nat,X: real,D4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( X != zero_zero_real )
       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
           => ( ( ord_less_real @ zero_zero_real @ X )
             => ( D4
                = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
             => ( ( ord_less_real @ X @ zero_zero_real )
               => ( D4
                  = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
           => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
               => ( D4
                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
             => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D4 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).

% DERIV_real_root_generic
thf(fact_9968_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 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
                  & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
       => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).

% DERIV_neg_imp_decreasing
thf(fact_9969_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 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
       => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).

% DERIV_pos_imp_increasing
thf(fact_9970_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_9971_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 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ zero_zero_real @ Y5 ) ) ) )
       => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).

% DERIV_nonneg_imp_nondecreasing
thf(fact_9972_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 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ Y5 @ zero_zero_real ) ) ) )
       => ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).

% DERIV_nonpos_imp_nonincreasing
thf(fact_9973_DERIV__neg__dec__right,axiom,
    ! [F: real > real,L: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H3: real] :
                ( ( ord_less_real @ zero_zero_real @ H3 )
               => ( ( ord_less_real @ H3 @ D3 )
                 => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H3 ) ) @ ( F @ X ) ) ) ) ) ) ) ).

% DERIV_neg_dec_right
thf(fact_9974_DERIV__pos__inc__right,axiom,
    ! [F: real > real,L: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H3: real] :
                ( ( ord_less_real @ zero_zero_real @ H3 )
               => ( ( ord_less_real @ H3 @ D3 )
                 => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H3 ) ) ) ) ) ) ) ) ).

% DERIV_pos_inc_right
thf(fact_9975_DERIV__pos__inc__left,axiom,
    ! [F: real > real,L: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H3: real] :
                ( ( ord_less_real @ zero_zero_real @ H3 )
               => ( ( ord_less_real @ H3 @ D3 )
                 => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H3 ) ) @ ( F @ X ) ) ) ) ) ) ) ).

% DERIV_pos_inc_left
thf(fact_9976_DERIV__neg__dec__left,axiom,
    ! [F: real > real,L: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H3: real] :
                ( ( ord_less_real @ zero_zero_real @ H3 )
               => ( ( ord_less_real @ H3 @ D3 )
                 => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H3 ) ) ) ) ) ) ) ) ).

% DERIV_neg_dec_left
thf(fact_9977_DERIV__const__ratio__const,axiom,
    ! [A: real,B: real,F: real > real,K: real] :
      ( ( A != B )
     => ( ! [X5: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
       => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
          = ( times_times_real @ ( minus_minus_real @ B @ A ) @ K ) ) ) ) ).

% DERIV_const_ratio_const
thf(fact_9978_DERIV__isconst__all,axiom,
    ! [F: real > real,X: real,Y: real] :
      ( ! [X5: real] : ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
     => ( ( F @ X )
        = ( F @ Y ) ) ) ).

% DERIV_isconst_all
thf(fact_9979_has__real__derivative__pos__inc__right,axiom,
    ! [F: real > real,L: real,X: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S3 ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H3: real] :
                ( ( ord_less_real @ zero_zero_real @ H3 )
               => ( ( member_real @ ( plus_plus_real @ X @ H3 ) @ S3 )
                 => ( ( ord_less_real @ H3 @ D3 )
                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H3 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
thf(fact_9980_has__real__derivative__neg__dec__right,axiom,
    ! [F: real > real,L: real,X: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S3 ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H3: real] :
                ( ( ord_less_real @ zero_zero_real @ H3 )
               => ( ( member_real @ ( plus_plus_real @ X @ H3 ) @ S3 )
                 => ( ( ord_less_real @ H3 @ D3 )
                   => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H3 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
thf(fact_9981_has__real__derivative__neg__dec__left,axiom,
    ! [F: real > real,L: real,X: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S3 ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H3: real] :
                ( ( ord_less_real @ zero_zero_real @ H3 )
               => ( ( member_real @ ( minus_minus_real @ X @ H3 ) @ S3 )
                 => ( ( ord_less_real @ H3 @ D3 )
                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H3 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
thf(fact_9982_has__real__derivative__pos__inc__left,axiom,
    ! [F: real > real,L: real,X: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ S3 ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H3: real] :
                ( ( ord_less_real @ zero_zero_real @ H3 )
               => ( ( member_real @ ( minus_minus_real @ X @ H3 ) @ S3 )
                 => ( ( ord_less_real @ H3 @ D3 )
                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H3 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
thf(fact_9983_DERIV__local__const,axiom,
    ! [F: real > real,L: real,X: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y4: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
             => ( ( F @ X )
                = ( F @ Y4 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
thf(fact_9984_MVT2,axiom,
    ! [A: real,B: real,F: real > real,F5: 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 @ ( F5 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
       => ? [Z4: real] :
            ( ( ord_less_real @ A @ Z4 )
            & ( ord_less_real @ Z4 @ B )
            & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
              = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F5 @ Z4 ) ) ) ) ) ) ).

% MVT2
thf(fact_9985_DERIV__ln,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_ln
thf(fact_9986_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_9987_DERIV__local__max,axiom,
    ! [F: real > real,L: real,X: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y4: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
thf(fact_9988_DERIV__local__min,axiom,
    ! [F: real > real,L: real,X: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y4: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
thf(fact_9989_DERIV__ln__divide,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_ln_divide
thf(fact_9990_DERIV__pow,axiom,
    ! [N: nat,X: real,S: set_real] :
      ( has_fi5821293074295781190e_real
      @ ^ [X6: real] : ( power_power_real @ X6 @ N )
      @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
      @ ( topolo2177554685111907308n_real @ X @ S ) ) ).

% DERIV_pow
thf(fact_9991_DERIV__fun__pow,axiom,
    ! [G: real > real,M: real,X: real,N: nat] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( has_fi5821293074295781190e_real
        @ ^ [X6: real] : ( power_power_real @ ( G @ X6 ) @ N )
        @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_fun_pow
thf(fact_9992_has__real__derivative__powr,axiom,
    ! [Z2: real,R2: real] :
      ( ( ord_less_real @ zero_zero_real @ Z2 )
     => ( has_fi5821293074295781190e_real
        @ ^ [Z3: real] : ( powr_real @ Z3 @ R2 )
        @ ( times_times_real @ R2 @ ( powr_real @ Z2 @ ( minus_minus_real @ R2 @ one_one_real ) ) )
        @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) ).

% has_real_derivative_powr
thf(fact_9993_DERIV__log,axiom,
    ! [X: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_log
thf(fact_9994_DERIV__fun__powr,axiom,
    ! [G: real > real,M: real,X: real,R2: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
       => ( has_fi5821293074295781190e_real
          @ ^ [X6: real] : ( powr_real @ ( G @ X6 ) @ R2 )
          @ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_fun_powr
thf(fact_9995_DERIV__powr,axiom,
    ! [G: real > real,M: real,X: real,F: real > real,R2: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
       => ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( has_fi5821293074295781190e_real
            @ ^ [X6: real] : ( powr_real @ ( G @ X6 ) @ ( F @ X6 ) )
            @ ( times_times_real @ ( powr_real @ ( G @ X ) @ ( F @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G @ X ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X ) ) @ ( G @ X ) ) ) )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_powr
thf(fact_9996_artanh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).

% artanh_real_has_field_derivative
thf(fact_9997_DERIV__real__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_real_sqrt
thf(fact_9998_DERIV__arctan,axiom,
    ! [X: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ).

% DERIV_arctan
thf(fact_9999_arsinh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ).

% arsinh_real_has_field_derivative
thf(fact_10000_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D4: real] :
      ( ( X != zero_zero_real )
     => ( ( ( ord_less_real @ zero_zero_real @ X )
         => ( D4
            = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( ( ord_less_real @ X @ zero_zero_real )
           => ( D4
              = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
         => ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_real_sqrt_generic
thf(fact_10001_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).

% arcosh_real_has_field_derivative
thf(fact_10002_DERIV__real__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_real_root
thf(fact_10003_DERIV__arccos,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_arccos
thf(fact_10004_DERIV__arcsin,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_arcsin
thf(fact_10005_Maclaurin__all__le__objl,axiom,
    ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
      ( ( ( ( Diff @ zero_zero_nat )
          = F )
        & ! [M3: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
     => ? [T6: real] :
          ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
          & ( ( F @ X )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M4 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M4 ) ) @ ( power_power_real @ X @ M4 ) )
                @ ( set_ord_lessThan_nat @ N ) )
              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).

% Maclaurin_all_le_objl
thf(fact_10006_Maclaurin__all__le,axiom,
    ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M3: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
       => ? [T6: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
            & ( ( F @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M4 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M4 ) ) @ ( power_power_real @ X @ M4 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_all_le
thf(fact_10007_DERIV__odd__real__root,axiom,
    ! [N: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( X != zero_zero_real )
       => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_odd_real_root
thf(fact_10008_Maclaurin,axiom,
    ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M3: nat,T6: real] :
                ( ( ( ord_less_nat @ M3 @ N )
                  & ( ord_less_eq_real @ zero_zero_real @ T6 )
                  & ( ord_less_eq_real @ T6 @ H2 ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
           => ? [T6: real] :
                ( ( ord_less_real @ zero_zero_real @ T6 )
                & ( ord_less_real @ T6 @ H2 )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M4 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M4 ) ) @ ( power_power_real @ H2 @ M4 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin
thf(fact_10009_Maclaurin2,axiom,
    ! [H2: real,Diff: nat > real > real,F: real > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M3: nat,T6: real] :
              ( ( ( ord_less_nat @ M3 @ N )
                & ( ord_less_eq_real @ zero_zero_real @ T6 )
                & ( ord_less_eq_real @ T6 @ H2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ? [T6: real] :
              ( ( ord_less_real @ zero_zero_real @ T6 )
              & ( ord_less_eq_real @ T6 @ H2 )
              & ( ( F @ H2 )
                = ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [M4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M4 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M4 ) ) @ ( power_power_real @ H2 @ M4 ) )
                    @ ( set_ord_lessThan_nat @ N ) )
                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).

% Maclaurin2
thf(fact_10010_Maclaurin__minus,axiom,
    ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ H2 @ zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M3: nat,T6: real] :
                ( ( ( ord_less_nat @ M3 @ N )
                  & ( ord_less_eq_real @ H2 @ T6 )
                  & ( ord_less_eq_real @ T6 @ zero_zero_real ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
           => ? [T6: real] :
                ( ( ord_less_real @ H2 @ T6 )
                & ( ord_less_real @ T6 @ zero_zero_real )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M4 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M4 ) ) @ ( power_power_real @ H2 @ M4 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin_minus
thf(fact_10011_Maclaurin__all__lt,axiom,
    ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( X != zero_zero_real )
         => ( ! [M3: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
           => ? [T6: real] :
                ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
                & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
                & ( ( F @ X )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M4 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M4 ) ) @ ( power_power_real @ X @ M4 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin_all_lt
thf(fact_10012_Maclaurin__bi__le,axiom,
    ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M3: nat,T6: real] :
            ( ( ( ord_less_nat @ M3 @ N )
              & ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) ) )
           => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
       => ? [T6: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
            & ( ( F @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M4 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M4 ) ) @ ( power_power_real @ X @ M4 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_bi_le
thf(fact_10013_Taylor,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M3: nat,T6: real] :
              ( ( ( ord_less_nat @ M3 @ N )
                & ( ord_less_eq_real @ A @ T6 )
                & ( ord_less_eq_real @ T6 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B )
             => ( ( ord_less_eq_real @ A @ X )
               => ( ( ord_less_eq_real @ X @ B )
                 => ( ( X != C )
                   => ? [T6: real] :
                        ( ( ( ord_less_real @ X @ C )
                         => ( ( ord_less_real @ X @ T6 )
                            & ( ord_less_real @ T6 @ C ) ) )
                        & ( ~ ( ord_less_real @ X @ C )
                         => ( ( ord_less_real @ C @ T6 )
                            & ( ord_less_real @ T6 @ X ) ) )
                        & ( ( F @ X )
                          = ( plus_plus_real
                            @ ( groups6591440286371151544t_real
                              @ ^ [M4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M4 @ C ) @ ( semiri2265585572941072030t_real @ M4 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M4 ) )
                              @ ( set_ord_lessThan_nat @ N ) )
                            @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
thf(fact_10014_Taylor__up,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M3: nat,T6: real] :
              ( ( ( ord_less_nat @ M3 @ N )
                & ( ord_less_eq_real @ A @ T6 )
                & ( ord_less_eq_real @ T6 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_real @ C @ B )
             => ? [T6: real] :
                  ( ( ord_less_real @ C @ T6 )
                  & ( ord_less_real @ T6 @ B )
                  & ( ( F @ B )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M4 @ C ) @ ( semiri2265585572941072030t_real @ M4 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M4 ) )
                        @ ( set_ord_lessThan_nat @ N ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).

% Taylor_up
thf(fact_10015_Taylor__down,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M3: nat,T6: real] :
              ( ( ( ord_less_nat @ M3 @ N )
                & ( ord_less_eq_real @ A @ T6 )
                & ( ord_less_eq_real @ T6 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B )
             => ? [T6: real] :
                  ( ( ord_less_real @ A @ T6 )
                  & ( ord_less_real @ T6 @ C )
                  & ( ( F @ A )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M4 @ C ) @ ( semiri2265585572941072030t_real @ M4 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M4 ) )
                        @ ( set_ord_lessThan_nat @ N ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).

% Taylor_down
thf(fact_10016_Maclaurin__lemma2,axiom,
    ! [N: nat,H2: real,Diff: nat > real > real,K: nat,B2: real] :
      ( ! [M3: nat,T6: real] :
          ( ( ( ord_less_nat @ M3 @ N )
            & ( ord_less_eq_real @ zero_zero_real @ T6 )
            & ( ord_less_eq_real @ T6 @ H2 ) )
         => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
     => ( ( N
          = ( suc @ K ) )
       => ! [M2: nat,T7: real] :
            ( ( ( ord_less_nat @ M2 @ N )
              & ( ord_less_eq_real @ zero_zero_real @ T7 )
              & ( ord_less_eq_real @ T7 @ H2 ) )
           => ( 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 @ N @ M2 ) ) )
                    @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N @ M2 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) )
              @ ( minus_minus_real @ ( Diff @ ( suc @ M2 ) @ T7 )
                @ ( 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 @ T7 @ P5 ) )
                    @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) )
                  @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ T7 @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) ) ) ) )
              @ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) ) ) ) ).

% Maclaurin_lemma2
thf(fact_10017_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( has_fi5821293074295781190e_real
        @ ^ [X10: 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 @ X10 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
        @ ( suminf_real
          @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_arctan_series
thf(fact_10018_DERIV__power__series_H,axiom,
    ! [R3: real,F: nat > real,X0: real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R3 ) @ R3 ) )
         => ( summable_real
            @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( F @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X5 @ N3 ) ) ) )
     => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R3 ) @ R3 ) )
       => ( ( ord_less_real @ zero_zero_real @ R3 )
         => ( has_fi5821293074295781190e_real
            @ ^ [X6: real] :
                ( suminf_real
                @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X6 @ ( suc @ N3 ) ) ) )
            @ ( suminf_real
              @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( F @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X0 @ N3 ) ) )
            @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).

% DERIV_power_series'
thf(fact_10019_DERIV__isconst3,axiom,
    ! [A: real,B: real,X: real,Y: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( member_real @ X @ ( 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 @ X )
              = ( F @ Y ) ) ) ) ) ) ).

% DERIV_isconst3
thf(fact_10020_DERIV__series_H,axiom,
    ! [F: real > nat > real,F5: real > nat > real,X0: real,A: real,B: real,L4: nat > real] :
      ( ! [N2: nat] :
          ( has_fi5821293074295781190e_real
          @ ^ [X6: real] : ( F @ X6 @ N2 )
          @ ( F5 @ X0 @ N2 )
          @ ( 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 @ ( F5 @ X0 ) )
           => ( ( summable_real @ L4 )
             => ( ! [N2: nat,X5: real,Y4: real] :
                    ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A @ B ) )
                   => ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A @ B ) )
                     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X5 @ N2 ) @ ( F @ Y4 @ N2 ) ) ) @ ( times_times_real @ ( L4 @ N2 ) @ ( abs_abs_real @ ( minus_minus_real @ X5 @ Y4 ) ) ) ) ) )
               => ( has_fi5821293074295781190e_real
                  @ ^ [X6: real] : ( suminf_real @ ( F @ X6 ) )
                  @ ( suminf_real @ ( F5 @ X0 ) )
                  @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_series'
thf(fact_10021_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_10022_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
      = ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% atLeastSucLessThan_greaterThanLessThan
thf(fact_10023_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 )
       => ? [R: real] :
            ( ( ord_less_real @ zero_zero_real @ R )
            & ! [X3: real] :
                ( ( ( X3 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X3 ) ) @ R ) )
               => ( ord_less_real @ zero_zero_real @ ( F @ X3 ) ) ) ) ) ) ).

% LIM_fun_gt_zero
thf(fact_10024_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 )
       => ? [R: real] :
            ( ( ord_less_real @ zero_zero_real @ R )
            & ! [X3: real] :
                ( ( ( X3 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X3 ) ) @ R ) )
               => ( ( F @ X3 )
                 != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
thf(fact_10025_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 )
       => ? [R: real] :
            ( ( ord_less_real @ zero_zero_real @ R )
            & ! [X3: real] :
                ( ( ( X3 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X3 ) ) @ R ) )
               => ( ord_less_real @ ( F @ X3 ) @ zero_zero_real ) ) ) ) ) ).

% LIM_fun_less_zero
thf(fact_10026_isCont__ln,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ln_ln_real ) ) ).

% isCont_ln
thf(fact_10027_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I3: nat,J: nat] :
      ( ( ord_less_nat @ ( suc @ I3 ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I3 @ J ) )
        = ( cons_nat @ ( suc @ I3 ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I3 ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanLessThan
thf(fact_10028_LIM__cos__div__sin,axiom,
    ( filterlim_real_real
    @ ^ [X6: real] : ( divide_divide_real @ ( cos_real @ X6 ) @ ( sin_real @ X6 ) )
    @ ( 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_10029_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I3: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I3 ) ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I3 @ J ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I3 @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_10030_DERIV__inverse__function,axiom,
    ! [F: real > real,D4: real,G: real > real,X: real,A: real,B: real] :
      ( ( has_fi5821293074295781190e_real @ F @ D4 @ ( topolo2177554685111907308n_real @ ( G @ X ) @ top_top_set_real ) )
     => ( ( D4 != zero_zero_real )
       => ( ( ord_less_real @ A @ X )
         => ( ( ord_less_real @ X @ B )
           => ( ! [Y4: real] :
                  ( ( ord_less_real @ A @ Y4 )
                 => ( ( ord_less_real @ Y4 @ B )
                   => ( ( F @ ( G @ Y4 ) )
                      = Y4 ) ) )
             => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ G )
               => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D4 ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_inverse_function
thf(fact_10031_LIM__less__bound,axiom,
    ! [B: real,X: real,F: real > real] :
      ( ( ord_less_real @ B @ X )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ B @ X ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) ) ) ) ).

% LIM_less_bound
thf(fact_10032_greaterThanLessThan__upto,axiom,
    ( set_or5832277885323065728an_int
    = ( ^ [I: int,J2: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ ( minus_minus_int @ J2 @ one_one_int ) ) ) ) ) ).

% greaterThanLessThan_upto
thf(fact_10033_isCont__inverse__function,axiom,
    ! [D: real,X: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ D )
     => ( ! [Z4: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X ) ) @ D )
           => ( ( G @ ( F @ Z4 ) )
              = Z4 ) )
       => ( ! [Z4: real] :
              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X ) ) @ D )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) )
         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ).

% isCont_inverse_function
thf(fact_10034_GMVT_H,axiom,
    ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F5: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [Z4: real] :
            ( ( ord_less_eq_real @ A @ Z4 )
           => ( ( ord_less_eq_real @ Z4 @ B )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) ) )
       => ( ! [Z4: real] :
              ( ( ord_less_eq_real @ A @ Z4 )
             => ( ( ord_less_eq_real @ Z4 @ B )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ G ) ) )
         => ( ! [Z4: real] :
                ( ( ord_less_real @ A @ Z4 )
               => ( ( ord_less_real @ Z4 @ B )
                 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
           => ( ! [Z4: real] :
                  ( ( ord_less_real @ A @ Z4 )
                 => ( ( ord_less_real @ Z4 @ B )
                   => ( has_fi5821293074295781190e_real @ F @ ( F5 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
             => ? [C2: real] :
                  ( ( ord_less_real @ A @ C2 )
                  & ( ord_less_real @ C2 @ B )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C2 ) )
                    = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F5 @ C2 ) ) ) ) ) ) ) ) ) ).

% GMVT'
thf(fact_10035_summable__Leibniz_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
         => ! [N7: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) @ one_one_nat ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) ) ) ) ) ) ) ) ).

% summable_Leibniz(3)
thf(fact_10036_summable__Leibniz_I2_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
         => ! [N7: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% summable_Leibniz(2)
thf(fact_10037_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
thf(fact_10038_filterlim__Suc,axiom,
    filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).

% filterlim_Suc
thf(fact_10039_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_10040_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( filterlim_nat_nat
        @ ^ [X6: nat] : ( times_times_nat @ X6 @ C )
        @ at_top_nat
        @ at_top_nat ) ) ).

% mult_nat_right_at_top
thf(fact_10041_nested__sequence__unique,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ! [N2: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N2 ) ) @ ( G @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
         => ( ( filterlim_nat_real
              @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) )
              @ ( topolo2815343760600316023s_real @ zero_zero_real )
              @ at_top_nat )
           => ? [L3: real] :
                ( ! [N7: nat] : ( ord_less_eq_real @ ( F @ N7 ) @ L3 )
                & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L3 ) @ at_top_nat )
                & ! [N7: nat] : ( ord_less_eq_real @ L3 @ ( G @ N7 ) )
                & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L3 ) @ at_top_nat ) ) ) ) ) ) ).

% nested_sequence_unique
thf(fact_10042_LIMSEQ__inverse__zero,axiom,
    ! [X9: nat > real] :
      ( ! [R: real] :
        ? [N9: nat] :
        ! [N2: nat] :
          ( ( ord_less_eq_nat @ N9 @ N2 )
         => ( ord_less_real @ R @ ( X9 @ N2 ) ) )
     => ( filterlim_nat_real
        @ ^ [N3: nat] : ( inverse_inverse_real @ ( X9 @ N3 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_zero
thf(fact_10043_lim__inverse__n_H,axiom,
    ( filterlim_nat_real
    @ ^ [N3: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N3 ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% lim_inverse_n'
thf(fact_10044_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( filterlim_nat_real
        @ ^ [N3: nat] : ( root @ N3 @ C )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat ) ) ).

% LIMSEQ_root_const
thf(fact_10045_LIMSEQ__inverse__real__of__nat,axiom,
    ( filterlim_nat_real
    @ ^ [N3: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat
thf(fact_10046_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N3: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add
thf(fact_10047_increasing__LIMSEQ,axiom,
    ! [F: nat > real,L: real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ L )
       => ( ! [E: real] :
              ( ( ord_less_real @ zero_zero_real @ E )
             => ? [N7: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N7 ) @ E ) ) )
         => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).

% increasing_LIMSEQ
thf(fact_10048_LIMSEQ__realpow__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( filterlim_nat_real @ ( power_power_real @ X ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).

% LIMSEQ_realpow_zero
thf(fact_10049_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( filterlim_nat_real
        @ ^ [N3: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N3 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_divide_realpow_zero
thf(fact_10050_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_10051_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_10052_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( filterlim_nat_real
        @ ^ [N3: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N3 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_realpow_zero
thf(fact_10053_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N3: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_10054_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N3: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_10055_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
          @ ^ [N3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( A @ N3 ) ) ) ) ) ).

% summable_Leibniz(1)
thf(fact_10056_summable,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( summable_real
            @ ^ [N3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( A @ N3 ) ) ) ) ) ) ).

% summable
thf(fact_10057_cos__diff__limit__1,axiom,
    ! [Theta: nat > real,Theta2: real] :
      ( ( filterlim_nat_real
        @ ^ [J2: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J2 ) @ Theta2 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ~ ! [K2: nat > int] :
            ~ ( filterlim_nat_real
              @ ^ [J2: nat] : ( minus_minus_real @ ( Theta @ J2 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
              @ ( topolo2815343760600316023s_real @ Theta2 )
              @ at_top_nat ) ) ).

% cos_diff_limit_1
thf(fact_10058_cos__limit__1,axiom,
    ! [Theta: nat > real] :
      ( ( filterlim_nat_real
        @ ^ [J2: nat] : ( cos_real @ ( Theta @ J2 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ? [K2: nat > int] :
          ( filterlim_nat_real
          @ ^ [J2: nat] : ( minus_minus_real @ ( Theta @ J2 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
          @ ( topolo2815343760600316023s_real @ zero_zero_real )
          @ at_top_nat ) ) ).

% cos_limit_1
thf(fact_10059_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
          @ ^ [N3: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(4)
thf(fact_10060_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( filterlim_nat_real
        @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% zeroseq_arctan_series
thf(fact_10061_summable__Leibniz_H_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( filterlim_nat_real
            @ ^ [N3: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(3)
thf(fact_10062_summable__Leibniz_H_I2_J,axiom,
    ! [A: nat > real,N: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( ord_less_eq_real
            @ ( groups6591440286371151544t_real
              @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
            @ ( suminf_real
              @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) ) ) ) ) ) ) ).

% summable_Leibniz'(2)
thf(fact_10063_sums__alternating__upper__lower,axiom,
    ! [A: nat > real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
         => ? [L3: real] :
              ( ! [N7: nat] :
                  ( ord_less_eq_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) ) )
                  @ L3 )
              & ( filterlim_nat_real
                @ ^ [N3: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
                @ ( topolo2815343760600316023s_real @ L3 )
                @ at_top_nat )
              & ! [N7: nat] :
                  ( ord_less_eq_real @ L3
                  @ ( groups6591440286371151544t_real
                    @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) @ one_one_nat ) ) ) )
              & ( filterlim_nat_real
                @ ^ [N3: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
                @ ( topolo2815343760600316023s_real @ L3 )
                @ at_top_nat ) ) ) ) ) ).

% sums_alternating_upper_lower
thf(fact_10064_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
          @ ^ [N3: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(5)
thf(fact_10065_summable__Leibniz_H_I4_J,axiom,
    ! [A: nat > real,N: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( ord_less_eq_real
            @ ( suminf_real
              @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) ) )
            @ ( groups6591440286371151544t_real
              @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).

% summable_Leibniz'(4)
thf(fact_10066_summable__Leibniz_H_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( filterlim_nat_real
            @ ^ [N3: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(5)
thf(fact_10067_real__bounded__linear,axiom,
    ( real_V5970128139526366754l_real
    = ( ^ [F6: real > real] :
        ? [C4: real] :
          ( F6
          = ( ^ [X6: real] : ( times_times_real @ X6 @ C4 ) ) ) ) ) ).

% real_bounded_linear
thf(fact_10068_ln__x__over__x__tendsto__0,axiom,
    ( filterlim_real_real
    @ ^ [X6: real] : ( divide_divide_real @ ( ln_ln_real @ X6 ) @ X6 )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_real ) ).

% ln_x_over_x_tendsto_0
thf(fact_10069_tendsto__power__div__exp__0,axiom,
    ! [K: nat] :
      ( filterlim_real_real
      @ ^ [X6: real] : ( divide_divide_real @ ( power_power_real @ X6 @ K ) @ ( exp_real @ X6 ) )
      @ ( topolo2815343760600316023s_real @ zero_zero_real )
      @ at_top_real ) ).

% tendsto_power_div_exp_0
thf(fact_10070_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_10071_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: real > real,Flim: real] :
      ( ! [X5: real] :
          ( ( ord_less_eq_real @ B @ X5 )
         => ? [Y5: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
              & ( ord_less_real @ Y5 @ 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_10072_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_10073_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F: real > real,F3: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F3 )
       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
         => ( filterlim_real_real
            @ ^ [X6: real] : ( power_power_real @ ( F @ X6 ) @ N )
            @ at_top_real
            @ F3 ) ) ) ) ).

% filterlim_pow_at_bot_even
thf(fact_10074_exp__at__bot,axiom,
    filterlim_real_real @ exp_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_bot_real ).

% exp_at_bot
thf(fact_10075_filterlim__inverse__at__bot__neg,axiom,
    filterlim_real_real @ inverse_inverse_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5984915006950818249n_real @ zero_zero_real ) ) ).

% filterlim_inverse_at_bot_neg
thf(fact_10076_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: real > real,Flim: real] :
      ( ! [X5: real] :
          ( ( ord_less_eq_real @ X5 @ B )
         => ? [Y5: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
              & ( ord_less_real @ zero_zero_real @ Y5 ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
       => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).

% DERIV_pos_imp_increasing_at_bot
thf(fact_10077_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F: real > real,F3: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F3 )
       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
         => ( filterlim_real_real
            @ ^ [X6: real] : ( power_power_real @ ( F @ X6 ) @ N )
            @ at_bot_real
            @ F3 ) ) ) ) ).

% filterlim_pow_at_bot_odd
thf(fact_10078_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_10079_tendsto__exp__limit__at__right,axiom,
    ! [X: real] :
      ( filterlim_real_real
      @ ^ [Y6: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X @ Y6 ) ) @ ( divide_divide_real @ one_one_real @ Y6 ) )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
      @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).

% tendsto_exp_limit_at_right
thf(fact_10080_filterlim__inverse__at__right__top,axiom,
    filterlim_real_real @ inverse_inverse_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) @ at_top_real ).

% filterlim_inverse_at_right_top
thf(fact_10081_filterlim__inverse__at__top__right,axiom,
    filterlim_real_real @ inverse_inverse_real @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).

% filterlim_inverse_at_top_right
thf(fact_10082_ln__at__0,axiom,
    filterlim_real_real @ ln_ln_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).

% ln_at_0
thf(fact_10083_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_10084_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_10085_lhopital__left__at__top,axiom,
    ! [G: real > real,X: real,G2: real > real,F: real > real,F5: real > real,Y: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
     => ( ( eventually_real
          @ ^ [X6: real] :
              ( ( G2 @ X6 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X6: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X6: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
           => ( ( filterlim_real_real
                @ ^ [X6: real] : ( divide_divide_real @ ( F5 @ X6 ) @ ( G2 @ X6 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
             => ( filterlim_real_real
                @ ^ [X6: real] : ( divide_divide_real @ ( F @ X6 ) @ ( G @ X6 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top
thf(fact_10086_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_10087_greaterThan__0,axiom,
    ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
    = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).

% greaterThan_0
thf(fact_10088_eventually__at__right__to__0,axiom,
    ! [P: real > $o,A: real] :
      ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
      = ( eventually_real
        @ ^ [X6: real] : ( P @ ( plus_plus_real @ X6 @ A ) )
        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% eventually_at_right_to_0
thf(fact_10089_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_10090_eventually__at__right__to__top,axiom,
    ! [P: real > $o] :
      ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
      = ( eventually_real
        @ ^ [X6: real] : ( P @ ( inverse_inverse_real @ X6 ) )
        @ at_top_real ) ) ).

% eventually_at_right_to_top
thf(fact_10091_eventually__at__top__to__right,axiom,
    ! [P: real > $o] :
      ( ( eventually_real @ P @ at_top_real )
      = ( eventually_real
        @ ^ [X6: real] : ( P @ ( inverse_inverse_real @ X6 ) )
        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% eventually_at_top_to_right
thf(fact_10092_lhopital,axiom,
    ! [F: real > real,X: real,G: real > real,G2: real > real,F5: real > real,F3: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X6: real] :
                ( ( G @ X6 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X6: real] :
                  ( ( G2 @ X6 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
           => ( ( eventually_real
                @ ^ [X6: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
             => ( ( eventually_real
                  @ ^ [X6: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
               => ( ( filterlim_real_real
                    @ ^ [X6: real] : ( divide_divide_real @ ( F5 @ X6 ) @ ( G2 @ X6 ) )
                    @ F3
                    @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                 => ( filterlim_real_real
                    @ ^ [X6: real] : ( divide_divide_real @ ( F @ X6 ) @ ( G @ X6 ) )
                    @ F3
                    @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ) ).

% lhopital
thf(fact_10093_lhospital__at__top__at__top,axiom,
    ! [G: real > real,G2: real > real,F: real > real,F5: real > real,X: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
     => ( ( eventually_real
          @ ^ [X6: real] :
              ( ( G2 @ X6 )
             != zero_zero_real )
          @ at_top_real )
       => ( ( eventually_real
            @ ^ [X6: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
            @ at_top_real )
         => ( ( eventually_real
              @ ^ [X6: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
              @ at_top_real )
           => ( ( filterlim_real_real
                @ ^ [X6: real] : ( divide_divide_real @ ( F5 @ X6 ) @ ( G2 @ X6 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ at_top_real )
             => ( filterlim_real_real
                @ ^ [X6: real] : ( divide_divide_real @ ( F @ X6 ) @ ( G @ X6 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ at_top_real ) ) ) ) ) ) ).

% lhospital_at_top_at_top
thf(fact_10094_lhopital__at__top,axiom,
    ! [G: real > real,X: real,G2: real > real,F: real > real,F5: real > real,Y: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( eventually_real
          @ ^ [X6: real] :
              ( ( G2 @ X6 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X6: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X6: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X6: real] : ( divide_divide_real @ ( F5 @ X6 ) @ ( G2 @ X6 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X6: real] : ( divide_divide_real @ ( F @ X6 ) @ ( G @ X6 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top
thf(fact_10095_lhopital__right,axiom,
    ! [F: real > real,X: real,G: real > real,G2: real > real,F5: real > real,F3: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X6: real] :
                ( ( G @ X6 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X6: real] :
                  ( ( G2 @ X6 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
           => ( ( eventually_real
                @ ^ [X6: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
             => ( ( eventually_real
                  @ ^ [X6: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X6: real] : ( divide_divide_real @ ( F5 @ X6 ) @ ( G2 @ X6 ) )
                    @ F3
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
                 => ( filterlim_real_real
                    @ ^ [X6: real] : ( divide_divide_real @ ( F @ X6 ) @ ( G @ X6 ) )
                    @ F3
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ) ) ).

% lhopital_right
thf(fact_10096_lhopital__right__0,axiom,
    ! [F0: real > real,G0: real > real,G2: real > real,F5: real > real,F3: filter_real] :
      ( ( filterlim_real_real @ F0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
     => ( ( filterlim_real_real @ G0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
       => ( ( eventually_real
            @ ^ [X6: real] :
                ( ( G0 @ X6 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
         => ( ( eventually_real
              @ ^ [X6: real] :
                  ( ( G2 @ X6 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
           => ( ( eventually_real
                @ ^ [X6: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F5 @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
             => ( ( eventually_real
                  @ ^ [X6: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G2 @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X6: real] : ( divide_divide_real @ ( F5 @ X6 ) @ ( G2 @ X6 ) )
                    @ F3
                    @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
                 => ( filterlim_real_real
                    @ ^ [X6: real] : ( divide_divide_real @ ( F0 @ X6 ) @ ( G0 @ X6 ) )
                    @ F3
                    @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).

% lhopital_right_0
thf(fact_10097_lhopital__left,axiom,
    ! [F: real > real,X: real,G: real > real,G2: real > real,F5: real > real,F3: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X6: real] :
                ( ( G @ X6 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X6: real] :
                  ( ( G2 @ X6 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
           => ( ( eventually_real
                @ ^ [X6: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
             => ( ( eventually_real
                  @ ^ [X6: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X6: real] : ( divide_divide_real @ ( F5 @ X6 ) @ ( G2 @ X6 ) )
                    @ F3
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
                 => ( filterlim_real_real
                    @ ^ [X6: real] : ( divide_divide_real @ ( F @ X6 ) @ ( G @ X6 ) )
                    @ F3
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ) ) ).

% lhopital_left
thf(fact_10098_lhopital__right__0__at__top,axiom,
    ! [G: real > real,G2: real > real,F: real > real,F5: real > real,X: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
     => ( ( eventually_real
          @ ^ [X6: real] :
              ( ( G2 @ X6 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
       => ( ( eventually_real
            @ ^ [X6: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
         => ( ( eventually_real
              @ ^ [X6: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
           => ( ( filterlim_real_real
                @ ^ [X6: real] : ( divide_divide_real @ ( F5 @ X6 ) @ ( G2 @ X6 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
             => ( filterlim_real_real
                @ ^ [X6: real] : ( divide_divide_real @ ( F @ X6 ) @ ( G @ X6 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).

% lhopital_right_0_at_top
thf(fact_10099_lhopital__right__at__top,axiom,
    ! [G: real > real,X: real,G2: real > real,F: real > real,F5: real > real,Y: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
     => ( ( eventually_real
          @ ^ [X6: real] :
              ( ( G2 @ X6 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X6: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X6: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
           => ( ( filterlim_real_real
                @ ^ [X6: real] : ( divide_divide_real @ ( F5 @ X6 ) @ ( G2 @ X6 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
             => ( filterlim_real_real
                @ ^ [X6: real] : ( divide_divide_real @ ( F @ X6 ) @ ( G @ X6 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ).

% lhopital_right_at_top
thf(fact_10100_eventually__sequentially__Suc,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat
        @ ^ [I: nat] : ( P @ ( suc @ I ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_Suc
thf(fact_10101_eventually__sequentially__seg,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat
        @ ^ [N3: nat] : ( P @ ( plus_plus_nat @ N3 @ K ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_seg
thf(fact_10102_sequentially__offset,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat @ P @ at_top_nat )
     => ( eventually_nat
        @ ^ [I: nat] : ( P @ ( plus_plus_nat @ I @ K ) )
        @ at_top_nat ) ) ).

% sequentially_offset
thf(fact_10103_Bseq__realpow,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( bfun_nat_real @ ( power_power_real @ X ) @ at_top_nat ) ) ) ).

% Bseq_realpow
thf(fact_10104_Gcd__eq__Max,axiom,
    ! [M7: set_nat] :
      ( ( finite_finite_nat @ M7 )
     => ( ( M7 != bot_bot_set_nat )
       => ( ~ ( member_nat @ zero_zero_nat @ M7 )
         => ( ( gcd_Gcd_nat @ M7 )
            = ( lattic8265883725875713057ax_nat
              @ ( comple7806235888213564991et_nat
                @ ( image_nat_set_nat
                  @ ^ [M4: nat] :
                      ( collect_nat
                      @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M4 ) )
                  @ M7 ) ) ) ) ) ) ) ).

% Gcd_eq_Max
thf(fact_10105_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ N ) ) )
        = N ) ) ).

% Max_divisors_self_nat
thf(fact_10106_Sup__nat__def,axiom,
    ( complete_Sup_Sup_nat
    = ( ^ [X4: set_nat] : ( if_nat @ ( X4 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X4 ) ) ) ) ).

% Sup_nat_def
thf(fact_10107_card__le__Suc__Max,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ord_less_eq_nat @ ( finite_card_nat @ S3 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S3 ) ) ) ) ).

% card_le_Suc_Max
thf(fact_10108_divide__nat__def,axiom,
    ( divide_divide_nat
    = ( ^ [M4: nat,N3: nat] :
          ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat
          @ ( lattic8265883725875713057ax_nat
            @ ( collect_nat
              @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N3 ) @ M4 ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_10109_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( gcd_gcd_nat @ M @ N )
        = ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D2: nat] :
                ( ( dvd_dvd_nat @ D2 @ M )
                & ( dvd_dvd_nat @ D2 @ N ) ) ) ) ) ) ).

% gcd_is_Max_divisors_nat
thf(fact_10110_Max__divisors__self__int,axiom,
    ! [N: int] :
      ( ( N != zero_zero_int )
     => ( ( lattic8263393255366662781ax_int
          @ ( collect_int
            @ ^ [D2: int] : ( dvd_dvd_int @ D2 @ N ) ) )
        = ( abs_abs_int @ N ) ) ) ).

% Max_divisors_self_int
thf(fact_10111_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
      = ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% atLeastSucAtMost_greaterThanAtMost
thf(fact_10112_gcd__is__Max__divisors__int,axiom,
    ! [N: int,M: int] :
      ( ( N != zero_zero_int )
     => ( ( gcd_gcd_int @ M @ N )
        = ( lattic8263393255366662781ax_int
          @ ( collect_int
            @ ^ [D2: int] :
                ( ( dvd_dvd_int @ D2 @ M )
                & ( dvd_dvd_int @ D2 @ N ) ) ) ) ) ) ).

% gcd_is_Max_divisors_int
thf(fact_10113_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I3: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( suc @ I3 ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I3 @ J ) )
        = ( cons_nat @ ( suc @ I3 ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I3 ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanAtMost
thf(fact_10114_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I3: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I3 ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I3 @ J ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I3 @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_10115_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_valid @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Y
            = ( Xa2 != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList4 @ Summary3 ) )
             => ( Y
                = ( ~ ( ( Deg2 = Xa2 )
                      & ! [X6: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                         => ( vEBT_VEBT_valid @ X6 @ ( 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 @ TreeList4 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X4 )
                          & ! [X6: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                             => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X4 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I: nat] :
                                  ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I ) @ X4 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X6: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                                   => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X4 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList4 @ Ma3 )
                                  & ! [X6: nat] :
                                      ( ( ord_less_nat @ X6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList4 @ X6 )
                                       => ( ( ord_less_nat @ Mi3 @ X6 )
                                          & ( ord_less_eq_nat @ X6 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
thf(fact_10116_atLeast__0,axiom,
    ( ( set_ord_atLeast_nat @ zero_zero_nat )
    = top_top_set_nat ) ).

% atLeast_0
thf(fact_10117_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( set_or1210151606488870762an_nat @ K ) ) ).

% atLeast_Suc_greaterThan
thf(fact_10118_greaterThanAtMost__upto,axiom,
    ( set_or6656581121297822940st_int
    = ( ^ [I: int,J2: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J2 ) ) ) ) ).

% greaterThanAtMost_upto
thf(fact_10119_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_10120_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList @ Summary ) @ Deg3 )
      = ( ( Deg = Deg3 )
        & ! [X6: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList ) )
           => ( vEBT_VEBT_valid @ X6 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        & ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( case_o184042715313410164at_nat
          @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X4 )
            & ! [X6: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList ) )
               => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X4 ) ) )
          @ ( produc6081775807080527818_nat_o
            @ ^ [Mi3: nat,Ma3: nat] :
                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                & ! [I: nat] :
                    ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                   => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I ) @ X4 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I ) ) )
                & ( ( Mi3 = Ma3 )
                 => ! [X6: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X4 ) ) )
                & ( ( Mi3 != Ma3 )
                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
                    & ! [X6: nat] :
                        ( ( ord_less_nat @ X6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X6 )
                         => ( ( ord_less_nat @ Mi3 @ X6 )
                            & ( ord_less_eq_nat @ X6 @ Ma3 ) ) ) ) ) ) ) )
          @ Mima2 ) ) ) ).

% VEBT_internal.valid'.simps(2)
thf(fact_10121_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa2 = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList4 @ Summary3 ) )
             => ( ( Deg2 = Xa2 )
                & ! [X5: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                   => ( 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 @ TreeList4 )
                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                & ( case_o184042715313410164at_nat
                  @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X4 )
                    & ! [X6: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                       => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X4 ) ) )
                  @ ( produc6081775807080527818_nat_o
                    @ ^ [Mi3: nat,Ma3: nat] :
                        ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                        & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                        & ! [I: nat] :
                            ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                           => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I ) @ X4 ) )
                              = ( vEBT_V8194947554948674370ptions @ Summary3 @ I ) ) )
                        & ( ( Mi3 = Ma3 )
                         => ! [X6: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                             => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X4 ) ) )
                        & ( ( Mi3 != Ma3 )
                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList4 @ Ma3 )
                            & ! [X6: nat] :
                                ( ( ord_less_nat @ X6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList4 @ X6 )
                                 => ( ( ord_less_nat @ Mi3 @ X6 )
                                    & ( ord_less_eq_nat @ X6 @ Ma3 ) ) ) ) ) ) ) )
                  @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
thf(fact_10122_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa2 != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList4 @ Summary3 ) )
             => ~ ( ( Deg2 = Xa2 )
                  & ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                     => ( vEBT_VEBT_valid @ X3 @ ( 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 @ TreeList4 )
                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( case_o184042715313410164at_nat
                    @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X4 )
                      & ! [X6: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                         => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X4 ) ) )
                    @ ( produc6081775807080527818_nat_o
                      @ ^ [Mi3: nat,Ma3: nat] :
                          ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                          & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                          & ! [I: nat] :
                              ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                             => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I ) @ X4 ) )
                                = ( vEBT_V8194947554948674370ptions @ Summary3 @ I ) ) )
                          & ( ( Mi3 = Ma3 )
                           => ! [X6: vEBT_VEBT] :
                                ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                               => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X4 ) ) )
                          & ( ( Mi3 != Ma3 )
                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList4 @ Ma3 )
                              & ! [X6: nat] :
                                  ( ( ord_less_nat @ X6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList4 @ X6 )
                                   => ( ( ord_less_nat @ Mi3 @ X6 )
                                      & ( ord_less_eq_nat @ X6 @ Ma3 ) ) ) ) ) ) ) )
                    @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
thf(fact_10123_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_valid @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Y
                  = ( Xa2 = one_one_nat ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList4 @ Summary3 ) )
               => ( ( Y
                    = ( ( Deg2 = Xa2 )
                      & ! [X6: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                         => ( vEBT_VEBT_valid @ X6 @ ( 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 @ TreeList4 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X4 )
                          & ! [X6: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                             => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X4 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I: nat] :
                                  ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I ) @ X4 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X6: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                                   => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X4 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList4 @ Ma3 )
                                  & ! [X6: nat] :
                                      ( ( ord_less_nat @ X6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList4 @ X6 )
                                       => ( ( ord_less_nat @ Mi3 @ X6 )
                                          & ( ord_less_eq_nat @ X6 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList4 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
thf(fact_10124_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
               => ( Xa2 != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList4 @ Summary3 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList4 @ Summary3 ) @ Xa2 ) )
                 => ~ ( ( Deg2 = Xa2 )
                      & ! [X3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                         => ( vEBT_VEBT_valid @ X3 @ ( 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 @ TreeList4 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X4 )
                          & ! [X6: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                             => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X4 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I: nat] :
                                  ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I ) @ X4 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X6: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                                   => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X4 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList4 @ Ma3 )
                                  & ! [X6: nat] :
                                      ( ( ord_less_nat @ X6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList4 @ X6 )
                                       => ( ( ord_less_nat @ Mi3 @ X6 )
                                          & ( ord_less_eq_nat @ X6 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
thf(fact_10125_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
               => ( Xa2 = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList4: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList4 @ Summary3 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList4 @ Summary3 ) @ Xa2 ) )
                 => ( ( Deg2 = Xa2 )
                    & ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                       => ( 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 @ TreeList4 )
                      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( case_o184042715313410164at_nat
                      @ ( ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X4 )
                        & ! [X6: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                           => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X4 ) ) )
                      @ ( produc6081775807080527818_nat_o
                        @ ^ [Mi3: nat,Ma3: nat] :
                            ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                            & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                            & ! [I: nat] :
                                ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                               => ( ( ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I ) @ X4 ) )
                                  = ( vEBT_V8194947554948674370ptions @ Summary3 @ I ) ) )
                            & ( ( Mi3 = Ma3 )
                             => ! [X6: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                                 => ~ ? [X4: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X4 ) ) )
                            & ( ( Mi3 != Ma3 )
                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList4 @ Ma3 )
                                & ! [X6: nat] :
                                    ( ( ord_less_nat @ X6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList4 @ X6 )
                                     => ( ( ord_less_nat @ Mi3 @ X6 )
                                        & ( ord_less_eq_nat @ X6 @ Ma3 ) ) ) ) ) ) ) )
                      @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
thf(fact_10126_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,C2: real] :
                  ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
                  & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
                  & ( ord_less_real @ A @ C2 )
                  & ( ord_less_real @ C2 @ B )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
                    = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).

% GMVT
thf(fact_10127_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 ) ) ) )
         => ? [L3: real,Z4: real] :
              ( ( ord_less_real @ A @ Z4 )
              & ( ord_less_real @ Z4 @ B )
              & ( has_fi5821293074295781190e_real @ F @ L3 @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) )
              & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
                = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L3 ) ) ) ) ) ) ).

% MVT
thf(fact_10128_Rolle__deriv,axiom,
    ! [A: real,B: real,F: real > real,F5: real > real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ A )
          = ( F @ B ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
         => ( ! [X5: real] :
                ( ( ord_less_real @ A @ X5 )
               => ( ( ord_less_real @ X5 @ B )
                 => ( has_de1759254742604945161l_real @ F @ ( F5 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
           => ? [Z4: real] :
                ( ( ord_less_real @ A @ Z4 )
                & ( ord_less_real @ Z4 @ B )
                & ( ( F5 @ Z4 )
                  = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).

% Rolle_deriv
thf(fact_10129_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 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
         => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).

% DERIV_pos_imp_increasing_open
thf(fact_10130_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 )
             => ? [Y5: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
                  & ( ord_less_real @ Y5 @ 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_10131_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_10132_DERIV__isconst2,axiom,
    ! [A: real,B: real,F: real > real,X: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [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 @ X )
           => ( ( ord_less_eq_real @ X @ B )
             => ( ( F @ X )
                = ( F @ A ) ) ) ) ) ) ) ).

% DERIV_isconst2
thf(fact_10133_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 ) ) ) )
           => ? [Z4: real] :
                ( ( ord_less_real @ A @ Z4 )
                & ( ord_less_real @ Z4 @ B )
                & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) ) ) ) ) ).

% Rolle
thf(fact_10134_uniformity__real__def,axiom,
    ( topolo1511823702728130853y_real
    = ( comple2936214249959783750l_real
      @ ( image_2178119161166701260l_real
        @ ^ [E3: real] :
            ( princi6114159922880469582l_real
            @ ( collec3799799289383736868l_real
              @ ( produc5414030515140494994real_o
                @ ^ [X6: real,Y6: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X6 @ Y6 ) @ E3 ) ) ) )
        @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% uniformity_real_def
thf(fact_10135_uniformity__complex__def,axiom,
    ( topolo896644834953643431omplex
    = ( comple8358262395181532106omplex
      @ ( image_5971271580939081552omplex
        @ ^ [E3: real] :
            ( princi3496590319149328850omplex
            @ ( collec8663557070575231912omplex
              @ ( produc6771430404735790350plex_o
                @ ^ [X6: complex,Y6: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X6 @ Y6 ) @ E3 ) ) ) )
        @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% uniformity_complex_def
thf(fact_10136_mono__Suc,axiom,
    order_mono_nat_nat @ suc ).

% mono_Suc
thf(fact_10137_mono__times__nat,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).

% mono_times_nat
thf(fact_10138_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( order_mono_nat_nat
        @ ^ [M4: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M4 ) @ M4 ) ) ) ).

% mono_ge2_power_minus_self
thf(fact_10139_inj__sgn__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( inj_on_real_real
        @ ^ [Y6: real] : ( times_times_real @ ( sgn_sgn_real @ Y6 ) @ ( power_power_real @ ( abs_abs_real @ Y6 ) @ N ) )
        @ top_top_set_real ) ) ).

% inj_sgn_power
thf(fact_10140_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_10141_inj__Suc,axiom,
    ! [N5: set_nat] : ( inj_on_nat_nat @ suc @ N5 ) ).

% inj_Suc
thf(fact_10142_inj__on__diff__nat,axiom,
    ! [N5: set_nat,K: nat] :
      ( ! [N2: nat] :
          ( ( member_nat @ N2 @ N5 )
         => ( ord_less_eq_nat @ K @ N2 ) )
     => ( inj_on_nat_nat
        @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ K )
        @ N5 ) ) ).

% inj_on_diff_nat
thf(fact_10143_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_10144_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_10145_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_10146_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_10147_min__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( suc @ ( ord_min_nat @ M @ N ) ) ) ).

% min_Suc_Suc
thf(fact_10148_min__0L,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% min_0L
thf(fact_10149_min__0R,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_10150_min__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_min_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_min_nat @ N @ ( pred_numeral @ K ) ) ) ) ).

% min_Suc_numeral
thf(fact_10151_min__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N ) ) ) ).

% min_numeral_Suc
thf(fact_10152_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
thf(fact_10153_concat__bit__assoc__sym,axiom,
    ! [M: nat,N: nat,K: int,L: int,R2: int] :
      ( ( bit_concat_bit @ M @ ( bit_concat_bit @ N @ K @ L ) @ R2 )
      = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N ) @ L @ R2 ) ) ) ).

% concat_bit_assoc_sym
thf(fact_10154_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q2 ) )
      = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).

% nat_mult_min_right
thf(fact_10155_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q2 )
      = ( ord_min_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).

% nat_mult_min_left
thf(fact_10156_min__diff,axiom,
    ! [M: nat,I3: nat,N: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M @ I3 ) @ ( minus_minus_nat @ N @ I3 ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I3 ) ) ).

% min_diff
thf(fact_10157_take__bit__concat__bit__eq,axiom,
    ! [M: nat,N: nat,K: int,L: int] :
      ( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N @ K @ L ) )
      = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ L ) ) ) ).

% take_bit_concat_bit_eq
thf(fact_10158_min__Suc1,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_min_nat @ ( suc @ N ) @ M )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M5: nat] : ( suc @ ( ord_min_nat @ N @ M5 ) )
        @ M ) ) ).

% min_Suc1
thf(fact_10159_min__Suc2,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_min_nat @ M @ ( suc @ N ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M5: nat] : ( suc @ ( ord_min_nat @ M5 @ N ) )
        @ M ) ) ).

% min_Suc2
thf(fact_10160_min__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q2 )
      = zero_z5237406670263579293d_enat ) ).

% min_enat_simps(3)
thf(fact_10161_min__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_mi8085742599997312461d_enat @ Q2 @ zero_z5237406670263579293d_enat )
      = zero_z5237406670263579293d_enat ) ).

% min_enat_simps(2)
thf(fact_10162_inf__enat__def,axiom,
    inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).

% inf_enat_def
thf(fact_10163_powr__real__of__int_H,axiom,
    ! [X: real,N: int] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( X != zero_zero_real )
          | ( ord_less_int @ zero_zero_int @ N ) )
       => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
          = ( power_int_real @ X @ N ) ) ) ) ).

% powr_real_of_int'
thf(fact_10164_pred__nat__def,axiom,
    ( pred_nat
    = ( collec3392354462482085612at_nat
      @ ( produc6081775807080527818_nat_o
        @ ^ [M4: nat,N3: nat] :
            ( N3
            = ( suc @ M4 ) ) ) ) ) ).

% pred_nat_def
thf(fact_10165_less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
      = ( ord_less_nat @ M @ N ) ) ).

% less_eq
thf(fact_10166_Rats__eq__int__div__nat,axiom,
    ( field_5140801741446780682s_real
    = ( collect_real
      @ ^ [Uu3: real] :
        ? [I: int,N3: nat] :
          ( ( Uu3
            = ( divide_divide_real @ ( ring_1_of_int_real @ I ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
          & ( N3 != zero_zero_nat ) ) ) ) ).

% Rats_eq_int_div_nat
thf(fact_10167_Rats__abs__iff,axiom,
    ! [X: real] :
      ( ( member_real @ ( abs_abs_real @ X ) @ field_5140801741446780682s_real )
      = ( member_real @ X @ field_5140801741446780682s_real ) ) ).

% Rats_abs_iff
thf(fact_10168_Rats__no__top__le,axiom,
    ! [X: real] :
    ? [X5: real] :
      ( ( member_real @ X5 @ field_5140801741446780682s_real )
      & ( ord_less_eq_real @ X @ X5 ) ) ).

% Rats_no_top_le
thf(fact_10169_Rats__no__bot__less,axiom,
    ! [X: real] :
    ? [X5: real] :
      ( ( member_real @ X5 @ field_5140801741446780682s_real )
      & ( ord_less_real @ X5 @ X ) ) ).

% Rats_no_bot_less
thf(fact_10170_Rats__dense__in__real,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ? [X5: real] :
          ( ( member_real @ X5 @ field_5140801741446780682s_real )
          & ( ord_less_real @ X @ X5 )
          & ( ord_less_real @ X5 @ Y ) ) ) ).

% Rats_dense_in_real
thf(fact_10171_Rats__eq__int__div__int,axiom,
    ( field_5140801741446780682s_real
    = ( collect_real
      @ ^ [Uu3: real] :
        ? [I: int,J2: int] :
          ( ( Uu3
            = ( divide_divide_real @ ( ring_1_of_int_real @ I ) @ ( ring_1_of_int_real @ J2 ) ) )
          & ( J2 != zero_zero_int ) ) ) ) ).

% Rats_eq_int_div_int
thf(fact_10172_complex__is__Nat__iff,axiom,
    ! [Z2: complex] :
      ( ( member_complex @ Z2 @ semiri3842193898606819883omplex )
      = ( ( ( im @ Z2 )
          = zero_zero_real )
        & ? [I: nat] :
            ( ( re @ Z2 )
            = ( semiri5074537144036343181t_real @ I ) ) ) ) ).

% complex_is_Nat_iff
thf(fact_10173_pos__deriv__imp__strict__mono,axiom,
    ! [F: real > real,F5: real > real] :
      ( ! [X5: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
     => ( ! [X5: real] : ( ord_less_real @ zero_zero_real @ ( F5 @ X5 ) )
       => ( order_7092887310737990675l_real @ F ) ) ) ).

% pos_deriv_imp_strict_mono
thf(fact_10174_strict__mono__imp__increasing,axiom,
    ! [F: nat > nat,N: nat] :
      ( ( order_5726023648592871131at_nat @ F )
     => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).

% strict_mono_imp_increasing
thf(fact_10175_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
              @ ^ [X6: nat] : ( F @ ( G @ X6 ) )
              @ at_top_nat )
            = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).

% nonneg_incseq_Bseq_subseq_iff
thf(fact_10176_upt__rec__numeral,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_10177_remdups__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( remdups_nat @ ( upt @ M @ N ) )
      = ( upt @ M @ N ) ) ).

% remdups_upt
thf(fact_10178_tl__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( tl_nat @ ( upt @ M @ N ) )
      = ( upt @ ( suc @ M ) @ N ) ) ).

% tl_upt
thf(fact_10179_hd__upt,axiom,
    ! [I3: nat,J: nat] :
      ( ( ord_less_nat @ I3 @ J )
     => ( ( hd_nat @ ( upt @ I3 @ J ) )
        = I3 ) ) ).

% hd_upt
thf(fact_10180_drop__upt,axiom,
    ! [M: nat,I3: nat,J: nat] :
      ( ( drop_nat @ M @ ( upt @ I3 @ J ) )
      = ( upt @ ( plus_plus_nat @ I3 @ M ) @ J ) ) ).

% drop_upt
thf(fact_10181_length__upt,axiom,
    ! [I3: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I3 @ J ) )
      = ( minus_minus_nat @ J @ I3 ) ) ).

% length_upt
thf(fact_10182_take__upt,axiom,
    ! [I3: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ M ) @ N )
     => ( ( take_nat @ M @ ( upt @ I3 @ N ) )
        = ( upt @ I3 @ ( plus_plus_nat @ I3 @ M ) ) ) ) ).

% take_upt
thf(fact_10183_upt__conv__Nil,axiom,
    ! [J: nat,I3: nat] :
      ( ( ord_less_eq_nat @ J @ I3 )
     => ( ( upt @ I3 @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_10184_sorted__list__of__set__range,axiom,
    ! [M: nat,N: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_or4665077453230672383an_nat @ M @ N ) )
      = ( upt @ M @ N ) ) ).

% sorted_list_of_set_range
thf(fact_10185_upt__eq__Nil__conv,axiom,
    ! [I3: nat,J: nat] :
      ( ( ( upt @ I3 @ J )
        = nil_nat )
      = ( ( J = zero_zero_nat )
        | ( ord_less_eq_nat @ J @ I3 ) ) ) ).

% upt_eq_Nil_conv
thf(fact_10186_nth__upt,axiom,
    ! [I3: nat,K: nat,J: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ J )
     => ( ( nth_nat @ ( upt @ I3 @ J ) @ K )
        = ( plus_plus_nat @ I3 @ K ) ) ) ).

% nth_upt
thf(fact_10187_upt__conv__Cons,axiom,
    ! [I3: nat,J: nat] :
      ( ( ord_less_nat @ I3 @ J )
     => ( ( upt @ I3 @ J )
        = ( cons_nat @ I3 @ ( upt @ ( suc @ I3 ) @ J ) ) ) ) ).

% upt_conv_Cons
thf(fact_10188_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N3 ) ) ) ) ) ).

% atMost_upto
thf(fact_10189_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N: nat,Ns: list_nat,Q2: nat] :
      ( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
        = ( upt @ M @ Q2 ) )
      = ( ( cons_nat @ N @ Ns )
        = ( upt @ ( suc @ M ) @ Q2 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_10190_greaterThanAtMost__upt,axiom,
    ( set_or6659071591806873216st_nat
    = ( ^ [N3: nat,M4: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ ( suc @ M4 ) ) ) ) ) ).

% greaterThanAtMost_upt
thf(fact_10191_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N3 ) ) ) ) ).

% atLeast_upt
thf(fact_10192_greaterThanLessThan__upt,axiom,
    ( set_or5834768355832116004an_nat
    = ( ^ [N3: nat,M4: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ M4 ) ) ) ) ).

% greaterThanLessThan_upt
thf(fact_10193_atLeastAtMost__upt,axiom,
    ( set_or1269000886237332187st_nat
    = ( ^ [N3: nat,M4: nat] : ( set_nat2 @ ( upt @ N3 @ ( suc @ M4 ) ) ) ) ) ).

% atLeastAtMost_upt
thf(fact_10194_atLeastLessThan__upt,axiom,
    ( set_or4665077453230672383an_nat
    = ( ^ [I: nat,J2: nat] : ( set_nat2 @ ( upt @ I @ J2 ) ) ) ) ).

% atLeastLessThan_upt
thf(fact_10195_distinct__upt,axiom,
    ! [I3: nat,J: nat] : ( distinct_nat @ ( upt @ I3 @ J ) ) ).

% distinct_upt
thf(fact_10196_upt__0,axiom,
    ! [I3: nat] :
      ( ( upt @ I3 @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_10197_upt__add__eq__append,axiom,
    ! [I3: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ( upt @ I3 @ ( plus_plus_nat @ J @ K ) )
        = ( append_nat @ ( upt @ I3 @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_10198_upt__eq__Cons__conv,axiom,
    ! [I3: nat,J: nat,X: nat,Xs2: list_nat] :
      ( ( ( upt @ I3 @ J )
        = ( cons_nat @ X @ Xs2 ) )
      = ( ( ord_less_nat @ I3 @ J )
        & ( I3 = X )
        & ( ( upt @ ( plus_plus_nat @ I3 @ one_one_nat ) @ J )
          = Xs2 ) ) ) ).

% upt_eq_Cons_conv
thf(fact_10199_upt__rec,axiom,
    ( upt
    = ( ^ [I: nat,J2: nat] : ( if_list_nat @ ( ord_less_nat @ I @ J2 ) @ ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J2 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_10200_upt__Suc__append,axiom,
    ! [I3: nat,J: nat] :
      ( ( ord_less_eq_nat @ I3 @ J )
     => ( ( upt @ I3 @ ( suc @ J ) )
        = ( append_nat @ ( upt @ I3 @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).

% upt_Suc_append
thf(fact_10201_upt__Suc,axiom,
    ! [I3: nat,J: nat] :
      ( ( ( ord_less_eq_nat @ I3 @ J )
       => ( ( upt @ I3 @ ( suc @ J ) )
          = ( append_nat @ ( upt @ I3 @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I3 @ J )
       => ( ( upt @ I3 @ ( suc @ J ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_10202_map__add__upt,axiom,
    ! [N: nat,M: nat] :
      ( ( map_nat_nat
        @ ^ [I: nat] : ( plus_plus_nat @ I @ N )
        @ ( upt @ zero_zero_nat @ M ) )
      = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).

% map_add_upt
thf(fact_10203_map__Suc__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
      = ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% map_Suc_upt
thf(fact_10204_map__decr__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( map_nat_nat
        @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( upt @ M @ N ) ) ).

% map_decr_upt
thf(fact_10205_Divides_Oadjust__div__def,axiom,
    ( adjust_div
    = ( produc8211389475949308722nt_int
      @ ^ [Q4: int,R5: int] : ( plus_plus_int @ Q4 @ ( zero_n2684676970156552555ol_int @ ( R5 != zero_zero_int ) ) ) ) ) ).

% Divides.adjust_div_def
thf(fact_10206_card__length__sum__list__rec,axiom,
    ! [M: nat,N5: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M )
     => ( ( finite_card_list_nat
          @ ( collect_list_nat
            @ ^ [L2: list_nat] :
                ( ( ( size_size_list_nat @ L2 )
                  = M )
                & ( ( groups4561878855575611511st_nat @ L2 )
                  = N5 ) ) ) )
        = ( 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 )
                    = N5 ) ) ) )
          @ ( 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 )
                    = N5 ) ) ) ) ) ) ) ).

% card_length_sum_list_rec
thf(fact_10207_card__length__sum__list,axiom,
    ! [M: nat,N5: nat] :
      ( ( finite_card_list_nat
        @ ( collect_list_nat
          @ ^ [L2: list_nat] :
              ( ( ( size_size_list_nat @ L2 )
                = M )
              & ( ( groups4561878855575611511st_nat @ L2 )
                = N5 ) ) ) )
      = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N5 @ M ) @ one_one_nat ) @ N5 ) ) ).

% card_length_sum_list
thf(fact_10208_sorted__wrt__upt,axiom,
    ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).

% sorted_wrt_upt
thf(fact_10209_sorted__upt,axiom,
    ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).

% sorted_upt
thf(fact_10210_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I3: nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
     => ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ns ) )
       => ( ord_less_eq_nat @ I3 @ ( nth_nat @ Ns @ I3 ) ) ) ) ).

% sorted_wrt_less_idx
thf(fact_10211_sorted__upto,axiom,
    ! [M: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N ) ) ).

% sorted_upto
thf(fact_10212_sorted__wrt__upto,axiom,
    ! [I3: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I3 @ J ) ) ).

% sorted_wrt_upto

% Helper facts (42)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( if_num @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( if_num @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( if_rat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( if_rat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( if_real @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( if_real @ $true @ X @ Y )
      = X ) ).

thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: real > $o] :
      ( ( P @ ( fChoice_real @ P ) )
      = ( ? [X4: real] : ( P @ X4 ) ) ) ).

thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( if_complex @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( if_complex @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( if_Extended_enat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( if_Extended_enat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( if_set_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( if_set_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( if_list_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( if_list_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: int > int,Y: int > int] :
      ( ( if_int_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: int > int,Y: int > int] :
      ( ( if_int_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ( if_option_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ( if_option_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X: option_num,Y: option_num] :
      ( ( if_option_num @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X: option_num,Y: option_num] :
      ( ( if_option_num @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
      = X ) ).

% Conjectures (2)
thf(conj_0,hypothesis,
    ord_less_nat @ i @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ).

thf(conj_1,conjecture,
    ( ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList2 @ i ) @ x )
    = ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ i ) @ x ) ) ).

%------------------------------------------------------------------------------