%------------------------------------------------------------------------------
% File     : ITP289^1 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_DelImperative 00402_025871
% 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    : 0095_VEBT_DelImperative_00402_025871 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11335 (5630 unt;1084 typ;   0 def)
%            Number of atoms       : 29158 (12644 equ;   0 cnn)
%            Maximal formula atoms :   71 (   2 avg)
%            Number of connectives : 126233 (3180   ~; 578   |;1794   &;109555   @)
%                                         (   0 <=>;11126  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   39 (   6 avg)
%            Number of types       :  119 ( 118 usr)
%            Number of type conns  : 3594 (3594   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  969 ( 966 usr;  78 con; 0-4 aty)
%            Number of variables   : 26776 (1481   ^;24374   !; 921   ?;26776   :)
% 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 22:13:02.899
%------------------------------------------------------------------------------
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    snga_assn_o: array_o > list_o > assn ).

thf(sy_c_Assertions_Osnga__assn_001t__Nat__Onat,type,
    snga_assn_nat: array_nat > list_nat > assn ).

thf(sy_c_Assertions_Osnga__assn_001t__Option__Ooption_It__Nat__Onat_J,type,
    snga_assn_option_nat: array_option_nat > list_option_nat > assn ).

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thf(sy_c_Automation_OFI__QUERY,type,
    fI_QUERY: assn > assn > assn > $o ).

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

thf(sy_c_Bit__Operations_Oand__int__rel,type,
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thf(sy_c_Bit__Operations_Oconcat__bit,type,
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thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
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thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Code____Numeral__Ointeger,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__Int__Oint,type,
    bit_ri631733984087533419it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__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,
    bit_se1409905431419307370or_int: int > int > int ).

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__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__bits__class_Obit_001t__Nat__Onat,type,
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thf(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftl_001t__Nat__Onat,type,
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thf(sy_c_Bits__Integer_Obin__last__integer,type,
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thf(sy_c_Bits__Integer_Obin__rest__integer,type,
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thf(sy_c_Code__Numeral_Odivmod__integer,type,
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thf(sy_c_Code__Numeral_Odup,type,
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thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
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thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
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thf(sy_c_Code__Numeral_Onat__of__integer,type,
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thf(sy_c_Code__Numeral_Onum__of__integer,type,
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thf(sy_c_Complex_OArg,type,
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thf(sy_c_Complex_Ocis,type,
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thf(sy_c_Complex_Ocnj,type,
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thf(sy_c_Complex_Ocomplex_OComplex,type,
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thf(sy_c_Complex_Ocomplex_OIm,type,
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thf(sy_c_Complex_Ocomplex_ORe,type,
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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_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,
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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,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Nat__Onat,type,
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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,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Nat__Onat,type,
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thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Complex__Ocomplex,type,
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thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
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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,
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thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
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thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
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thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
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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__Int__Oint,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_I_Eo_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__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__Nat__Onat_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__Real__Oreal_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,
    minus_811609699411566653omplex: set_complex > set_complex > set_complex ).

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

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

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Num__Onum_J,type,
    minus_minus_set_num: set_num > set_num > set_num ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    minus_1052850069191792384nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Rat__Orat_J,type,
    minus_minus_set_rat: set_rat > set_rat > set_rat ).

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

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    minus_5127226145743854075T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT ).

thf(sy_c_Groups_Oone__class_Oone_001t__Assertions__Oassn,type,
    one_one_assn: assn ).

thf(sy_c_Groups_Oone__class_Oone_001t__Code____Numeral__Ointeger,type,
    one_one_Code_integer: code_integer ).

thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
    one_one_complex: complex ).

thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
    one_one_int: int ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Rat__Orat,type,
    one_one_rat: rat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
    one_one_real: real ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Code____Numeral__Ointeger,type,
    plus_p5714425477246183910nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
    plus_plus_complex: complex > complex > complex ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nat__Oenat,type,
    plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
    plus_plus_int: int > int > int ).

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

thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
    plus_plus_num: num > num > num ).

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

thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
    plus_plus_real: real > real > real ).

thf(sy_c_Groups_Osgn__class_Osgn_001t__Code____Numeral__Ointeger,type,
    sgn_sgn_Code_integer: code_integer > code_integer ).

thf(sy_c_Groups_Osgn__class_Osgn_001t__Complex__Ocomplex,type,
    sgn_sgn_complex: complex > complex ).

thf(sy_c_Groups_Osgn__class_Osgn_001t__Int__Oint,type,
    sgn_sgn_int: int > int ).

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

thf(sy_c_Groups_Osgn__class_Osgn_001t__Real__Oreal,type,
    sgn_sgn_real: real > real ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Assertions__Oassn,type,
    times_times_assn: assn > assn > assn ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Code____Numeral__Ointeger,type,
    times_3573771949741848930nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
    times_times_complex: complex > complex > complex ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Nat__Oenat,type,
    times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
    times_times_int: int > int > int ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
    times_times_nat: nat > nat > nat ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
    times_times_num: num > num > num ).

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

thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
    times_times_real: real > real > real ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Complex__Ocomplex_M_Eo_J,type,
    uminus1680532995456772888plex_o: ( complex > $o ) > complex > $o ).

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

thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Nat__Onat_M_Eo_J,type,
    uminus_uminus_nat_o: ( nat > $o ) > nat > $o ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    uminus7117520113953359693_int_o: ( product_prod_int_int > $o ) > product_prod_int_int > $o ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Real__Oreal_M_Eo_J,type,
    uminus_uminus_real_o: ( real > $o ) > real > $o ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    uminus2746543603091002386VEBT_o: ( vEBT_VEBT > $o ) > vEBT_VEBT > $o ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Code____Numeral__Ointeger,type,
    uminus1351360451143612070nteger: code_integer > code_integer ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
    uminus1482373934393186551omplex: complex > complex ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
    uminus_uminus_int: int > int ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Rat__Orat,type,
    uminus_uminus_rat: rat > rat ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
    uminus_uminus_real: real > real ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_I_Eo_J,type,
    uminus_uminus_set_o: set_o > set_o ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    uminus8566677241136511917omplex: set_complex > set_complex ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Int__Oint_J,type,
    uminus1532241313380277803et_int: set_int > set_int ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
    uminus5710092332889474511et_nat: set_nat > set_nat ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    uminus6221592323253981072nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Real__Oreal_J,type,
    uminus612125837232591019t_real: set_real > set_real ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    uminus8041839845116263051T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Code____Numeral__Ointeger,type,
    zero_z3403309356797280102nteger: code_integer ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
    zero_zero_complex: complex ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
    zero_z5237406670263579293d_enat: extended_enat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
    zero_zero_int: int ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Rat__Orat,type,
    zero_zero_rat: rat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
    zero_zero_real: real ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Int__Oint,type,
    groups4538972089207619220nt_int: ( int > int ) > set_int > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
    groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
    groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Int__Oint,type,
    groups1705073143266064639nt_int: ( int > int ) > set_int > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Int__Oint,type,
    groups705719431365010083at_int: ( nat > int ) > set_nat > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Nat__Onat,type,
    groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).

thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001_Eo_001t__Int__Oint,type,
    groups9116527308978886569_o_int: ( $o > int ) > int > list_o > int ).

thf(sy_c_HOL_OThe_001t__Int__Oint,type,
    the_int: ( int > $o ) > int ).

thf(sy_c_HOL_OThe_001t__Real__Oreal,type,
    the_real: ( real > $o ) > real ).

thf(sy_c_Heap_Oarray_Osize__array_001t__VEBT____BuildupMemImp__OVEBTi,type,
    size_a6397454172108246045_VEBTi: ( vEBT_VEBTi > nat ) > array_VEBT_VEBTi > nat ).

thf(sy_c_Heap__Time__Monad_Oreturn_001_Eo,type,
    heap_Time_return_o: $o > heap_Time_Heap_o ).

thf(sy_c_Heap__Time__Monad_Oreturn_001t__Nat__Onat,type,
    heap_Time_return_nat: nat > heap_Time_Heap_nat ).

thf(sy_c_Heap__Time__Monad_Oreturn_001t__Option__Ooption_It__Nat__Onat_J,type,
    heap_T3487192422709364219on_nat: option_nat > heap_T2636463487746394924on_nat ).

thf(sy_c_Heap__Time__Monad_Oreturn_001t__VEBT____BuildupMemImp__OVEBTi,type,
    heap_T3630416162098727440_VEBTi: vEBT_VEBTi > heap_T8145700208782473153_VEBTi ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001_Eo,type,
    hoare_hoare_triple_o: assn > heap_Time_Heap_o > ( $o > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__Nat__Onat,type,
    hoare_3067605981109127869le_nat: assn > heap_Time_Heap_nat > ( nat > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__Option__Ooption_It__Nat__Onat_J,type,
    hoare_7629718768684598413on_nat: assn > heap_T2636463487746394924on_nat > ( option_nat > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__VEBT____BuildupMemImp__OVEBTi,type,
    hoare_1429296392585015714_VEBTi: assn > heap_T8145700208782473153_VEBTi > ( vEBT_VEBTi > assn ) > $o ).

thf(sy_c_If_001t__Assertions__Oassn,type,
    if_assn: $o > assn > assn > assn ).

thf(sy_c_If_001t__Code____Numeral__Ointeger,type,
    if_Code_integer: $o > code_integer > code_integer > code_integer ).

thf(sy_c_If_001t__Complex__Ocomplex,type,
    if_complex: $o > complex > complex > complex ).

thf(sy_c_If_001t__Int__Oint,type,
    if_int: $o > int > int > int ).

thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
    if_list_int: $o > list_int > list_int > list_int ).

thf(sy_c_If_001t__Nat__Onat,type,
    if_nat: $o > nat > nat > nat ).

thf(sy_c_If_001t__Num__Onum,type,
    if_num: $o > num > num > num ).

thf(sy_c_If_001t__Option__Ooption_It__Nat__Onat_J,type,
    if_option_nat: $o > option_nat > option_nat > option_nat ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_If_001t__Rat__Orat,type,
    if_rat: $o > rat > rat > rat ).

thf(sy_c_If_001t__Real__Oreal,type,
    if_real: $o > real > real > real ).

thf(sy_c_If_001t__Set__Oset_It__Int__Oint_J,type,
    if_set_int: $o > set_int > set_int > set_int ).

thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
    if_set_nat: $o > set_nat > set_nat > set_nat ).

thf(sy_c_If_001t__VEBT____Definitions__OVEBT,type,
    if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).

thf(sy_c_Int_Oint__ge__less__than,type,
    int_ge_less_than: int > set_Pr958786334691620121nt_int ).

thf(sy_c_Int_Oint__ge__less__than2,type,
    int_ge_less_than2: int > set_Pr958786334691620121nt_int ).

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

thf(sy_c_Int_Oring__1__class_OInts_001t__Complex__Ocomplex,type,
    ring_1_Ints_complex: set_complex ).

thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
    ring_1_Ints_real: set_real ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
    ring_18347121197199848620nteger: int > code_integer ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Complex__Ocomplex,type,
    ring_17405671764205052669omplex: int > complex ).

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,
    ring_1_of_int_rat: int > rat ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
    ring_1_of_int_real: int > real ).

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

thf(sy_c_Least__significant__bit_Olsb__class_Olsb_001t__Code____Numeral__Ointeger,type,
    least_7544222001954398261nteger: code_integer > $o ).

thf(sy_c_Least__significant__bit_Olsb__class_Olsb_001t__Int__Oint,type,
    least_4859182151741483524sb_int: int > $o ).

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

thf(sy_c_List_Ofoldr_001_Eo_001t__Nat__Onat,type,
    foldr_o_nat: ( $o > nat > nat ) > list_o > nat > nat ).

thf(sy_c_List_Ofoldr_001t__Nat__Onat_001t__Nat__Onat,type,
    foldr_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).

thf(sy_c_List_Ofoldr_001t__Real__Oreal_001t__Nat__Onat,type,
    foldr_real_nat: ( real > nat > nat ) > list_real > nat > nat ).

thf(sy_c_List_Ofoldr_001t__Real__Oreal_001t__Real__Oreal,type,
    foldr_real_real: ( real > real > real ) > list_real > real > real ).

thf(sy_c_List_Ofoldr_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat,type,
    foldr_VEBT_VEBTi_nat: ( vEBT_VEBTi > nat > nat ) > list_VEBT_VEBTi > nat > nat ).

thf(sy_c_List_Ofoldr_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    foldr_VEBT_VEBT_nat: ( vEBT_VEBT > nat > nat ) > list_VEBT_VEBT > nat > nat ).

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

thf(sy_c_List_Olist_OCons_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_Omap_001_Eo_001t__Nat__Onat,type,
    map_o_nat: ( $o > nat ) > list_o > list_nat ).

thf(sy_c_List_Olist_Omap_001_Eo_001t__Real__Oreal,type,
    map_o_real: ( $o > real ) > list_o > list_real ).

thf(sy_c_List_Olist_Omap_001_Eo_001t__VEBT____BuildupMemImp__OVEBTi,type,
    map_o_VEBT_VEBTi: ( $o > vEBT_VEBTi ) > list_o > list_VEBT_VEBTi ).

thf(sy_c_List_Olist_Omap_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    map_o_VEBT_VEBT: ( $o > vEBT_VEBT ) > list_o > list_VEBT_VEBT ).

thf(sy_c_List_Olist_Omap_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    map_complex_complex: ( complex > complex ) > list_complex > list_complex ).

thf(sy_c_List_Olist_Omap_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    map_complex_real: ( complex > real ) > list_complex > list_real ).

thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Int__Oint,type,
    map_int_int: ( int > int ) > list_int > list_int ).

thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Real__Oreal,type,
    map_int_real: ( int > real ) > list_int > list_real ).

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

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Real__Oreal,type,
    map_nat_real: ( nat > real ) > list_nat > list_real ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    map_nat_VEBT_VEBT: ( nat > vEBT_VEBT ) > list_nat > list_VEBT_VEBT ).

thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Nat__Onat,type,
    map_real_nat: ( real > nat ) > list_real > list_nat ).

thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Real__Oreal,type,
    map_real_real: ( real > real ) > list_real > list_real ).

thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__VEBT____BuildupMemImp__OVEBTi,type,
    map_real_VEBT_VEBTi: ( real > vEBT_VEBTi ) > list_real > list_VEBT_VEBTi ).

thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    map_real_VEBT_VEBT: ( real > vEBT_VEBT ) > list_real > list_VEBT_VEBT ).

thf(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat,type,
    map_VEBT_VEBTi_nat: ( vEBT_VEBTi > nat ) > list_VEBT_VEBTi > list_nat ).

thf(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__Real__Oreal,type,
    map_VEBT_VEBTi_real: ( vEBT_VEBTi > real ) > list_VEBT_VEBTi > list_real ).

thf(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    map_VE483055756984248624_VEBTi: ( vEBT_VEBTi > vEBT_VEBTi ) > list_VEBT_VEBTi > list_VEBT_VEBTi ).

thf(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    map_VE7998069337340375161T_VEBT: ( vEBT_VEBTi > vEBT_VEBT ) > list_VEBT_VEBTi > list_VEBT_VEBT ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    map_VEBT_VEBT_nat: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > list_nat ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
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thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    map_VE8901447254227204932T_VEBT: ( vEBT_VEBT > vEBT_VEBT ) > list_VEBT_VEBT > list_VEBT_VEBT ).

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

thf(sy_c_List_Olist_Oset_001t__Code____Numeral__Ointeger,type,
    set_Code_integer2: list_Code_integer > set_Code_integer ).

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__VEBT____BuildupMemImp__OVEBTi,type,
    set_VEBT_VEBTi2: list_VEBT_VEBTi > set_VEBT_VEBTi ).

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

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

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

thf(sy_c_List_Olist__update_001t__Complex__Ocomplex,type,
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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__VEBT____BuildupMemImp__OVEBTi,type,
    list_u6098035379799741383_VEBTi: list_VEBT_VEBTi > nat > vEBT_VEBTi > list_VEBT_VEBTi ).

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,
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thf(sy_c_List_Onth_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
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thf(sy_c_List_Onth_001t__Int__Oint,type,
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thf(sy_c_List_Onth_001t__Nat__Onat,type,
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thf(sy_c_List_Onth_001t__Num__Onum,type,
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thf(sy_c_List_Onth_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_List_Onth_001t__Real__Oreal,type,
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thf(sy_c_List_Onth_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Onth_001tf__c____11__058ATP,type,
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thf(sy_c_List_Oproduct_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
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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__Real__Oreal_001_Eo,type,
    product_real_o: list_real > list_o > list_P3595434254542482545real_o ).

thf(sy_c_List_Oproduct_001t__Real__Oreal_001t__Nat__Onat,type,
    product_real_nat: list_real > list_nat > list_P6834414599653733731al_nat ).

thf(sy_c_List_Oproduct_001t__Real__Oreal_001t__Real__Oreal,type,
    product_real_real: list_real > list_real > list_P8689742595348180415l_real ).

thf(sy_c_List_Oproduct_001t__Real__Oreal_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_List_Oproduct_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
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thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
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thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Oupto__aux,type,
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thf(sy_c_List_Oupto__rel,type,
    upto_rel: product_prod_int_int > product_prod_int_int > $o ).

thf(sy_c_Nat_OSuc,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
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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__Int__Oint,type,
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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,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
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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__Option__Ooption_It__Nat__Onat_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_M_Eo_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____BuildupMemImp__OVEBTi_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__Uint32__Ouint32,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Nat__Bijection_Oset__decode,type,
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thf(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: set_nat > nat ).

thf(sy_c_Nat__Bijection_Otriangle,type,
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thf(sy_c_NthRoot_Oroot,type,
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thf(sy_c_NthRoot_Osqrt,type,
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thf(sy_c_Num_OBitM,type,
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thf(sy_c_Num_Oinc,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
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thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
    neg_nu6075765906172075777c_real: real > real ).

thf(sy_c_Num_Onum_OBit0,type,
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thf(sy_c_Num_Onum_OBit1,type,
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thf(sy_c_Num_Onum_OOne,type,
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thf(sy_c_Num_Onum_Osize__num,type,
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
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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,
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thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
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thf(sy_c_Num_Opow,type,
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thf(sy_c_Num_Opred__numeral,type,
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thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
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thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    none_P533106815845188193et_nat: option936205604648967762et_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
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thf(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
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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,
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thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    some_P624177172695371229et_nat: produc3658429121746597890et_nat > option936205604648967762et_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    some_P4184893108420464158nt_int: product_prod_int_int > option4624381673175914239nt_int ).

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

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    some_P6201964756284913402um_num: product_prod_num_num > option2661157926820139483um_num ).

thf(sy_c_Option_Ooption_OSome_001t__Rat__Orat,type,
    some_rat: rat > option_rat ).

thf(sy_c_Option_Ooption_OSome_001t__Real__Oreal,type,
    some_real: real > option_real ).

thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Nat__Onat_J,type,
    some_set_nat: set_nat > option_set_nat ).

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

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__Complex__Ocomplex_M_Eo_J,type,
    bot_bot_complex_o: complex > $o ).

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

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

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    bot_bo8147686125503663512_int_o: product_prod_int_int > $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_001t__Assertions__Oassn,type,
    bot_bot_assn: assn ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Option__Ooption_It__Nat__Onat_J,type,
    bot_bot_option_nat: option_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__Code____Numeral__Ointeger_J,type,
    bot_bo3990330152332043303nteger: set_Code_integer ).

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__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
    bot_bot_set_num: set_num ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int ).

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__VEBT____Definitions__OVEBT_M_Eo_J,type,
    ord_less_VEBT_VEBT_o: ( vEBT_VEBT > $o ) > ( vEBT_VEBT > $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__Option__Ooption_It__Int__Oint_J,type,
    ord_less_option_int: option_int > option_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Nat__Onat_J,type,
    ord_less_option_nat: option_nat > option_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Num__Onum_J,type,
    ord_less_option_num: option_num > option_num > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Rat__Orat_J,type,
    ord_less_option_rat: option_rat > option_rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Real__Oreal_J,type,
    ord_less_option_real: option_real > option_real > $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__Code____Numeral__Ointeger_J,type,
    ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o ).

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__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_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    ord_le3480810397992357184T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $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__VEBT____Definitions__OVEBT_M_Eo_J,type,
    ord_le418104280809901481VEBT_o: ( vEBT_VEBT > $o ) > ( vEBT_VEBT > $o ) > $o ).

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__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__Option__Ooption_It__Int__Oint_J,type,
    ord_le1736525451366464988on_int: option_int > option_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Nat__Onat_J,type,
    ord_le5914376470875661696on_nat: option_nat > option_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Num__Onum_J,type,
    ord_le6622620407824499402on_num: option_num > option_num > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Rat__Orat_J,type,
    ord_le2406147912482264968on_rat: option_rat > option_rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_le2843612097646854710et_nat: option_set_nat > option_set_nat > $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__Code____Numeral__Ointeger_J,type,
    ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o ).

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

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

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__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____BuildupMemImp__OVEBTi_J,type,
    ord_le6592769550269828683_VEBTi: set_VEBT_VEBTi > set_VEBT_VEBTi > $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__Int__Oint,type,
    ord_max_int: int > int > int ).

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Orderings_Otop__class_Otop_001t__Assertions__Oassn,type,
    top_top_assn: assn ).

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__Numeral____Type__Onum0_J,type,
    top_to3689904424835650196l_num0: set_Numeral_num0 ).

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__Oliteral_J,type,
    top_top_set_literal: set_literal ).

thf(sy_c_Power_Opower__class_Opower_001t__Assertions__Oassn,type,
    power_power_assn: assn > nat > assn ).

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

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

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

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

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

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

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_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__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc3209952032786966637at_nat: ( nat > nat > nat ) > produc7248412053542808358at_nat > produc4471711990508489141at_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__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__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,
    produc3994169339658061776at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > produc6121120109295599847at_nat > produc5491161045314408544at_nat ).

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,
    produc2899441246263362727at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > produc6121120109295599847at_nat > produc5542196010084753463at_nat ).

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,
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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__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,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__Real__Oreal,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,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__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_Rat_OFrct,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_Refine__Imp__Hol_Orefines_001_Eo,type,
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thf(sy_c_Refine__Imp__Hol_Orefines_001t__List__Olist_I_Eo_J,type,
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thf(sy_c_Refine__Imp__Hol_Orefines_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,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__Assertions__Oassn,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
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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__Int__Oint,type,
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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
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thf(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
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thf(sy_c_Series_Osuminf_001t__Int__Oint,type,
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thf(sy_c_Series_Osuminf_001t__Nat__Onat,type,
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thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
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thf(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
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thf(sy_c_Series_Osummable_001t__Int__Oint,type,
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thf(sy_c_Series_Osummable_001t__Nat__Onat,type,
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thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
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thf(sy_c_Series_Osums_001t__Real__Oreal,type,
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thf(sy_c_Set_OCollect_001_Eo,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_I_Eo_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__Real__Oreal_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Set_Oimage_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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thf(sy_c_Set_Oinsert_001_Eo,type,
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thf(sy_c_Set_Oinsert_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
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thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
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thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
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thf(sy_c_Set_Oinsert_001t__Rat__Orat,type,
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thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
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thf(sy_c_Set_Oinsert_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_Set_Oinsert_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001_Eo,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Num__Onum,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Rat__Orat,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Real__Oreal,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
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thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
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thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Code____Numeral__Ointeger,type,
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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__Code____Numeral__Ointeger,type,
    set_or4266950643985792945nteger: code_integer > code_integer > set_Code_integer ).

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_Signed__Division_Osigned__division__class_Osigned__divide_001t__Int__Oint,type,
    signed6714573509424544716de_int: int > int > int ).

thf(sy_c_Signed__Division_Osigned__division__class_Osigned__modulo_001t__Int__Oint,type,
    signed6292675348222524329lo_int: int > int > int ).

thf(sy_c_Time__Reasoning_OTBOUND_001_Eo,type,
    time_TBOUND_o: heap_Time_Heap_o > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_I_Eo_J,type,
    time_TBOUND_list_o: heap_T844314716496656296list_o > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__Nat__Onat_J,type,
    time_TBOUND_list_nat: heap_T290393402774840812st_nat > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
    time_T3808005469503390304on_nat: heap_T5317711798761887292on_nat > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    time_T8149879359713347829_VEBTi: heap_T4980287057938770641_VEBTi > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__Nat__Onat,type,
    time_TBOUND_nat: heap_Time_Heap_nat > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__Option__Ooption_It__Nat__Onat_J,type,
    time_T8353473612707095248on_nat: heap_T2636463487746394924on_nat > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_T5737551269749752165_VEBTi: heap_T8145700208782473153_VEBTi > nat > $o ).

thf(sy_c_Time__Reasoning_Ohtt_001_Eo,type,
    time_htt_o: assn > heap_Time_Heap_o > ( $o > assn ) > nat > $o ).

thf(sy_c_Time__Reasoning_Ohtt_001t__Nat__Onat,type,
    time_htt_nat: assn > heap_Time_Heap_nat > ( nat > assn ) > nat > $o ).

thf(sy_c_Time__Reasoning_Ohtt_001t__Option__Ooption_It__Nat__Onat_J,type,
    time_htt_option_nat: assn > heap_T2636463487746394924on_nat > ( option_nat > assn ) > nat > $o ).

thf(sy_c_Time__Reasoning_Ohtt_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_htt_VEBT_VEBTi: assn > heap_T8145700208782473153_VEBTi > ( vEBT_VEBTi > assn ) > nat > $o ).

thf(sy_c_Time__Reasoning_Otime_001_Eo,type,
    time_time_o: heap_Time_Heap_o > heap_e7401611519738050253t_unit > nat ).

thf(sy_c_Time__Reasoning_Otime_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    time_t3534373299052942712_VEBTi: heap_T4980287057938770641_VEBTi > heap_e7401611519738050253t_unit > nat ).

thf(sy_c_Time__Reasoning_Otime_001t__Nat__Onat,type,
    time_time_nat: heap_Time_Heap_nat > heap_e7401611519738050253t_unit > nat ).

thf(sy_c_Time__Reasoning_Otime_001t__Option__Ooption_It__Nat__Onat_J,type,
    time_time_option_nat: heap_T2636463487746394924on_nat > heap_e7401611519738050253t_unit > nat ).

thf(sy_c_Time__Reasoning_Otime_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_time_VEBT_VEBTi: heap_T8145700208782473153_VEBTi > heap_e7401611519738050253t_unit > nat ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Transcendental_Opi,type,
    pi: real ).

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

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

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

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

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

thf(sy_c_Transcendental_Otan_001t__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_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____2,type,
    type_l31302759751748492nite_2: itself_finite_2 > nat ).

thf(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____3,type,
    type_l31302759751748493nite_3: itself_finite_3 > nat ).

thf(sy_c_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J,type,
    type_l796852477590012082l_num1: itself8794530163899892676l_num1 > nat ).

thf(sy_c_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Onum0,type,
    type_l4264026598287037464l_num0: itself_Numeral_num0 > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t,type,
    vEBT_T_i_n_s_e_r_t: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H,type,
    vEBT_T_i_n_s_e_r_t2: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H__rel,type,
    vEBT_T5076183648494686801_t_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t__rel,type,
    vEBT_T9217963907923527482_t_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t,type,
    vEBT_T_m_a_x_t: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t__rel,type,
    vEBT_T_m_a_x_t_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r,type,
    vEBT_T_m_e_m_b_e_r: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H,type,
    vEBT_T_m_e_m_b_e_r2: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H__rel,type,
    vEBT_T8099345112685741742_r_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r__rel,type,
    vEBT_T5837161174952499735_r_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l,type,
    vEBT_T_m_i_n_N_u_l_l: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l__rel,type,
    vEBT_T5462971552011256508_l_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t,type,
    vEBT_T_m_i_n_t: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t__rel,type,
    vEBT_T_m_i_n_t_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d,type,
    vEBT_T_p_r_e_d: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H,type,
    vEBT_T_p_r_e_d2: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H__rel,type,
    vEBT_T_p_r_e_d_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d__rel,type,
    vEBT_T_p_r_e_d_rel2: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c,type,
    vEBT_T_s_u_c_c: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H,type,
    vEBT_T_s_u_c_c2: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H__rel,type,
    vEBT_T_s_u_c_c_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c__rel,type,
    vEBT_T_s_u_c_c_rel2: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi,type,
    vEBT_V441764108873111860ildupi: nat > nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H,type,
    vEBT_V9176841429113362141ildupi: nat > int ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H__rel,type,
    vEBT_V3352910403632780892pi_rel: nat > nat > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi__rel,type,
    vEBT_V2957053500504383685pi_rel: nat > nat > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb,type,
    vEBT_VEBT_Tb: nat > int ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H,type,
    vEBT_VEBT_Tb2: nat > nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H__rel,type,
    vEBT_VEBT_Tb_rel: nat > nat > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb__rel,type,
    vEBT_VEBT_Tb_rel2: nat > nat > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ohighi,type,
    vEBT_VEBT_highi: nat > nat > heap_Time_Heap_nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Olowi,type,
    vEBT_VEBT_lowi: nat > nat > heap_Time_Heap_nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OminNulli,type,
    vEBT_VEBT_minNulli: vEBT_VEBTi > heap_Time_Heap_o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OminNulli__rel,type,
    vEBT_V5740978063120863272li_rel: vEBT_VEBTi > vEBT_VEBTi > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001_Eo,type,
    vEBT_V2326993469660664182atei_o: nat > heap_Time_Heap_o > heap_T844314716496656296list_o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__Nat__Onat,type,
    vEBT_V7726092123322077554ei_nat: nat > heap_Time_Heap_nat > heap_T290393402774840812st_nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__Option__Ooption_It__Nat__Onat_J,type,
    vEBT_V792416675989592002on_nat: nat > heap_T2636463487746394924on_nat > heap_T5317711798761887292on_nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_V1859673955506687831_VEBTi: nat > heap_T8145700208782473153_VEBTi > heap_T4980287057938770641_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__buildupi_H,type,
    vEBT_V739175172307565963ildupi: nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__inserti_H,type,
    vEBT_V3964819847710782039nserti: vEBT_VEBT > vEBT_VEBTi > nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__memberi_H,type,
    vEBT_V854960066525838166emberi: vEBT_VEBT > vEBT_VEBTi > nat > heap_Time_Heap_o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_OLeafi,type,
    vEBT_Leafi: $o > $o > vEBT_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_ONodei,type,
    vEBT_Nodei: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > vEBT_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    vEBT_c6104975476656191286Heap_o: ( option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_o ) > ( $o > $o > heap_Time_Heap_o ) > vEBT_VEBTi > heap_Time_Heap_o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
    vEBT_c1335663792808957512ap_nat: ( option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_nat ) > ( $o > $o > heap_Time_Heap_nat ) > vEBT_VEBTi > heap_Time_Heap_nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
    vEBT_c6250501799366334488on_nat: ( option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat ) > ( $o > $o > heap_T2636463487746394924on_nat ) > vEBT_VEBTi > heap_T2636463487746394924on_nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    vEBT_c6028912655521741485_VEBTi: ( option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi ) > ( $o > $o > heap_T8145700208782473153_VEBTi ) > vEBT_VEBTi > heap_T8145700208782473153_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Nat__Onat,type,
    vEBT_case_VEBTi_nat: ( option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > nat ) > ( $o > $o > nat ) > vEBT_VEBTi > nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Osize__VEBTi,type,
    vEBT_size_VEBTi: vEBT_VEBTi > nat ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw,type,
    vEBT_vebt_assn_raw: vEBT_VEBT > vEBT_VEBTi > assn ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw__rel,type,
    vEBT_v8524038756793281170aw_rel: produc3625547720036274456_VEBTi > produc3625547720036274456_VEBTi > $o ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__buildupi,type,
    vEBT_vebt_buildupi: nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__inserti,type,
    vEBT_vebt_inserti: vEBT_VEBTi > nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__maxti,type,
    vEBT_vebt_maxti: vEBT_VEBTi > heap_T2636463487746394924on_nat ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__maxti__rel,type,
    vEBT_vebt_maxti_rel: vEBT_VEBTi > vEBT_VEBTi > $o ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__memberi,type,
    vEBT_vebt_memberi: vEBT_VEBTi > nat > heap_Time_Heap_o ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__minti,type,
    vEBT_vebt_minti: vEBT_VEBTi > heap_T2636463487746394924on_nat ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__minti__rel,type,
    vEBT_vebt_minti_rel: vEBT_VEBTi > vEBT_VEBTi > $o ).

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

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

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

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

thf(sy_c_VEBT__Definitions_OVEBT__internal_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_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__DelImperative_OVEBT__internal_Ovebt__deletei_H,type,
    vEBT_V1365221501068881998eletei: vEBT_VEBT > vEBT_VEBTi > nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e,type,
    vEBT_T_d_e_l_e_t_e: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e__rel,type,
    vEBT_T8441311223069195367_e_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H,type,
    vEBT_V1232361888498592333_e_t_e: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H__rel,type,
    vEBT_V6368547301243506412_e_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_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__Height_OVEBT__internal_Oheight,type,
    vEBT_VEBT_height: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
    vEBT_VEBT_height_rel: vEBT_VEBT > vEBT_VEBT > $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__List__Assn_OlistI__assn_001_Eo_001t__Nat__Onat,type,
    vEBT_L2281750874075065672_o_nat: set_nat > ( $o > nat > assn ) > list_o > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001_Eo_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L6286945158656146733_VEBTi: set_nat > ( $o > vEBT_VEBTi > assn ) > list_o > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L1319876754960170684T_VEBT: set_nat > ( $o > vEBT_VEBT > assn ) > list_o > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__Nat__Onat_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L7489483478785760935_VEBTi: set_nat > ( nat > vEBT_VEBTi > assn ) > list_nat > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L8511957252848910786T_VEBT: set_nat > ( nat > vEBT_VEBT > assn ) > list_nat > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001_Eo,type,
    vEBT_L7980206306069228746real_o: set_nat > ( real > $o > assn ) > list_real > list_o > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001t__Nat__Onat,type,
    vEBT_L234762979517870878al_nat: set_nat > ( real > nat > assn ) > list_real > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001t__Real__Oreal,type,
    vEBT_L5184575500739366650l_real: set_nat > ( real > real > assn ) > list_real > list_real > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L7851252805511451907_VEBTi: set_nat > ( real > vEBT_VEBTi > assn ) > list_real > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L3095048238742455910T_VEBT: set_nat > ( real > vEBT_VEBT > assn ) > list_real > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L886525131989349516_VEBTi: set_nat > ( vEBT_VEBTi > vEBT_VEBTi > assn ) > list_VEBT_VEBTi > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L2497118539674116125T_VEBT: set_nat > ( vEBT_VEBTi > vEBT_VEBT > assn ) > list_VEBT_VEBTi > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    vEBT_L7058566406413635588VEBT_o: set_nat > ( vEBT_VEBT > $o > assn ) > list_VEBT_VEBT > list_o > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    vEBT_L8650695023172932196BT_nat: set_nat > ( vEBT_VEBT > nat > assn ) > list_VEBT_VEBT > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    vEBT_L4281036506115550016T_real: set_nat > ( vEBT_VEBT > real > assn ) > list_VEBT_VEBT > list_real > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L1528199826722428489_VEBTi: set_nat > ( vEBT_VEBT > vEBT_VEBTi > assn ) > list_VEBT_VEBT > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L3204528365124325536T_VEBT: set_nat > ( vEBT_VEBT > vEBT_VEBT > assn ) > list_VEBT_VEBT > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_OlistI__assn_001tf__c____11__058ATP_001tf__d____11__058ATP,type,
    vEBT_L375988980963497884911_ATP: set_nat > ( c_11_ATP > d_11_ATP > assn ) > list_c_11_ATP > list_d_11_ATP > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001_Eo_001t__Nat__Onat,type,
    vEBT_L4785011123346445925_o_nat: ( $o > nat > assn ) > list_o > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001_Eo_001t__Real__Oreal,type,
    vEBT_L4725278957065240257o_real: ( $o > real > assn ) > list_o > list_real > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001_Eo_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L3704169666673096010_VEBTi: ( $o > vEBT_VEBTi > assn ) > list_o > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L1750719106661372127T_VEBT: ( $o > vEBT_VEBT > assn ) > list_o > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    vEBT_L4260503343685368993omplex: ( complex > complex > assn ) > list_complex > list_complex > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__Int__Oint,type,
    vEBT_L134985006839036959ex_int: ( complex > int > assn ) > list_complex > list_int > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__Nat__Onat,type,
    vEBT_L137475477348087235ex_nat: ( complex > nat > assn ) > list_complex > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    vEBT_L2479436891206192927x_real: ( complex > real > assn ) > list_complex > list_real > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Complex__Ocomplex_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L8524933119956041985T_VEBT: ( complex > vEBT_VEBT > assn ) > list_complex > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__Complex__Ocomplex,type,
    vEBT_L6716599654974302751omplex: ( int > complex > assn ) > list_int > list_complex > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__Int__Oint,type,
    vEBT_L74593716426352029nt_int: ( int > int > assn ) > list_int > list_int > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__Nat__Onat,type,
    vEBT_L77084186935402305nt_nat: ( int > nat > assn ) > list_int > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__Real__Oreal,type,
    vEBT_L8288995350762215837t_real: ( int > real > assn ) > list_int > list_real > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L1664421287176695555T_VEBT: ( int > vEBT_VEBT > assn ) > list_int > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__Real__Oreal,type,
    vEBT_L6102073776069194049t_real: ( nat > real > assn ) > list_nat > list_real > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L4387162340545308618_VEBTi: ( nat > vEBT_VEBTi > assn ) > list_nat > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L8158188754432654943T_VEBT: ( nat > vEBT_VEBT > assn ) > list_nat > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001_Eo,type,
    vEBT_L6234343332106409831real_o: ( real > $o > assn ) > list_real > list_o > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__Nat__Onat,type,
    vEBT_L1446010312343316929al_nat: ( real > nat > assn ) > list_real > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__Real__Oreal,type,
    vEBT_L1930518968523514909l_real: ( real > real > assn ) > list_real > list_real > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L9060850011106065574_VEBTi: ( real > vEBT_VEBTi > assn ) > list_real > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L4595930785310033027T_VEBT: ( real > vEBT_VEBT > assn ) > list_real > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__Real__Oreal,type,
    vEBT_L8937798142398754470i_real: ( vEBT_VEBTi > real > assn ) > list_VEBT_VEBTi > list_real > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L7265847600308530106T_VEBT: ( vEBT_VEBTi > vEBT_VEBT > assn ) > list_VEBT_VEBTi > list_VEBT_VEBT > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    vEBT_L7489408758114837031VEBT_o: ( vEBT_VEBT > $o > assn ) > list_VEBT_VEBT > list_o > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    vEBT_L8296926524756676353BT_nat: ( vEBT_VEBT > nat > assn ) > list_VEBT_VEBT > list_nat > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    vEBT_L5781919052683127133T_real: ( vEBT_VEBT > real > assn ) > list_VEBT_VEBT > list_real > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L6296928887356842470_VEBTi: ( vEBT_VEBT > vEBT_VEBTi > assn ) > list_VEBT_VEBT > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    vEBT_L1279224858307276611T_VEBT: ( vEBT_VEBT > vEBT_VEBT > assn ) > list_VEBT_VEBT > list_VEBT_VEBT > assn ).

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__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__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d,type,
    vEBT_V8646137997579335489_i_l_d: nat > nat ).

thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p,type,
    vEBT_V8346862874174094_d_u_p: nat > nat ).

thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p__rel,type,
    vEBT_V1247956027447740395_p_rel: nat > nat > $o ).

thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d__rel,type,
    vEBT_V5144397997797733112_d_rel: nat > nat > $o ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt,type,
    vEBT_VEBT_cnt: vEBT_VEBT > real ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H,type,
    vEBT_VEBT_cnt2: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H__rel,type,
    vEBT_VEBT_cnt_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt__rel,type,
    vEBT_VEBT_cnt_rel2: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ospace,type,
    vEBT_VEBT_space: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ospace_H,type,
    vEBT_VEBT_space2: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ospace_H__rel,type,
    vEBT_VEBT_space_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
    vEBT_VEBT_space_rel2: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__predi_H,type,
    vEBT_VEBT_vebt_predi: vEBT_VEBT > vEBT_VEBTi > nat > heap_T2636463487746394924on_nat ).

thf(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__succi_H,type,
    vEBT_VEBT_vebt_succi: vEBT_VEBT > vEBT_VEBTi > nat > heap_T2636463487746394924on_nat ).

thf(sy_c_VEBT__SuccPredImperative_Ovebt__predi,type,
    vEBT_vebt_predi: vEBT_VEBTi > nat > heap_T2636463487746394924on_nat ).

thf(sy_c_VEBT__SuccPredImperative_Ovebt__succi,type,
    vEBT_vebt_succi: vEBT_VEBTi > nat > heap_T2636463487746394924on_nat ).

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__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__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    accp_P7675410724331315407_VEBTi: ( produc3625547720036274456_VEBTi > produc3625547720036274456_VEBTi > $o ) > produc3625547720036274456_VEBTi > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__VEBT____BuildupMemImp__OVEBTi,type,
    accp_VEBT_VEBTi: ( vEBT_VEBTi > vEBT_VEBTi > $o ) > vEBT_VEBTi > $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_fChoice_001t__Real__Oreal,type,
    fChoice_real: ( real > $o ) > real ).

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

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

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

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

thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
    member_list_o: list_o > set_list_o > $o ).

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

thf(sy_c_member_001t__List__Olist_It__Real__Oreal_J,type,
    member_list_real: list_real > set_list_real > $o ).

thf(sy_c_member_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    member8117914210271334748_VEBTi: list_VEBT_VEBTi > set_list_VEBT_VEBTi > $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__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    member157494554546826820nteger: produc8923325533196201883nteger > set_Pr4811707699266497531nteger > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member6260224972018164377et_nat: produc3658429121746597890et_nat > set_Pr3948176798113811640et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    member7279096912039735102um_num: product_prod_num_num > set_Pr8218934625190621173um_num > $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____BuildupMemImp__OVEBTi,type,
    member_VEBT_VEBTi: vEBT_VEBTi > set_VEBT_VEBTi > $o ).

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

thf(sy_v_A__11_058ATP,type,
    a_11_ATP: c_11_ATP > d_11_ATP > assn ).

thf(sy_v_F__11_058ATP,type,
    f_11_ATP: assn ).

thf(sy_v_I__11_058ATP,type,
    i_11_ATP: set_nat ).

thf(sy_v_aktnode____,type,
    aktnode: vEBT_VEBT ).

thf(sy_v_i__11_058ATP,type,
    i_11_ATP2: nat ).

thf(sy_v_ma____,type,
    ma: nat ).

thf(sy_v_mi____,type,
    mi: nat ).

thf(sy_v_minew____,type,
    minew: nat ).

thf(sy_v_na____,type,
    na: nat ).

thf(sy_v_newnode____,type,
    newnode: vEBT_VEBT ).

thf(sy_v_summary____,type,
    summary: vEBT_VEBT ).

thf(sy_v_tia____,type,
    tia: vEBT_VEBTi ).

thf(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

thf(sy_v_tree__is______,type,
    tree_is: list_VEBT_VEBTi ).

thf(sy_v_uu__16_058ATP,type,
    uu_16_ATP: list_c_11_ATP ).

thf(sy_v_uua__16_058ATP,type,
    uua_16_ATP: nat ).

thf(sy_v_va____,type,
    va: nat ).

thf(sy_v_x11______,type,
    x11: option4927543243414619207at_nat ).

thf(sy_v_x13______,type,
    x13: array_VEBT_VEBTi ).

thf(sy_v_x14______,type,
    x14: vEBT_VEBTi ).

thf(sy_v_xa____,type,
    xa: nat ).

thf(sy_v_xb______,type,
    xb: vEBT_VEBTi ).

thf(sy_v_xi__11_058ATP,type,
    xi_11_ATP: d_11_ATP ).

thf(sy_v_xnew____,type,
    xnew: nat ).

thf(sy_v_xs__11_058ATP,type,
    xs_11_ATP: list_c_11_ATP ).

thf(sy_v_xsi__11_058ATP,type,
    xsi_11_ATP: list_d_11_ATP ).

thf(sy_v_y______,type,
    y: nat ).

% Relevant facts (10206)
thf(fact_0_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N: nat] :
          ( X
         != ( plus_plus_nat @ N @ N ) )
     => ~ ! [N: nat] :
            ( X
           != ( plus_plus_nat @ N @ ( suc @ N ) ) ) ) ).

% even_odd_cases
thf(fact_1_groupy,axiom,
    ! [A: assn,B: assn,C: assn,D: assn,X2: assn] :
      ( ( entails @ ( times_times_assn @ ( times_times_assn @ A @ B ) @ ( times_times_assn @ C @ D ) ) @ X2 )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ A @ B ) @ C ) @ D ) @ X2 ) ) ).

% groupy
thf(fact_2_midextr,axiom,
    ! [P: assn,Q: assn,Q2: assn,R: assn,X2: assn] :
      ( ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ P @ Q ) @ Q2 ) @ R ) @ X2 )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ P @ R ) @ Q ) @ Q2 ) @ X2 ) ) ).

% midextr
thf(fact_3_swappa,axiom,
    ! [B: assn,A: assn,C: assn,X2: assn] :
      ( ( entails @ ( times_times_assn @ ( times_times_assn @ B @ A ) @ C ) @ X2 )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ A @ B ) @ C ) @ X2 ) ) ).

% swappa
thf(fact_4_power__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( power_power_nat @ X @ Y )
        = Z )
      = ( ( vEBT_VEBT_power @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z ) ) ) ).

% power_shift
thf(fact_5_bit__split__inv,axiom,
    ! [X: nat,D2: nat] :
      ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D2 ) @ ( vEBT_VEBT_low @ X @ D2 ) @ D2 )
      = X ) ).

% bit_split_inv
thf(fact_6_pow__sum,axiom,
    ! [A2: nat,B2: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A2 @ B2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ).

% pow_sum
thf(fact_7_mulcomm,axiom,
    ! [I: nat,Va: nat] :
      ( ( times_times_nat @ I @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
      = ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ I ) ) ).

% mulcomm
thf(fact_8_high__def,axiom,
    ( vEBT_VEBT_high
    = ( ^ [X3: nat,N2: nat] : ( divide_divide_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% high_def
thf(fact_9_high__bound__aux,axiom,
    ! [Ma: nat,N3: nat,M: nat] :
      ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N3 @ M ) ) )
     => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% high_bound_aux
thf(fact_10_high__inv,axiom,
    ! [X: nat,N3: nat,Y: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
     => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) @ X ) @ N3 )
        = Y ) ) ).

% high_inv
thf(fact_11_low__inv,axiom,
    ! [X: nat,N3: nat,Y: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
     => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) @ X ) @ N3 )
        = X ) ) ).

% low_inv
thf(fact_12_minewdef,axiom,
    ( minew
    = ( 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 @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) ) ).

% minewdef
thf(fact_13_bit__concat__def,axiom,
    ( vEBT_VEBT_bit_concat
    = ( ^ [H: nat,L: nat,D3: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D3 ) ) @ L ) ) ) ).

% bit_concat_def
thf(fact_14_xndef,axiom,
    ( xnew
    = ( 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 @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) ) ).

% xndef
thf(fact_15_newnodedef,axiom,
    ( newnode
    = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( 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 @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% newnodedef
thf(fact_16_local_Oext,axiom,
    ! [Y: nat,TreeList: list_VEBT_VEBT,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ Y @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( entails @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList @ Y ) @ ( nth_VEBT_VEBTi @ Tree_is @ Y ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList @ Y ) @ ( nth_VEBT_VEBTi @ Tree_is @ Y ) ) ) ) ) ).

% local.ext
thf(fact_17_aktnodedef,axiom,
    ( ( ma != mi )
   => ( ( ord_less_eq_nat @ xa @ ma )
     => ( aktnode
        = ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% aktnodedef
thf(fact_18__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062aktnode_O_A_I_092_060lbrakk_062ma_A_092_060noteq_062_Ami_059_Ax_A_092_060le_062_Ama_092_060rbrakk_062_A_092_060Longrightarrow_062_Aaktnode_A_061_AtreeList_A_B_Ahigh_A_I2_A_K_A2_A_094_A_Iva_Adiv_A2_J_A_K_Athe_A_Ivebt__mint_Asummary_J_A_L_Athe_A_Ivebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__mint_Asummary_J_J_J_J_A_ISuc_A_Iva_Adiv_A2_J_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Aktnode: vEBT_VEBT] :
        ~ ( ( ma != mi )
         => ( ( ord_less_eq_nat @ xa @ ma )
           => ( Aktnode
              = ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>aktnode. (\<lbrakk>ma \<noteq> mi; x \<le> ma\<rbrakk> \<Longrightarrow> aktnode = treeList ! high (2 * 2 ^ (va div 2) * the (vebt_mint summary) + the (vebt_mint (treeList ! the (vebt_mint summary)))) (Suc (va div 2))) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_19_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_20_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_21_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_22_sum__power2__eq__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_z3403309356797280102nteger )
      = ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_23_zero__less__power2,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A2 != zero_z3403309356797280102nteger ) ) ).

% zero_less_power2
thf(fact_24_zero__less__power2,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A2 != zero_zero_real ) ) ).

% zero_less_power2
thf(fact_25_zero__less__power2,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A2 != zero_zero_rat ) ) ).

% zero_less_power2
thf(fact_26_zero__less__power2,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A2 != zero_zero_int ) ) ).

% zero_less_power2
thf(fact_27_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_28_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_29_add__2__eq__Suc,axiom,
    ! [N3: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
      = ( suc @ ( suc @ N3 ) ) ) ).

% add_2_eq_Suc
thf(fact_30_add__2__eq__Suc_H,axiom,
    ! [N3: nat] :
      ( ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( suc @ N3 ) ) ) ).

% add_2_eq_Suc'
thf(fact_31_zero__eq__power2,axiom,
    ! [A2: rat] :
      ( ( ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_rat )
      = ( A2 = zero_zero_rat ) ) ).

% zero_eq_power2
thf(fact_32_zero__eq__power2,axiom,
    ! [A2: nat] :
      ( ( ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat )
      = ( A2 = zero_zero_nat ) ) ).

% zero_eq_power2
thf(fact_33_zero__eq__power2,axiom,
    ! [A2: real] :
      ( ( ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_real )
      = ( A2 = zero_zero_real ) ) ).

% zero_eq_power2
thf(fact_34_zero__eq__power2,axiom,
    ! [A2: int] :
      ( ( ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% zero_eq_power2
thf(fact_35_zero__eq__power2,axiom,
    ! [A2: complex] :
      ( ( ( power_power_complex @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_complex )
      = ( A2 = zero_zero_complex ) ) ).

% zero_eq_power2
thf(fact_36_zero__eq__power2,axiom,
    ! [A2: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% zero_eq_power2
thf(fact_37_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList2: list_VEBT_VEBT,Y: vEBT_VEBT,X: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
       => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Y @ X ) @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) ) ) ) ).

% tcd
thf(fact_38_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList2: list_real,Y: vEBT_VEBT,X: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( size_size_list_real @ TreeList2 ) )
       => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Y @ X ) @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) ) ) ) ).

% tcd
thf(fact_39_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList2: list_o,Y: vEBT_VEBT,X: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( size_size_list_o @ TreeList2 ) )
       => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Y @ X ) @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) ) ) ) ).

% tcd
thf(fact_40_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList2: list_nat,Y: vEBT_VEBT,X: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( size_size_list_nat @ TreeList2 ) )
       => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Y @ X ) @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) ) ) ) ).

% tcd
thf(fact_41_tcd,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,TreeList2: list_VEBT_VEBTi,Y: vEBT_VEBT,X: vEBT_VEBTi,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( size_s7982070591426661849_VEBTi @ TreeList2 ) )
       => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Y @ X ) @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ I @ Y ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ I @ X ) ) ) ) ) ) ).

% tcd
thf(fact_42_recomp,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList @ I ) @ ( nth_VEBT_VEBTi @ Tree_is @ I ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) ) ).

% recomp
thf(fact_43_repack,axiom,
    ! [I: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,Rest: assn,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList @ I ) @ ( nth_VEBT_VEBTi @ Tree_is @ I ) ) @ Rest ) @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ Rest @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) ) ).

% repack
thf(fact_44_max__in__set__def,axiom,
    ( vEBT_VEBT_max_in_set
    = ( ^ [Xs: set_nat,X3: nat] :
          ( ( member_nat @ X3 @ Xs )
          & ! [Y2: nat] :
              ( ( member_nat @ Y2 @ Xs )
             => ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ) ).

% max_in_set_def
thf(fact_45_min__in__set__def,axiom,
    ( vEBT_VEBT_min_in_set
    = ( ^ [Xs: set_nat,X3: nat] :
          ( ( member_nat @ X3 @ Xs )
          & ! [Y2: nat] :
              ( ( member_nat @ Y2 @ Xs )
             => ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ).

% min_in_set_def
thf(fact_46_numeral__eq__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ( numera6690914467698888265omplex @ M )
        = ( numera6690914467698888265omplex @ N3 ) )
      = ( M = N3 ) ) ).

% numeral_eq_iff
thf(fact_47_numeral__eq__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ( numeral_numeral_real @ M )
        = ( numeral_numeral_real @ N3 ) )
      = ( M = N3 ) ) ).

% numeral_eq_iff
thf(fact_48_numeral__eq__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ( numeral_numeral_rat @ M )
        = ( numeral_numeral_rat @ N3 ) )
      = ( M = N3 ) ) ).

% numeral_eq_iff
thf(fact_49_numeral__eq__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ( numeral_numeral_nat @ M )
        = ( numeral_numeral_nat @ N3 ) )
      = ( M = N3 ) ) ).

% numeral_eq_iff
thf(fact_50_numeral__eq__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ( numeral_numeral_int @ M )
        = ( numeral_numeral_int @ N3 ) )
      = ( M = N3 ) ) ).

% numeral_eq_iff
thf(fact_51_assnle,axiom,
    ! [TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] : ( entails @ ( times_times_assn @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) ).

% assnle
thf(fact_52_numeral__le__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N3 ) )
      = ( ord_less_eq_num @ M @ N3 ) ) ).

% numeral_le_iff
thf(fact_53_numeral__le__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N3 ) )
      = ( ord_less_eq_num @ M @ N3 ) ) ).

% numeral_le_iff
thf(fact_54_numeral__le__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N3 ) )
      = ( ord_less_eq_num @ M @ N3 ) ) ).

% numeral_le_iff
thf(fact_55_numeral__le__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N3 ) )
      = ( ord_less_eq_num @ M @ N3 ) ) ).

% numeral_le_iff
thf(fact_56_numeral__less__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N3 ) )
      = ( ord_less_num @ M @ N3 ) ) ).

% numeral_less_iff
thf(fact_57_numeral__less__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N3 ) )
      = ( ord_less_num @ M @ N3 ) ) ).

% numeral_less_iff
thf(fact_58_numeral__less__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N3 ) )
      = ( ord_less_num @ M @ N3 ) ) ).

% numeral_less_iff
thf(fact_59_numeral__less__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N3 ) )
      = ( ord_less_num @ M @ N3 ) ) ).

% numeral_less_iff
thf(fact_60_add__numeral__left,axiom,
    ! [V: num,W: num,Z: complex] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_61_add__numeral__left,axiom,
    ! [V: num,W: num,Z: real] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_62_add__numeral__left,axiom,
    ! [V: num,W: num,Z: rat] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_63_add__numeral__left,axiom,
    ! [V: num,W: num,Z: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_64_add__numeral__left,axiom,
    ! [V: num,W: num,Z: int] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_65_numeral__plus__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N3 ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N3 ) ) ) ).

% numeral_plus_numeral
thf(fact_66_numeral__plus__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N3 ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M @ N3 ) ) ) ).

% numeral_plus_numeral
thf(fact_67_numeral__plus__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N3 ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N3 ) ) ) ).

% numeral_plus_numeral
thf(fact_68_numeral__plus__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N3 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) ).

% numeral_plus_numeral
thf(fact_69_numeral__plus__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N3 ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N3 ) ) ) ).

% numeral_plus_numeral
thf(fact_70_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_71_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
      = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_72_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_73_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_74_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_75_numeral__times__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N3 ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N3 ) ) ) ).

% numeral_times_numeral
thf(fact_76_numeral__times__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N3 ) )
      = ( numeral_numeral_real @ ( times_times_num @ M @ N3 ) ) ) ).

% numeral_times_numeral
thf(fact_77_numeral__times__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N3 ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M @ N3 ) ) ) ).

% numeral_times_numeral
thf(fact_78_numeral__times__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N3 ) )
      = ( numeral_numeral_nat @ ( times_times_num @ M @ N3 ) ) ) ).

% numeral_times_numeral
thf(fact_79_numeral__times__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N3 ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N3 ) ) ) ).

% numeral_times_numeral
thf(fact_80_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_81_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_82_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_83_distrib__right__numeral,axiom,
    ! [A2: complex,B2: complex,V: num] :
      ( ( times_times_complex @ ( plus_plus_complex @ A2 @ B2 ) @ ( numera6690914467698888265omplex @ V ) )
      = ( plus_plus_complex @ ( times_times_complex @ A2 @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B2 @ ( numera6690914467698888265omplex @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_84_distrib__right__numeral,axiom,
    ! [A2: real,B2: real,V: num] :
      ( ( times_times_real @ ( plus_plus_real @ A2 @ B2 ) @ ( numeral_numeral_real @ V ) )
      = ( plus_plus_real @ ( times_times_real @ A2 @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B2 @ ( numeral_numeral_real @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_85_distrib__right__numeral,axiom,
    ! [A2: rat,B2: rat,V: num] :
      ( ( times_times_rat @ ( plus_plus_rat @ A2 @ B2 ) @ ( numeral_numeral_rat @ V ) )
      = ( plus_plus_rat @ ( times_times_rat @ A2 @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B2 @ ( numeral_numeral_rat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_86_distrib__right__numeral,axiom,
    ! [A2: nat,B2: nat,V: num] :
      ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ ( numeral_numeral_nat @ V ) )
      = ( plus_plus_nat @ ( times_times_nat @ A2 @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B2 @ ( numeral_numeral_nat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_87_distrib__right__numeral,axiom,
    ! [A2: int,B2: int,V: num] :
      ( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ ( numeral_numeral_int @ V ) )
      = ( plus_plus_int @ ( times_times_int @ A2 @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B2 @ ( numeral_numeral_int @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_88_distrib__left__numeral,axiom,
    ! [V: num,B2: complex,C2: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B2 @ C2 ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B2 ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C2 ) ) ) ).

% distrib_left_numeral
thf(fact_89_distrib__left__numeral,axiom,
    ! [V: num,B2: real,C2: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B2 @ C2 ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B2 ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C2 ) ) ) ).

% distrib_left_numeral
thf(fact_90_distrib__left__numeral,axiom,
    ! [V: num,B2: rat,C2: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B2 @ C2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B2 ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C2 ) ) ) ).

% distrib_left_numeral
thf(fact_91_distrib__left__numeral,axiom,
    ! [V: num,B2: nat,C2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B2 @ C2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C2 ) ) ) ).

% distrib_left_numeral
thf(fact_92_distrib__left__numeral,axiom,
    ! [V: num,B2: int,C2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B2 @ C2 ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C2 ) ) ) ).

% distrib_left_numeral
thf(fact_93_mem__Collect__eq,axiom,
    ! [A2: real,P: real > $o] :
      ( ( member_real @ A2 @ ( collect_real @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_94_mem__Collect__eq,axiom,
    ! [A2: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( member_VEBT_VEBT @ A2 @ ( collect_VEBT_VEBT @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_95_mem__Collect__eq,axiom,
    ! [A2: nat,P: nat > $o] :
      ( ( member_nat @ A2 @ ( collect_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_96_mem__Collect__eq,axiom,
    ! [A2: int,P: int > $o] :
      ( ( member_int @ A2 @ ( collect_int @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_97_mem__Collect__eq,axiom,
    ! [A2: complex,P: complex > $o] :
      ( ( member_complex @ A2 @ ( collect_complex @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_98_mem__Collect__eq,axiom,
    ! [A2: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( member5262025264175285858nt_int @ A2 @ ( collec213857154873943460nt_int @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_99_Collect__mem__eq,axiom,
    ! [A: set_real] :
      ( ( collect_real
        @ ^ [X3: real] : ( member_real @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_100_Collect__mem__eq,axiom,
    ! [A: set_VEBT_VEBT] :
      ( ( collect_VEBT_VEBT
        @ ^ [X3: vEBT_VEBT] : ( member_VEBT_VEBT @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_101_Collect__mem__eq,axiom,
    ! [A: set_nat] :
      ( ( collect_nat
        @ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_102_Collect__mem__eq,axiom,
    ! [A: set_int] :
      ( ( collect_int
        @ ^ [X3: int] : ( member_int @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_103_Collect__mem__eq,axiom,
    ! [A: set_complex] :
      ( ( collect_complex
        @ ^ [X3: complex] : ( member_complex @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_104_Collect__mem__eq,axiom,
    ! [A: set_Pr958786334691620121nt_int] :
      ( ( collec213857154873943460nt_int
        @ ^ [X3: product_prod_int_int] : ( member5262025264175285858nt_int @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_105_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X4: nat] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_106_Collect__cong,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X4: int] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_int @ P )
        = ( collect_int @ Q ) ) ) ).

% Collect_cong
thf(fact_107_Collect__cong,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ! [X4: complex] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_complex @ P )
        = ( collect_complex @ Q ) ) ) ).

% Collect_cong
thf(fact_108_Collect__cong,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ! [X4: product_prod_int_int] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collec213857154873943460nt_int @ P )
        = ( collec213857154873943460nt_int @ Q ) ) ) ).

% Collect_cong
thf(fact_109_div__mult__mult1__if,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ( C2 = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) )
          = zero_zero_nat ) )
      & ( ( C2 != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) )
          = ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).

% div_mult_mult1_if
thf(fact_110_div__mult__mult1__if,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ( C2 = zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
          = zero_zero_int ) )
      & ( ( C2 != zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
          = ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% div_mult_mult1_if
thf(fact_111_div__mult__mult2,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) )
        = ( divide_divide_nat @ A2 @ B2 ) ) ) ).

% div_mult_mult2
thf(fact_112_div__mult__mult2,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( C2 != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) )
        = ( divide_divide_int @ A2 @ B2 ) ) ) ).

% div_mult_mult2
thf(fact_113_div__mult__mult1,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) )
        = ( divide_divide_nat @ A2 @ B2 ) ) ) ).

% div_mult_mult1
thf(fact_114_div__mult__mult1,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( C2 != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
        = ( divide_divide_int @ A2 @ B2 ) ) ) ).

% div_mult_mult1
thf(fact_115_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: complex,C2: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B2 @ C2 ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B2 ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C2 ) ) ) ).

% right_diff_distrib_numeral
thf(fact_116_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: real,C2: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B2 @ C2 ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B2 ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C2 ) ) ) ).

% right_diff_distrib_numeral
thf(fact_117_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: rat,C2: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B2 @ C2 ) )
      = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B2 ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C2 ) ) ) ).

% right_diff_distrib_numeral
thf(fact_118_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: int,C2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B2 @ C2 ) )
      = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C2 ) ) ) ).

% right_diff_distrib_numeral
thf(fact_119_left__diff__distrib__numeral,axiom,
    ! [A2: complex,B2: complex,V: num] :
      ( ( times_times_complex @ ( minus_minus_complex @ A2 @ B2 ) @ ( numera6690914467698888265omplex @ V ) )
      = ( minus_minus_complex @ ( times_times_complex @ A2 @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B2 @ ( numera6690914467698888265omplex @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_120_left__diff__distrib__numeral,axiom,
    ! [A2: real,B2: real,V: num] :
      ( ( times_times_real @ ( minus_minus_real @ A2 @ B2 ) @ ( numeral_numeral_real @ V ) )
      = ( minus_minus_real @ ( times_times_real @ A2 @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B2 @ ( numeral_numeral_real @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_121_left__diff__distrib__numeral,axiom,
    ! [A2: rat,B2: rat,V: num] :
      ( ( times_times_rat @ ( minus_minus_rat @ A2 @ B2 ) @ ( numeral_numeral_rat @ V ) )
      = ( minus_minus_rat @ ( times_times_rat @ A2 @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B2 @ ( numeral_numeral_rat @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_122_left__diff__distrib__numeral,axiom,
    ! [A2: int,B2: int,V: num] :
      ( ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ ( numeral_numeral_int @ V ) )
      = ( minus_minus_int @ ( times_times_int @ A2 @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B2 @ ( numeral_numeral_int @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_123_power__0__Suc,axiom,
    ! [N3: nat] :
      ( ( power_power_rat @ zero_zero_rat @ ( suc @ N3 ) )
      = zero_zero_rat ) ).

% power_0_Suc
thf(fact_124_power__0__Suc,axiom,
    ! [N3: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N3 ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_125_power__0__Suc,axiom,
    ! [N3: nat] :
      ( ( power_power_real @ zero_zero_real @ ( suc @ N3 ) )
      = zero_zero_real ) ).

% power_0_Suc
thf(fact_126_power__0__Suc,axiom,
    ! [N3: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N3 ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_127_power__0__Suc,axiom,
    ! [N3: nat] :
      ( ( power_power_complex @ zero_zero_complex @ ( suc @ N3 ) )
      = zero_zero_complex ) ).

% power_0_Suc
thf(fact_128_power__0__Suc,axiom,
    ! [N3: nat] :
      ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ ( suc @ N3 ) )
      = zero_z3403309356797280102nteger ) ).

% power_0_Suc
thf(fact_129_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_rat ) ).

% power_zero_numeral
thf(fact_130_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_nat ) ).

% power_zero_numeral
thf(fact_131_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
      = zero_zero_real ) ).

% power_zero_numeral
thf(fact_132_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
      = zero_zero_int ) ).

% power_zero_numeral
thf(fact_133_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
      = zero_zero_complex ) ).

% power_zero_numeral
thf(fact_134_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ ( numeral_numeral_nat @ K ) )
      = zero_z3403309356797280102nteger ) ).

% power_zero_numeral
thf(fact_135_Suc__numeral,axiom,
    ! [N3: num] :
      ( ( suc @ ( numeral_numeral_nat @ N3 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N3 @ one ) ) ) ).

% Suc_numeral
thf(fact_136_power__add__numeral2,axiom,
    ! [A2: complex,M: num,N3: num,B2: complex] :
      ( ( times_times_complex @ ( power_power_complex @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A2 @ ( numeral_numeral_nat @ N3 ) ) @ B2 ) )
      = ( times_times_complex @ ( power_power_complex @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_137_power__add__numeral2,axiom,
    ! [A2: code_integer,M: num,N3: num,B2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ N3 ) ) @ B2 ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_138_power__add__numeral2,axiom,
    ! [A2: real,M: num,N3: num,B2: real] :
      ( ( times_times_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ N3 ) ) @ B2 ) )
      = ( times_times_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_139_power__add__numeral2,axiom,
    ! [A2: rat,M: num,N3: num,B2: rat] :
      ( ( times_times_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ N3 ) ) @ B2 ) )
      = ( times_times_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_140_power__add__numeral2,axiom,
    ! [A2: nat,M: num,N3: num,B2: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ N3 ) ) @ B2 ) )
      = ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_141_power__add__numeral2,axiom,
    ! [A2: int,M: num,N3: num,B2: int] :
      ( ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ N3 ) ) @ B2 ) )
      = ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_142_power__add__numeral2,axiom,
    ! [A2: assn,M: num,N3: num,B2: assn] :
      ( ( times_times_assn @ ( power_power_assn @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_assn @ ( power_power_assn @ A2 @ ( numeral_numeral_nat @ N3 ) ) @ B2 ) )
      = ( times_times_assn @ ( power_power_assn @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) @ B2 ) ) ).

% power_add_numeral2
thf(fact_143_power__add__numeral,axiom,
    ! [A2: complex,M: num,N3: num] :
      ( ( times_times_complex @ ( power_power_complex @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A2 @ ( numeral_numeral_nat @ N3 ) ) )
      = ( power_power_complex @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) ) ).

% power_add_numeral
thf(fact_144_power__add__numeral,axiom,
    ! [A2: code_integer,M: num,N3: num] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ N3 ) ) )
      = ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) ) ).

% power_add_numeral
thf(fact_145_power__add__numeral,axiom,
    ! [A2: real,M: num,N3: num] :
      ( ( times_times_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ N3 ) ) )
      = ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) ) ).

% power_add_numeral
thf(fact_146_power__add__numeral,axiom,
    ! [A2: rat,M: num,N3: num] :
      ( ( times_times_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ N3 ) ) )
      = ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) ) ).

% power_add_numeral
thf(fact_147_power__add__numeral,axiom,
    ! [A2: nat,M: num,N3: num] :
      ( ( times_times_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ N3 ) ) )
      = ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) ) ).

% power_add_numeral
thf(fact_148_power__add__numeral,axiom,
    ! [A2: int,M: num,N3: num] :
      ( ( times_times_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ N3 ) ) )
      = ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) ) ).

% power_add_numeral
thf(fact_149_power__add__numeral,axiom,
    ! [A2: assn,M: num,N3: num] :
      ( ( times_times_assn @ ( power_power_assn @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_assn @ A2 @ ( numeral_numeral_nat @ N3 ) ) )
      = ( power_power_assn @ A2 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N3 ) ) ) ) ).

% power_add_numeral
thf(fact_150_power__Suc0__right,axiom,
    ! [A2: nat] :
      ( ( power_power_nat @ A2 @ ( suc @ zero_zero_nat ) )
      = A2 ) ).

% power_Suc0_right
thf(fact_151_power__Suc0__right,axiom,
    ! [A2: real] :
      ( ( power_power_real @ A2 @ ( suc @ zero_zero_nat ) )
      = A2 ) ).

% power_Suc0_right
thf(fact_152_power__Suc0__right,axiom,
    ! [A2: int] :
      ( ( power_power_int @ A2 @ ( suc @ zero_zero_nat ) )
      = A2 ) ).

% power_Suc0_right
thf(fact_153_power__Suc0__right,axiom,
    ! [A2: complex] :
      ( ( power_power_complex @ A2 @ ( suc @ zero_zero_nat ) )
      = A2 ) ).

% power_Suc0_right
thf(fact_154_power__Suc0__right,axiom,
    ! [A2: code_integer] :
      ( ( power_8256067586552552935nteger @ A2 @ ( suc @ zero_zero_nat ) )
      = A2 ) ).

% power_Suc0_right
thf(fact_155_div__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
      = M ) ).

% div_by_Suc_0
thf(fact_156_div__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ( divide_divide_nat @ M @ N3 )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_157_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_158_power__Suc__0,axiom,
    ! [N3: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N3 )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_159_nat__zero__less__power__iff,axiom,
    ! [X: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N3 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N3 = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_160_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A2: complex,B2: complex,W: num] :
      ( ( A2
        = ( divide1717551699836669952omplex @ B2 @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( ( ( numera6690914467698888265omplex @ W )
           != zero_zero_complex )
         => ( ( times_times_complex @ A2 @ ( numera6690914467698888265omplex @ W ) )
            = B2 ) )
        & ( ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_161_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( A2
        = ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ( numeral_numeral_real @ W )
           != zero_zero_real )
         => ( ( times_times_real @ A2 @ ( numeral_numeral_real @ W ) )
            = B2 ) )
        & ( ( ( numeral_numeral_real @ W )
            = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_162_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( ( A2
        = ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ( numeral_numeral_rat @ W )
           != zero_zero_rat )
         => ( ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W ) )
            = B2 ) )
        & ( ( ( numeral_numeral_rat @ W )
            = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_163_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: complex,W: num,A2: complex] :
      ( ( ( divide1717551699836669952omplex @ B2 @ ( numera6690914467698888265omplex @ W ) )
        = A2 )
      = ( ( ( ( numera6690914467698888265omplex @ W )
           != zero_zero_complex )
         => ( B2
            = ( times_times_complex @ A2 @ ( numera6690914467698888265omplex @ W ) ) ) )
        & ( ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_164_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: real,W: num,A2: real] :
      ( ( ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) )
        = A2 )
      = ( ( ( ( numeral_numeral_real @ W )
           != zero_zero_real )
         => ( B2
            = ( times_times_real @ A2 @ ( numeral_numeral_real @ W ) ) ) )
        & ( ( ( numeral_numeral_real @ W )
            = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_165_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( ( ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) )
        = A2 )
      = ( ( ( ( numeral_numeral_rat @ W )
           != zero_zero_rat )
         => ( B2
            = ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W ) ) ) )
        & ( ( ( numeral_numeral_rat @ W )
            = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_166_le__divide__eq__numeral1_I1_J,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( ord_less_eq_real @ A2 @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) )
      = ( ord_less_eq_real @ ( times_times_real @ A2 @ ( numeral_numeral_real @ W ) ) @ B2 ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_167_le__divide__eq__numeral1_I1_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) )
      = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W ) ) @ B2 ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_168_divide__le__eq__numeral1_I1_J,axiom,
    ! [B2: real,W: num,A2: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) @ A2 )
      = ( ord_less_eq_real @ B2 @ ( times_times_real @ A2 @ ( numeral_numeral_real @ W ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_169_divide__le__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) @ A2 )
      = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_170_less__divide__eq__numeral1_I1_J,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( ord_less_real @ A2 @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) )
      = ( ord_less_real @ ( times_times_real @ A2 @ ( numeral_numeral_real @ W ) ) @ B2 ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_171_less__divide__eq__numeral1_I1_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) )
      = ( ord_less_rat @ ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W ) ) @ B2 ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_172_divide__less__eq__numeral1_I1_J,axiom,
    ! [B2: real,W: num,A2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ ( numeral_numeral_real @ W ) ) @ A2 )
      = ( ord_less_real @ B2 @ ( times_times_real @ A2 @ ( numeral_numeral_real @ W ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_173_divide__less__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ ( numeral_numeral_rat @ W ) ) @ A2 )
      = ( ord_less_rat @ B2 @ ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_174_div__mult__self4,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C2 ) @ A2 ) @ B2 )
        = ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).

% div_mult_self4
thf(fact_175_div__mult__self4,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C2 ) @ A2 ) @ B2 )
        = ( plus_plus_int @ C2 @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% div_mult_self4
thf(fact_176_div__mult__self3,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C2 @ B2 ) @ A2 ) @ B2 )
        = ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).

% div_mult_self3
thf(fact_177_div__mult__self3,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C2 @ B2 ) @ A2 ) @ B2 )
        = ( plus_plus_int @ C2 @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% div_mult_self3
thf(fact_178_div__mult__self2,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) ) @ B2 )
        = ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).

% div_mult_self2
thf(fact_179_div__mult__self2,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ ( times_times_int @ B2 @ C2 ) ) @ B2 )
        = ( plus_plus_int @ C2 @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% div_mult_self2
thf(fact_180_div__mult__self1,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ C2 @ B2 ) ) @ B2 )
        = ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).

% div_mult_self1
thf(fact_181_div__mult__self1,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ ( times_times_int @ C2 @ B2 ) ) @ B2 )
        = ( plus_plus_int @ C2 @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% div_mult_self1
thf(fact_182_power__eq__0__iff,axiom,
    ! [A2: rat,N3: nat] :
      ( ( ( power_power_rat @ A2 @ N3 )
        = zero_zero_rat )
      = ( ( A2 = zero_zero_rat )
        & ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% power_eq_0_iff
thf(fact_183_power__eq__0__iff,axiom,
    ! [A2: nat,N3: nat] :
      ( ( ( power_power_nat @ A2 @ N3 )
        = zero_zero_nat )
      = ( ( A2 = zero_zero_nat )
        & ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% power_eq_0_iff
thf(fact_184_power__eq__0__iff,axiom,
    ! [A2: real,N3: nat] :
      ( ( ( power_power_real @ A2 @ N3 )
        = zero_zero_real )
      = ( ( A2 = zero_zero_real )
        & ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% power_eq_0_iff
thf(fact_185_power__eq__0__iff,axiom,
    ! [A2: int,N3: nat] :
      ( ( ( power_power_int @ A2 @ N3 )
        = zero_zero_int )
      = ( ( A2 = zero_zero_int )
        & ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% power_eq_0_iff
thf(fact_186_power__eq__0__iff,axiom,
    ! [A2: complex,N3: nat] :
      ( ( ( power_power_complex @ A2 @ N3 )
        = zero_zero_complex )
      = ( ( A2 = zero_zero_complex )
        & ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% power_eq_0_iff
thf(fact_187_power__eq__0__iff,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( ( power_8256067586552552935nteger @ A2 @ N3 )
        = zero_z3403309356797280102nteger )
      = ( ( A2 = zero_z3403309356797280102nteger )
        & ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% power_eq_0_iff
thf(fact_188_div__mult__self1__is__m,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( divide_divide_nat @ ( times_times_nat @ N3 @ M ) @ N3 )
        = M ) ) ).

% div_mult_self1_is_m
thf(fact_189_div__mult__self__is__m,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( divide_divide_nat @ ( times_times_nat @ M @ N3 ) @ N3 )
        = M ) ) ).

% div_mult_self_is_m
thf(fact_190_txe,axiom,
    ! [Y: nat,TreeList: list_VEBT_VEBT,Tree_is: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ Y @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList @ Y ) @ ( nth_VEBT_VEBTi @ Tree_is @ Y ) ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) ) ) ).

% txe
thf(fact_191_lesseq__shift,axiom,
    ( ord_less_eq_nat
    = ( ^ [X3: nat,Y2: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X3 ) @ ( some_nat @ Y2 ) ) ) ) ).

% lesseq_shift
thf(fact_192_power__mono__iff,axiom,
    ! [A2: real,B2: real,N3: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N3 )
         => ( ( ord_less_eq_real @ ( power_power_real @ A2 @ N3 ) @ ( power_power_real @ B2 @ N3 ) )
            = ( ord_less_eq_real @ A2 @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_193_power__mono__iff,axiom,
    ! [A2: code_integer,B2: code_integer,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N3 )
         => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ ( power_8256067586552552935nteger @ B2 @ N3 ) )
            = ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_194_power__mono__iff,axiom,
    ! [A2: rat,B2: rat,N3: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N3 )
         => ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N3 ) @ ( power_power_rat @ B2 @ N3 ) )
            = ( ord_less_eq_rat @ A2 @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_195_power__mono__iff,axiom,
    ! [A2: nat,B2: nat,N3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N3 )
         => ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N3 ) @ ( power_power_nat @ B2 @ N3 ) )
            = ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_196_power__mono__iff,axiom,
    ! [A2: int,B2: int,N3: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N3 )
         => ( ( ord_less_eq_int @ ( power_power_int @ A2 @ N3 ) @ ( power_power_int @ B2 @ N3 ) )
            = ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ) ).

% power_mono_iff
thf(fact_197_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_198_power2__eq__iff__nonneg,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_199_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_200_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_201_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_202_power2__less__eq__zero__iff,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
      = ( A2 = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
thf(fact_203_power2__less__eq__zero__iff,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_z3403309356797280102nteger )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% power2_less_eq_zero_iff
thf(fact_204_power2__less__eq__zero__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
      = ( A2 = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
thf(fact_205_power2__less__eq__zero__iff,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
thf(fact_206_local_Opower__def,axiom,
    ( vEBT_VEBT_power
    = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).

% local.power_def
thf(fact_207_add__One__commute,axiom,
    ! [N3: num] :
      ( ( plus_plus_num @ one @ N3 )
      = ( plus_plus_num @ N3 @ one ) ) ).

% add_One_commute
thf(fact_208_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_209_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_210_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_211_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_212_div__le__dividend,axiom,
    ! [M: nat,N3: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N3 ) @ M ) ).

% div_le_dividend
thf(fact_213_div__le__mono,axiom,
    ! [M: nat,N3: nat,K: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N3 @ K ) ) ) ).

% div_le_mono
thf(fact_214_not__numeral__le__zero,axiom,
    ! [N3: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N3 ) @ zero_zero_real ) ).

% not_numeral_le_zero
thf(fact_215_not__numeral__le__zero,axiom,
    ! [N3: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N3 ) @ zero_zero_rat ) ).

% not_numeral_le_zero
thf(fact_216_not__numeral__le__zero,axiom,
    ! [N3: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N3 ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_217_not__numeral__le__zero,axiom,
    ! [N3: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N3 ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_218_zero__le__numeral,axiom,
    ! [N3: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N3 ) ) ).

% zero_le_numeral
thf(fact_219_zero__le__numeral,axiom,
    ! [N3: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N3 ) ) ).

% zero_le_numeral
thf(fact_220_zero__le__numeral,axiom,
    ! [N3: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N3 ) ) ).

% zero_le_numeral
thf(fact_221_zero__le__numeral,axiom,
    ! [N3: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N3 ) ) ).

% zero_le_numeral
thf(fact_222_zero__le__power,axiom,
    ! [A2: real,N3: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ N3 ) ) ) ).

% zero_le_power
thf(fact_223_zero__le__power,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ).

% zero_le_power
thf(fact_224_zero__le__power,axiom,
    ! [A2: rat,N3: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ N3 ) ) ) ).

% zero_le_power
thf(fact_225_zero__le__power,axiom,
    ! [A2: nat,N3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A2 @ N3 ) ) ) ).

% zero_le_power
thf(fact_226_zero__le__power,axiom,
    ! [A2: int,N3: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ N3 ) ) ) ).

% zero_le_power
thf(fact_227_power__mono,axiom,
    ! [A2: real,B2: real,N3: nat] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
       => ( ord_less_eq_real @ ( power_power_real @ A2 @ N3 ) @ ( power_power_real @ B2 @ N3 ) ) ) ) ).

% power_mono
thf(fact_228_power__mono,axiom,
    ! [A2: code_integer,B2: code_integer,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ ( power_8256067586552552935nteger @ B2 @ N3 ) ) ) ) ).

% power_mono
thf(fact_229_power__mono,axiom,
    ! [A2: rat,B2: rat,N3: nat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
       => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N3 ) @ ( power_power_rat @ B2 @ N3 ) ) ) ) ).

% power_mono
thf(fact_230_power__mono,axiom,
    ! [A2: nat,B2: nat,N3: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
       => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N3 ) @ ( power_power_nat @ B2 @ N3 ) ) ) ) ).

% power_mono
thf(fact_231_power__mono,axiom,
    ! [A2: int,B2: int,N3: nat] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
       => ( ord_less_eq_int @ ( power_power_int @ A2 @ N3 ) @ ( power_power_int @ B2 @ N3 ) ) ) ) ).

% power_mono
thf(fact_232_Suc__div__le__mono,axiom,
    ! [M: nat,N3: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N3 ) @ ( divide_divide_nat @ ( suc @ M ) @ N3 ) ) ).

% Suc_div_le_mono
thf(fact_233_times__div__less__eq__dividend,axiom,
    ! [N3: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N3 @ ( divide_divide_nat @ M @ N3 ) ) @ M ) ).

% times_div_less_eq_dividend
thf(fact_234_div__times__less__eq__dividend,axiom,
    ! [M: nat,N3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N3 ) @ N3 ) @ M ) ).

% div_times_less_eq_dividend
thf(fact_235_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_236_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_237_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_238_power__less__imp__less__base,axiom,
    ! [A2: code_integer,N3: nat,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ ( power_8256067586552552935nteger @ B2 @ N3 ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ) ).

% power_less_imp_less_base
thf(fact_239_power__less__imp__less__base,axiom,
    ! [A2: real,N3: nat,B2: real] :
      ( ( ord_less_real @ ( power_power_real @ A2 @ N3 ) @ ( power_power_real @ B2 @ N3 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ A2 @ B2 ) ) ) ).

% power_less_imp_less_base
thf(fact_240_power__less__imp__less__base,axiom,
    ! [A2: rat,N3: nat,B2: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ A2 @ N3 ) @ ( power_power_rat @ B2 @ N3 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ A2 @ B2 ) ) ) ).

% power_less_imp_less_base
thf(fact_241_power__less__imp__less__base,axiom,
    ! [A2: nat,N3: nat,B2: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ A2 @ N3 ) @ ( power_power_nat @ B2 @ N3 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ A2 @ B2 ) ) ) ).

% power_less_imp_less_base
thf(fact_242_power__less__imp__less__base,axiom,
    ! [A2: int,N3: nat,B2: int] :
      ( ( ord_less_int @ ( power_power_int @ A2 @ N3 ) @ ( power_power_int @ B2 @ N3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ A2 @ B2 ) ) ) ).

% power_less_imp_less_base
thf(fact_243_power__le__imp__le__base,axiom,
    ! [A2: real,N3: nat,B2: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A2 @ ( suc @ N3 ) ) @ ( power_power_real @ B2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_eq_real @ A2 @ B2 ) ) ) ).

% power_le_imp_le_base
thf(fact_244_power__le__imp__le__base,axiom,
    ! [A2: code_integer,N3: nat,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ ( suc @ N3 ) ) @ ( power_8256067586552552935nteger @ B2 @ ( suc @ N3 ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ) ).

% power_le_imp_le_base
thf(fact_245_power__le__imp__le__base,axiom,
    ! [A2: rat,N3: nat,B2: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ ( suc @ N3 ) ) @ ( power_power_rat @ B2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).

% power_le_imp_le_base
thf(fact_246_power__le__imp__le__base,axiom,
    ! [A2: nat,N3: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ ( suc @ N3 ) ) @ ( power_power_nat @ B2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).

% power_le_imp_le_base
thf(fact_247_power__le__imp__le__base,axiom,
    ! [A2: int,N3: nat,B2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A2 @ ( suc @ N3 ) ) @ ( power_power_int @ B2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ A2 @ B2 ) ) ) ).

% power_le_imp_le_base
thf(fact_248_power__inject__base,axiom,
    ! [A2: real,N3: nat,B2: real] :
      ( ( ( power_power_real @ A2 @ ( suc @ N3 ) )
        = ( power_power_real @ B2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
thf(fact_249_power__inject__base,axiom,
    ! [A2: code_integer,N3: nat,B2: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A2 @ ( suc @ N3 ) )
        = ( power_8256067586552552935nteger @ B2 @ ( suc @ N3 ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
thf(fact_250_power__inject__base,axiom,
    ! [A2: rat,N3: nat,B2: rat] :
      ( ( ( power_power_rat @ A2 @ ( suc @ N3 ) )
        = ( power_power_rat @ B2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
thf(fact_251_power__inject__base,axiom,
    ! [A2: nat,N3: nat,B2: nat] :
      ( ( ( power_power_nat @ A2 @ ( suc @ N3 ) )
        = ( power_power_nat @ B2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
thf(fact_252_power__inject__base,axiom,
    ! [A2: int,N3: nat,B2: int] :
      ( ( ( power_power_int @ A2 @ ( suc @ N3 ) )
        = ( power_power_int @ B2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
         => ( A2 = B2 ) ) ) ) ).

% power_inject_base
thf(fact_253_div__greater__zero__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N3 ) )
      = ( ( ord_less_eq_nat @ N3 @ M )
        & ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% div_greater_zero_iff
thf(fact_254_div__le__mono2,axiom,
    ! [M: nat,N3: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N3 )
       => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N3 ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).

% div_le_mono2
thf(fact_255_nat__one__le__power,axiom,
    ! [I: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
     => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N3 ) ) ) ).

% nat_one_le_power
thf(fact_256_Suc__nat__number__of__add,axiom,
    ! [V: num,N3: nat] :
      ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N3 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N3 ) ) ).

% Suc_nat_number_of_add
thf(fact_257_power__eq__imp__eq__base,axiom,
    ! [A2: real,N3: nat,B2: real] :
      ( ( ( power_power_real @ A2 @ N3 )
        = ( power_power_real @ B2 @ N3 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
         => ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_258_power__eq__imp__eq__base,axiom,
    ! [A2: code_integer,N3: nat,B2: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A2 @ N3 )
        = ( power_8256067586552552935nteger @ B2 @ N3 ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
         => ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_259_power__eq__imp__eq__base,axiom,
    ! [A2: rat,N3: nat,B2: rat] :
      ( ( ( power_power_rat @ A2 @ N3 )
        = ( power_power_rat @ B2 @ N3 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
         => ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_260_power__eq__imp__eq__base,axiom,
    ! [A2: nat,N3: nat,B2: nat] :
      ( ( ( power_power_nat @ A2 @ N3 )
        = ( power_power_nat @ B2 @ N3 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
         => ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_261_power__eq__imp__eq__base,axiom,
    ! [A2: int,N3: nat,B2: int] :
      ( ( ( power_power_int @ A2 @ N3 )
        = ( power_power_int @ B2 @ N3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
         => ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( A2 = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_262_power__eq__iff__eq__base,axiom,
    ! [N3: nat,A2: real,B2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
         => ( ( ( power_power_real @ A2 @ N3 )
              = ( power_power_real @ B2 @ N3 ) )
            = ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_263_power__eq__iff__eq__base,axiom,
    ! [N3: nat,A2: code_integer,B2: code_integer] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
         => ( ( ( power_8256067586552552935nteger @ A2 @ N3 )
              = ( power_8256067586552552935nteger @ B2 @ N3 ) )
            = ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_264_power__eq__iff__eq__base,axiom,
    ! [N3: nat,A2: rat,B2: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
         => ( ( ( power_power_rat @ A2 @ N3 )
              = ( power_power_rat @ B2 @ N3 ) )
            = ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_265_power__eq__iff__eq__base,axiom,
    ! [N3: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
         => ( ( ( power_power_nat @ A2 @ N3 )
              = ( power_power_nat @ B2 @ N3 ) )
            = ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_266_power__eq__iff__eq__base,axiom,
    ! [N3: nat,A2: int,B2: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
         => ( ( ( power_power_int @ A2 @ N3 )
              = ( power_power_int @ B2 @ N3 ) )
            = ( A2 = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_267_power2__nat__le__imp__le,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N3 )
     => ( ord_less_eq_nat @ M @ N3 ) ) ).

% power2_nat_le_imp_le
thf(fact_268_power2__nat__le__eq__le,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% power2_nat_le_eq_le
thf(fact_269_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_270_div__nat__eqI,axiom,
    ! [N3: nat,Q3: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N3 @ Q3 ) @ M )
     => ( ( ord_less_nat @ M @ ( times_times_nat @ N3 @ ( suc @ Q3 ) ) )
       => ( ( divide_divide_nat @ M @ N3 )
          = Q3 ) ) ) ).

% div_nat_eqI
thf(fact_271_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q3 )
     => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N3 @ Q3 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q3 ) @ N3 ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_272_is__num__normalize_I1_J,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A2 @ B2 ) @ C2 )
      = ( plus_plus_real @ A2 @ ( plus_plus_real @ B2 @ C2 ) ) ) ).

% is_num_normalize(1)
thf(fact_273_is__num__normalize_I1_J,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C2 )
      = ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B2 @ C2 ) ) ) ).

% is_num_normalize(1)
thf(fact_274_is__num__normalize_I1_J,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 )
      = ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C2 ) ) ) ).

% is_num_normalize(1)
thf(fact_275_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: real,C2: real] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B2 @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C2 ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C2 ) ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_276_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B2 @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C2 ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_277_divide__le__eq__numeral_I1_J,axiom,
    ! [B2: real,C2: real,W: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C2 ) @ ( numeral_numeral_real @ W ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C2 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C2 ) @ B2 ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_278_divide__le__eq__numeral_I1_J,axiom,
    ! [B2: rat,C2: rat,W: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C2 ) @ ( numeral_numeral_rat @ W ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C2 ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_279_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_280_power2__le__imp__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ord_le3102999989581377725nteger @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_281_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_282_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_283_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_284_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_285_power2__eq__imp__eq,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_286_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_287_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_288_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_289_zero__le__power2,axiom,
    ! [A2: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_290_zero__le__power2,axiom,
    ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_291_zero__le__power2,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_292_zero__le__power2,axiom,
    ! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_293_power__strict__mono,axiom,
    ! [A2: code_integer,B2: code_integer,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N3 )
         => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ ( power_8256067586552552935nteger @ B2 @ N3 ) ) ) ) ) ).

% power_strict_mono
thf(fact_294_power__strict__mono,axiom,
    ! [A2: real,B2: real,N3: nat] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N3 )
         => ( ord_less_real @ ( power_power_real @ A2 @ N3 ) @ ( power_power_real @ B2 @ N3 ) ) ) ) ) ).

% power_strict_mono
thf(fact_295_power__strict__mono,axiom,
    ! [A2: rat,B2: rat,N3: nat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N3 )
         => ( ord_less_rat @ ( power_power_rat @ A2 @ N3 ) @ ( power_power_rat @ B2 @ N3 ) ) ) ) ) ).

% power_strict_mono
thf(fact_296_power__strict__mono,axiom,
    ! [A2: nat,B2: nat,N3: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N3 )
         => ( ord_less_nat @ ( power_power_nat @ A2 @ N3 ) @ ( power_power_nat @ B2 @ N3 ) ) ) ) ) ).

% power_strict_mono
thf(fact_297_power__strict__mono,axiom,
    ! [A2: int,B2: int,N3: nat] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
       => ( ( ord_less_nat @ zero_zero_nat @ N3 )
         => ( ord_less_int @ ( power_power_int @ A2 @ N3 ) @ ( power_power_int @ B2 @ N3 ) ) ) ) ) ).

% power_strict_mono
thf(fact_298_split__div_H,axiom,
    ! [P: nat > $o,M: nat,N3: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N3 ) )
      = ( ( ( N3 = zero_zero_nat )
          & ( P @ zero_zero_nat ) )
        | ? [Q4: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N3 @ Q4 ) @ M )
            & ( ord_less_nat @ M @ ( times_times_nat @ N3 @ ( suc @ Q4 ) ) )
            & ( P @ Q4 ) ) ) ) ).

% split_div'
thf(fact_299_power2__less__imp__less,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ord_le6747313008572928689nteger @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_300_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_301_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_302_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_303_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_304_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_305_sum__power2__le__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_z3403309356797280102nteger )
      = ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_306_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_307_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_308_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_309_sum__power2__ge__zero,axiom,
    ! [X: code_integer,Y: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_310_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_311_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_312_zero__le__even__power_H,axiom,
    ! [A2: real,N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% zero_le_even_power'
thf(fact_313_zero__le__even__power_H,axiom,
    ! [A2: code_integer,N3: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% zero_le_even_power'
thf(fact_314_zero__le__even__power_H,axiom,
    ! [A2: rat,N3: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% zero_le_even_power'
thf(fact_315_zero__le__even__power_H,axiom,
    ! [A2: int,N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% zero_le_even_power'
thf(fact_316_odd__0__le__power__imp__0__le,axiom,
    ! [A2: real,N3: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_317_odd__0__le__power__imp__0__le,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_318_odd__0__le__power__imp__0__le,axiom,
    ! [A2: rat,N3: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_319_odd__0__le__power__imp__0__le,axiom,
    ! [A2: int,N3: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_320_zero__neq__numeral,axiom,
    ! [N3: num] :
      ( zero_zero_complex
     != ( numera6690914467698888265omplex @ N3 ) ) ).

% zero_neq_numeral
thf(fact_321_zero__neq__numeral,axiom,
    ! [N3: num] :
      ( zero_zero_real
     != ( numeral_numeral_real @ N3 ) ) ).

% zero_neq_numeral
thf(fact_322_zero__neq__numeral,axiom,
    ! [N3: num] :
      ( zero_zero_rat
     != ( numeral_numeral_rat @ N3 ) ) ).

% zero_neq_numeral
thf(fact_323_zero__neq__numeral,axiom,
    ! [N3: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N3 ) ) ).

% zero_neq_numeral
thf(fact_324_zero__neq__numeral,axiom,
    ! [N3: num] :
      ( zero_zero_int
     != ( numeral_numeral_int @ N3 ) ) ).

% zero_neq_numeral
thf(fact_325_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).

% less_numeral_extra(3)
thf(fact_326_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_327_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_328_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_329_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: rat,N3: nat] :
      ( ( A2 != zero_zero_rat )
     => ( ( power_power_rat @ A2 @ N3 )
       != zero_zero_rat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_330_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: nat,N3: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( power_power_nat @ A2 @ N3 )
       != zero_zero_nat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_331_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: real,N3: nat] :
      ( ( A2 != zero_zero_real )
     => ( ( power_power_real @ A2 @ N3 )
       != zero_zero_real ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_332_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: int,N3: nat] :
      ( ( A2 != zero_zero_int )
     => ( ( power_power_int @ A2 @ N3 )
       != zero_zero_int ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_333_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: complex,N3: nat] :
      ( ( A2 != zero_zero_complex )
     => ( ( power_power_complex @ A2 @ N3 )
       != zero_zero_complex ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_334_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ A2 @ N3 )
       != zero_z3403309356797280102nteger ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_335_power__commuting__commutes,axiom,
    ! [X: complex,Y: complex,N3: nat] :
      ( ( ( times_times_complex @ X @ Y )
        = ( times_times_complex @ Y @ X ) )
     => ( ( times_times_complex @ ( power_power_complex @ X @ N3 ) @ Y )
        = ( times_times_complex @ Y @ ( power_power_complex @ X @ N3 ) ) ) ) ).

% power_commuting_commutes
thf(fact_336_power__commuting__commutes,axiom,
    ! [X: code_integer,Y: code_integer,N3: nat] :
      ( ( ( times_3573771949741848930nteger @ X @ Y )
        = ( times_3573771949741848930nteger @ Y @ X ) )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N3 ) @ Y )
        = ( times_3573771949741848930nteger @ Y @ ( power_8256067586552552935nteger @ X @ N3 ) ) ) ) ).

% power_commuting_commutes
thf(fact_337_power__commuting__commutes,axiom,
    ! [X: real,Y: real,N3: nat] :
      ( ( ( times_times_real @ X @ Y )
        = ( times_times_real @ Y @ X ) )
     => ( ( times_times_real @ ( power_power_real @ X @ N3 ) @ Y )
        = ( times_times_real @ Y @ ( power_power_real @ X @ N3 ) ) ) ) ).

% power_commuting_commutes
thf(fact_338_power__commuting__commutes,axiom,
    ! [X: rat,Y: rat,N3: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = ( times_times_rat @ Y @ X ) )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N3 ) @ Y )
        = ( times_times_rat @ Y @ ( power_power_rat @ X @ N3 ) ) ) ) ).

% power_commuting_commutes
thf(fact_339_power__commuting__commutes,axiom,
    ! [X: nat,Y: nat,N3: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = ( times_times_nat @ Y @ X ) )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N3 ) @ Y )
        = ( times_times_nat @ Y @ ( power_power_nat @ X @ N3 ) ) ) ) ).

% power_commuting_commutes
thf(fact_340_power__commuting__commutes,axiom,
    ! [X: int,Y: int,N3: nat] :
      ( ( ( times_times_int @ X @ Y )
        = ( times_times_int @ Y @ X ) )
     => ( ( times_times_int @ ( power_power_int @ X @ N3 ) @ Y )
        = ( times_times_int @ Y @ ( power_power_int @ X @ N3 ) ) ) ) ).

% power_commuting_commutes
thf(fact_341_power__commuting__commutes,axiom,
    ! [X: assn,Y: assn,N3: nat] :
      ( ( ( times_times_assn @ X @ Y )
        = ( times_times_assn @ Y @ X ) )
     => ( ( times_times_assn @ ( power_power_assn @ X @ N3 ) @ Y )
        = ( times_times_assn @ Y @ ( power_power_assn @ X @ N3 ) ) ) ) ).

% power_commuting_commutes
thf(fact_342_power__mult__distrib,axiom,
    ! [A2: complex,B2: complex,N3: nat] :
      ( ( power_power_complex @ ( times_times_complex @ A2 @ B2 ) @ N3 )
      = ( times_times_complex @ ( power_power_complex @ A2 @ N3 ) @ ( power_power_complex @ B2 @ N3 ) ) ) ).

% power_mult_distrib
thf(fact_343_power__mult__distrib,axiom,
    ! [A2: code_integer,B2: code_integer,N3: nat] :
      ( ( power_8256067586552552935nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ N3 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ ( power_8256067586552552935nteger @ B2 @ N3 ) ) ) ).

% power_mult_distrib
thf(fact_344_power__mult__distrib,axiom,
    ! [A2: real,B2: real,N3: nat] :
      ( ( power_power_real @ ( times_times_real @ A2 @ B2 ) @ N3 )
      = ( times_times_real @ ( power_power_real @ A2 @ N3 ) @ ( power_power_real @ B2 @ N3 ) ) ) ).

% power_mult_distrib
thf(fact_345_power__mult__distrib,axiom,
    ! [A2: rat,B2: rat,N3: nat] :
      ( ( power_power_rat @ ( times_times_rat @ A2 @ B2 ) @ N3 )
      = ( times_times_rat @ ( power_power_rat @ A2 @ N3 ) @ ( power_power_rat @ B2 @ N3 ) ) ) ).

% power_mult_distrib
thf(fact_346_power__mult__distrib,axiom,
    ! [A2: nat,B2: nat,N3: nat] :
      ( ( power_power_nat @ ( times_times_nat @ A2 @ B2 ) @ N3 )
      = ( times_times_nat @ ( power_power_nat @ A2 @ N3 ) @ ( power_power_nat @ B2 @ N3 ) ) ) ).

% power_mult_distrib
thf(fact_347_power__mult__distrib,axiom,
    ! [A2: int,B2: int,N3: nat] :
      ( ( power_power_int @ ( times_times_int @ A2 @ B2 ) @ N3 )
      = ( times_times_int @ ( power_power_int @ A2 @ N3 ) @ ( power_power_int @ B2 @ N3 ) ) ) ).

% power_mult_distrib
thf(fact_348_power__mult__distrib,axiom,
    ! [A2: assn,B2: assn,N3: nat] :
      ( ( power_power_assn @ ( times_times_assn @ A2 @ B2 ) @ N3 )
      = ( times_times_assn @ ( power_power_assn @ A2 @ N3 ) @ ( power_power_assn @ B2 @ N3 ) ) ) ).

% power_mult_distrib
thf(fact_349_power__commutes,axiom,
    ! [A2: complex,N3: nat] :
      ( ( times_times_complex @ ( power_power_complex @ A2 @ N3 ) @ A2 )
      = ( times_times_complex @ A2 @ ( power_power_complex @ A2 @ N3 ) ) ) ).

% power_commutes
thf(fact_350_power__commutes,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ A2 )
      = ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ).

% power_commutes
thf(fact_351_power__commutes,axiom,
    ! [A2: real,N3: nat] :
      ( ( times_times_real @ ( power_power_real @ A2 @ N3 ) @ A2 )
      = ( times_times_real @ A2 @ ( power_power_real @ A2 @ N3 ) ) ) ).

% power_commutes
thf(fact_352_power__commutes,axiom,
    ! [A2: rat,N3: nat] :
      ( ( times_times_rat @ ( power_power_rat @ A2 @ N3 ) @ A2 )
      = ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N3 ) ) ) ).

% power_commutes
thf(fact_353_power__commutes,axiom,
    ! [A2: nat,N3: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A2 @ N3 ) @ A2 )
      = ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N3 ) ) ) ).

% power_commutes
thf(fact_354_power__commutes,axiom,
    ! [A2: int,N3: nat] :
      ( ( times_times_int @ ( power_power_int @ A2 @ N3 ) @ A2 )
      = ( times_times_int @ A2 @ ( power_power_int @ A2 @ N3 ) ) ) ).

% power_commutes
thf(fact_355_power__commutes,axiom,
    ! [A2: assn,N3: nat] :
      ( ( times_times_assn @ ( power_power_assn @ A2 @ N3 ) @ A2 )
      = ( times_times_assn @ A2 @ ( power_power_assn @ A2 @ N3 ) ) ) ).

% power_commutes
thf(fact_356_power__divide,axiom,
    ! [A2: complex,B2: complex,N3: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ A2 @ B2 ) @ N3 )
      = ( divide1717551699836669952omplex @ ( power_power_complex @ A2 @ N3 ) @ ( power_power_complex @ B2 @ N3 ) ) ) ).

% power_divide
thf(fact_357_power__divide,axiom,
    ! [A2: real,B2: real,N3: nat] :
      ( ( power_power_real @ ( divide_divide_real @ A2 @ B2 ) @ N3 )
      = ( divide_divide_real @ ( power_power_real @ A2 @ N3 ) @ ( power_power_real @ B2 @ N3 ) ) ) ).

% power_divide
thf(fact_358_power__divide,axiom,
    ! [A2: rat,B2: rat,N3: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ A2 @ B2 ) @ N3 )
      = ( divide_divide_rat @ ( power_power_rat @ A2 @ N3 ) @ ( power_power_rat @ B2 @ N3 ) ) ) ).

% power_divide
thf(fact_359_power__mult,axiom,
    ! [A2: nat,M: nat,N3: nat] :
      ( ( power_power_nat @ A2 @ ( times_times_nat @ M @ N3 ) )
      = ( power_power_nat @ ( power_power_nat @ A2 @ M ) @ N3 ) ) ).

% power_mult
thf(fact_360_power__mult,axiom,
    ! [A2: real,M: nat,N3: nat] :
      ( ( power_power_real @ A2 @ ( times_times_nat @ M @ N3 ) )
      = ( power_power_real @ ( power_power_real @ A2 @ M ) @ N3 ) ) ).

% power_mult
thf(fact_361_power__mult,axiom,
    ! [A2: int,M: nat,N3: nat] :
      ( ( power_power_int @ A2 @ ( times_times_nat @ M @ N3 ) )
      = ( power_power_int @ ( power_power_int @ A2 @ M ) @ N3 ) ) ).

% power_mult
thf(fact_362_power__mult,axiom,
    ! [A2: complex,M: nat,N3: nat] :
      ( ( power_power_complex @ A2 @ ( times_times_nat @ M @ N3 ) )
      = ( power_power_complex @ ( power_power_complex @ A2 @ M ) @ N3 ) ) ).

% power_mult
thf(fact_363_power__mult,axiom,
    ! [A2: code_integer,M: nat,N3: nat] :
      ( ( power_8256067586552552935nteger @ A2 @ ( times_times_nat @ M @ N3 ) )
      = ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A2 @ M ) @ N3 ) ) ).

% power_mult
thf(fact_364_div__mult2__eq,axiom,
    ! [M: nat,N3: nat,Q3: nat] :
      ( ( divide_divide_nat @ M @ ( times_times_nat @ N3 @ Q3 ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ M @ N3 ) @ Q3 ) ) ).

% div_mult2_eq
thf(fact_365_not__numeral__less__zero,axiom,
    ! [N3: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N3 ) @ zero_zero_real ) ).

% not_numeral_less_zero
thf(fact_366_not__numeral__less__zero,axiom,
    ! [N3: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N3 ) @ zero_zero_rat ) ).

% not_numeral_less_zero
thf(fact_367_not__numeral__less__zero,axiom,
    ! [N3: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N3 ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_368_not__numeral__less__zero,axiom,
    ! [N3: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N3 ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_369_zero__less__numeral,axiom,
    ! [N3: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N3 ) ) ).

% zero_less_numeral
thf(fact_370_zero__less__numeral,axiom,
    ! [N3: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N3 ) ) ).

% zero_less_numeral
thf(fact_371_zero__less__numeral,axiom,
    ! [N3: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N3 ) ) ).

% zero_less_numeral
thf(fact_372_zero__less__numeral,axiom,
    ! [N3: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N3 ) ) ).

% zero_less_numeral
thf(fact_373_numeral__Bit0,axiom,
    ! [N3: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N3 ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N3 ) @ ( numera6690914467698888265omplex @ N3 ) ) ) ).

% numeral_Bit0
thf(fact_374_numeral__Bit0,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N3 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N3 ) @ ( numeral_numeral_real @ N3 ) ) ) ).

% numeral_Bit0
thf(fact_375_numeral__Bit0,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N3 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N3 ) @ ( numeral_numeral_rat @ N3 ) ) ) ).

% numeral_Bit0
thf(fact_376_numeral__Bit0,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N3 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N3 ) @ ( numeral_numeral_nat @ N3 ) ) ) ).

% numeral_Bit0
thf(fact_377_numeral__Bit0,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N3 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N3 ) @ ( numeral_numeral_int @ N3 ) ) ) ).

% numeral_Bit0
thf(fact_378_mult__numeral__1__right,axiom,
    ! [A2: complex] :
      ( ( times_times_complex @ A2 @ ( numera6690914467698888265omplex @ one ) )
      = A2 ) ).

% mult_numeral_1_right
thf(fact_379_mult__numeral__1__right,axiom,
    ! [A2: real] :
      ( ( times_times_real @ A2 @ ( numeral_numeral_real @ one ) )
      = A2 ) ).

% mult_numeral_1_right
thf(fact_380_mult__numeral__1__right,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ A2 @ ( numeral_numeral_rat @ one ) )
      = A2 ) ).

% mult_numeral_1_right
thf(fact_381_mult__numeral__1__right,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ A2 @ ( numeral_numeral_nat @ one ) )
      = A2 ) ).

% mult_numeral_1_right
thf(fact_382_mult__numeral__1__right,axiom,
    ! [A2: int] :
      ( ( times_times_int @ A2 @ ( numeral_numeral_int @ one ) )
      = A2 ) ).

% mult_numeral_1_right
thf(fact_383_mult__numeral__1,axiom,
    ! [A2: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A2 )
      = A2 ) ).

% mult_numeral_1
thf(fact_384_mult__numeral__1,axiom,
    ! [A2: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A2 )
      = A2 ) ).

% mult_numeral_1
thf(fact_385_mult__numeral__1,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A2 )
      = A2 ) ).

% mult_numeral_1
thf(fact_386_mult__numeral__1,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A2 )
      = A2 ) ).

% mult_numeral_1
thf(fact_387_mult__numeral__1,axiom,
    ! [A2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A2 )
      = A2 ) ).

% mult_numeral_1
thf(fact_388_zero__less__power,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ).

% zero_less_power
thf(fact_389_zero__less__power,axiom,
    ! [A2: real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A2 @ N3 ) ) ) ).

% zero_less_power
thf(fact_390_zero__less__power,axiom,
    ! [A2: rat,N3: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ N3 ) ) ) ).

% zero_less_power
thf(fact_391_zero__less__power,axiom,
    ! [A2: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A2 @ N3 ) ) ) ).

% zero_less_power
thf(fact_392_zero__less__power,axiom,
    ! [A2: int,N3: nat] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A2 @ N3 ) ) ) ).

% zero_less_power
thf(fact_393_divide__numeral__1,axiom,
    ! [A2: complex] :
      ( ( divide1717551699836669952omplex @ A2 @ ( numera6690914467698888265omplex @ one ) )
      = A2 ) ).

% divide_numeral_1
thf(fact_394_divide__numeral__1,axiom,
    ! [A2: real] :
      ( ( divide_divide_real @ A2 @ ( numeral_numeral_real @ one ) )
      = A2 ) ).

% divide_numeral_1
thf(fact_395_divide__numeral__1,axiom,
    ! [A2: rat] :
      ( ( divide_divide_rat @ A2 @ ( numeral_numeral_rat @ one ) )
      = A2 ) ).

% divide_numeral_1
thf(fact_396_power__Suc2,axiom,
    ! [A2: complex,N3: nat] :
      ( ( power_power_complex @ A2 @ ( suc @ N3 ) )
      = ( times_times_complex @ ( power_power_complex @ A2 @ N3 ) @ A2 ) ) ).

% power_Suc2
thf(fact_397_power__Suc2,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( power_8256067586552552935nteger @ A2 @ ( suc @ N3 ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ A2 ) ) ).

% power_Suc2
thf(fact_398_power__Suc2,axiom,
    ! [A2: real,N3: nat] :
      ( ( power_power_real @ A2 @ ( suc @ N3 ) )
      = ( times_times_real @ ( power_power_real @ A2 @ N3 ) @ A2 ) ) ).

% power_Suc2
thf(fact_399_power__Suc2,axiom,
    ! [A2: rat,N3: nat] :
      ( ( power_power_rat @ A2 @ ( suc @ N3 ) )
      = ( times_times_rat @ ( power_power_rat @ A2 @ N3 ) @ A2 ) ) ).

% power_Suc2
thf(fact_400_power__Suc2,axiom,
    ! [A2: nat,N3: nat] :
      ( ( power_power_nat @ A2 @ ( suc @ N3 ) )
      = ( times_times_nat @ ( power_power_nat @ A2 @ N3 ) @ A2 ) ) ).

% power_Suc2
thf(fact_401_power__Suc2,axiom,
    ! [A2: int,N3: nat] :
      ( ( power_power_int @ A2 @ ( suc @ N3 ) )
      = ( times_times_int @ ( power_power_int @ A2 @ N3 ) @ A2 ) ) ).

% power_Suc2
thf(fact_402_power__Suc2,axiom,
    ! [A2: assn,N3: nat] :
      ( ( power_power_assn @ A2 @ ( suc @ N3 ) )
      = ( times_times_assn @ ( power_power_assn @ A2 @ N3 ) @ A2 ) ) ).

% power_Suc2
thf(fact_403_power__Suc,axiom,
    ! [A2: complex,N3: nat] :
      ( ( power_power_complex @ A2 @ ( suc @ N3 ) )
      = ( times_times_complex @ A2 @ ( power_power_complex @ A2 @ N3 ) ) ) ).

% power_Suc
thf(fact_404_power__Suc,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( power_8256067586552552935nteger @ A2 @ ( suc @ N3 ) )
      = ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ).

% power_Suc
thf(fact_405_power__Suc,axiom,
    ! [A2: real,N3: nat] :
      ( ( power_power_real @ A2 @ ( suc @ N3 ) )
      = ( times_times_real @ A2 @ ( power_power_real @ A2 @ N3 ) ) ) ).

% power_Suc
thf(fact_406_power__Suc,axiom,
    ! [A2: rat,N3: nat] :
      ( ( power_power_rat @ A2 @ ( suc @ N3 ) )
      = ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N3 ) ) ) ).

% power_Suc
thf(fact_407_power__Suc,axiom,
    ! [A2: nat,N3: nat] :
      ( ( power_power_nat @ A2 @ ( suc @ N3 ) )
      = ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N3 ) ) ) ).

% power_Suc
thf(fact_408_power__Suc,axiom,
    ! [A2: int,N3: nat] :
      ( ( power_power_int @ A2 @ ( suc @ N3 ) )
      = ( times_times_int @ A2 @ ( power_power_int @ A2 @ N3 ) ) ) ).

% power_Suc
thf(fact_409_power__Suc,axiom,
    ! [A2: assn,N3: nat] :
      ( ( power_power_assn @ A2 @ ( suc @ N3 ) )
      = ( times_times_assn @ A2 @ ( power_power_assn @ A2 @ N3 ) ) ) ).

% power_Suc
thf(fact_410_power__add,axiom,
    ! [A2: complex,M: nat,N3: nat] :
      ( ( power_power_complex @ A2 @ ( plus_plus_nat @ M @ N3 ) )
      = ( times_times_complex @ ( power_power_complex @ A2 @ M ) @ ( power_power_complex @ A2 @ N3 ) ) ) ).

% power_add
thf(fact_411_power__add,axiom,
    ! [A2: code_integer,M: nat,N3: nat] :
      ( ( power_8256067586552552935nteger @ A2 @ ( plus_plus_nat @ M @ N3 ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ M ) @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ).

% power_add
thf(fact_412_power__add,axiom,
    ! [A2: real,M: nat,N3: nat] :
      ( ( power_power_real @ A2 @ ( plus_plus_nat @ M @ N3 ) )
      = ( times_times_real @ ( power_power_real @ A2 @ M ) @ ( power_power_real @ A2 @ N3 ) ) ) ).

% power_add
thf(fact_413_power__add,axiom,
    ! [A2: rat,M: nat,N3: nat] :
      ( ( power_power_rat @ A2 @ ( plus_plus_nat @ M @ N3 ) )
      = ( times_times_rat @ ( power_power_rat @ A2 @ M ) @ ( power_power_rat @ A2 @ N3 ) ) ) ).

% power_add
thf(fact_414_power__add,axiom,
    ! [A2: nat,M: nat,N3: nat] :
      ( ( power_power_nat @ A2 @ ( plus_plus_nat @ M @ N3 ) )
      = ( times_times_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N3 ) ) ) ).

% power_add
thf(fact_415_power__add,axiom,
    ! [A2: int,M: nat,N3: nat] :
      ( ( power_power_int @ A2 @ ( plus_plus_nat @ M @ N3 ) )
      = ( times_times_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N3 ) ) ) ).

% power_add
thf(fact_416_power__add,axiom,
    ! [A2: assn,M: nat,N3: nat] :
      ( ( power_power_assn @ A2 @ ( plus_plus_nat @ M @ N3 ) )
      = ( times_times_assn @ ( power_power_assn @ A2 @ M ) @ ( power_power_assn @ A2 @ N3 ) ) ) ).

% power_add
thf(fact_417_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( divide_divide_nat @ M @ N3 )
        = zero_zero_nat )
      = ( ( ord_less_nat @ M @ N3 )
        | ( N3 = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
thf(fact_418_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N3 ) )
       => ( ord_less_nat @ M @ N3 ) ) ) ).

% nat_power_less_imp_less
thf(fact_419_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N3: nat] :
      ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N3 ) )
     => ( ord_less_nat @ ( divide_divide_nat @ M @ N3 ) @ I ) ) ).

% less_mult_imp_div_less
thf(fact_420_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_421_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_422_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_423_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: complex,C2: complex] :
      ( ( ( numera6690914467698888265omplex @ W )
        = ( divide1717551699836669952omplex @ B2 @ C2 ) )
      = ( ( ( C2 != zero_zero_complex )
         => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C2 )
            = B2 ) )
        & ( ( C2 = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_424_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: real,C2: real] :
      ( ( ( numeral_numeral_real @ W )
        = ( divide_divide_real @ B2 @ C2 ) )
      = ( ( ( C2 != zero_zero_real )
         => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C2 )
            = B2 ) )
        & ( ( C2 = zero_zero_real )
         => ( ( numeral_numeral_real @ W )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_425_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: rat,C2: rat] :
      ( ( ( numeral_numeral_rat @ W )
        = ( divide_divide_rat @ B2 @ C2 ) )
      = ( ( ( C2 != zero_zero_rat )
         => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C2 )
            = B2 ) )
        & ( ( C2 = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_426_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: complex,C2: complex,W: num] :
      ( ( ( divide1717551699836669952omplex @ B2 @ C2 )
        = ( numera6690914467698888265omplex @ W ) )
      = ( ( ( C2 != zero_zero_complex )
         => ( B2
            = ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C2 ) ) )
        & ( ( C2 = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_427_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: real,C2: real,W: num] :
      ( ( ( divide_divide_real @ B2 @ C2 )
        = ( numeral_numeral_real @ W ) )
      = ( ( ( C2 != zero_zero_real )
         => ( B2
            = ( times_times_real @ ( numeral_numeral_real @ W ) @ C2 ) ) )
        & ( ( C2 = zero_zero_real )
         => ( ( numeral_numeral_real @ W )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_428_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: rat,C2: rat,W: num] :
      ( ( ( divide_divide_rat @ B2 @ C2 )
        = ( numeral_numeral_rat @ W ) )
      = ( ( ( C2 != zero_zero_rat )
         => ( B2
            = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C2 ) ) )
        & ( ( C2 = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_429_numeral__Bit0__div__2,axiom,
    ! [N3: num] :
      ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( numeral_numeral_nat @ N3 ) ) ).

% numeral_Bit0_div_2
thf(fact_430_numeral__Bit0__div__2,axiom,
    ! [N3: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( numeral_numeral_int @ N3 ) ) ).

% numeral_Bit0_div_2
thf(fact_431_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_432_zero__power,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( power_power_rat @ zero_zero_rat @ N3 )
        = zero_zero_rat ) ) ).

% zero_power
thf(fact_433_zero__power,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( power_power_nat @ zero_zero_nat @ N3 )
        = zero_zero_nat ) ) ).

% zero_power
thf(fact_434_zero__power,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( power_power_real @ zero_zero_real @ N3 )
        = zero_zero_real ) ) ).

% zero_power
thf(fact_435_zero__power,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( power_power_int @ zero_zero_int @ N3 )
        = zero_zero_int ) ) ).

% zero_power
thf(fact_436_zero__power,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( power_power_complex @ zero_zero_complex @ N3 )
        = zero_zero_complex ) ) ).

% zero_power
thf(fact_437_zero__power,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N3 )
        = zero_z3403309356797280102nteger ) ) ).

% zero_power
thf(fact_438_power__gt__expt,axiom,
    ! [N3: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N3 )
     => ( ord_less_nat @ K @ ( power_power_nat @ N3 @ K ) ) ) ).

% power_gt_expt
thf(fact_439_div__less__iff__less__mult,axiom,
    ! [Q3: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q3 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q3 ) @ N3 )
        = ( ord_less_nat @ M @ ( times_times_nat @ N3 @ Q3 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_440_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: real,C2: real] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B2 @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C2 ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C2 ) ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_441_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: rat,C2: rat] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B2 @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C2 ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_442_divide__less__eq__numeral_I1_J,axiom,
    ! [B2: real,C2: real,W: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ C2 ) @ ( numeral_numeral_real @ W ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ B2 @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C2 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C2 ) @ B2 ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_443_divide__less__eq__numeral_I1_J,axiom,
    ! [B2: rat,C2: rat,W: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C2 ) @ ( numeral_numeral_rat @ W ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ B2 @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C2 ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_444_left__add__twice,axiom,
    ! [A2: complex,B2: complex] :
      ( ( plus_plus_complex @ A2 @ ( plus_plus_complex @ A2 @ B2 ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A2 ) @ B2 ) ) ).

% left_add_twice
thf(fact_445_left__add__twice,axiom,
    ! [A2: real,B2: real] :
      ( ( plus_plus_real @ A2 @ ( plus_plus_real @ A2 @ B2 ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A2 ) @ B2 ) ) ).

% left_add_twice
thf(fact_446_left__add__twice,axiom,
    ! [A2: rat,B2: rat] :
      ( ( plus_plus_rat @ A2 @ ( plus_plus_rat @ A2 @ B2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A2 ) @ B2 ) ) ).

% left_add_twice
thf(fact_447_left__add__twice,axiom,
    ! [A2: nat,B2: nat] :
      ( ( plus_plus_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) @ B2 ) ) ).

% left_add_twice
thf(fact_448_left__add__twice,axiom,
    ! [A2: int,B2: int] :
      ( ( plus_plus_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) @ B2 ) ) ).

% left_add_twice
thf(fact_449_mult__2__right,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ Z @ Z ) ) ).

% mult_2_right
thf(fact_450_mult__2__right,axiom,
    ! [Z: real] :
      ( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ Z @ Z ) ) ).

% mult_2_right
thf(fact_451_mult__2__right,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2_right
thf(fact_452_mult__2__right,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2_right
thf(fact_453_mult__2__right,axiom,
    ! [Z: int] :
      ( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2_right
thf(fact_454_mult__2,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_complex @ Z @ Z ) ) ).

% mult_2
thf(fact_455_mult__2,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_real @ Z @ Z ) ) ).

% mult_2
thf(fact_456_mult__2,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2
thf(fact_457_mult__2,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2
thf(fact_458_mult__2,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2
thf(fact_459_zero__power2,axiom,
    ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_rat ) ).

% zero_power2
thf(fact_460_zero__power2,axiom,
    ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% zero_power2
thf(fact_461_zero__power2,axiom,
    ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_real ) ).

% zero_power2
thf(fact_462_zero__power2,axiom,
    ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% zero_power2
thf(fact_463_zero__power2,axiom,
    ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_complex ) ).

% zero_power2
thf(fact_464_zero__power2,axiom,
    ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% zero_power2
thf(fact_465_power2__eq__square,axiom,
    ! [A2: complex] :
      ( ( power_power_complex @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_complex @ A2 @ A2 ) ) ).

% power2_eq_square
thf(fact_466_power2__eq__square,axiom,
    ! [A2: code_integer] :
      ( ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_3573771949741848930nteger @ A2 @ A2 ) ) ).

% power2_eq_square
thf(fact_467_power2__eq__square,axiom,
    ! [A2: real] :
      ( ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_real @ A2 @ A2 ) ) ).

% power2_eq_square
thf(fact_468_power2__eq__square,axiom,
    ! [A2: rat] :
      ( ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_rat @ A2 @ A2 ) ) ).

% power2_eq_square
thf(fact_469_power2__eq__square,axiom,
    ! [A2: nat] :
      ( ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_nat @ A2 @ A2 ) ) ).

% power2_eq_square
thf(fact_470_power2__eq__square,axiom,
    ! [A2: int] :
      ( ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_int @ A2 @ A2 ) ) ).

% power2_eq_square
thf(fact_471_power2__eq__square,axiom,
    ! [A2: assn] :
      ( ( power_power_assn @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_assn @ A2 @ A2 ) ) ).

% power2_eq_square
thf(fact_472_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_473_power4__eq__xxxx,axiom,
    ! [X: code_integer] :
      ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_474_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_475_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_476_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_477_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_478_power4__eq__xxxx,axiom,
    ! [X: assn] :
      ( ( power_power_assn @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_479_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_480_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_481_power2__commute,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( power_8256067586552552935nteger @ ( minus_8373710615458151222nteger @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_8256067586552552935nteger @ ( minus_8373710615458151222nteger @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_482_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_483_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_484_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_485_power__even__eq,axiom,
    ! [A2: nat,N3: nat] :
      ( ( power_power_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( power_power_nat @ ( power_power_nat @ A2 @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_486_power__even__eq,axiom,
    ! [A2: real,N3: nat] :
      ( ( power_power_real @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( power_power_real @ ( power_power_real @ A2 @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_487_power__even__eq,axiom,
    ! [A2: int,N3: nat] :
      ( ( power_power_int @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( power_power_int @ ( power_power_int @ A2 @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_488_power__even__eq,axiom,
    ! [A2: complex,N3: nat] :
      ( ( power_power_complex @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( power_power_complex @ ( power_power_complex @ A2 @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_489_power__even__eq,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( power_8256067586552552935nteger @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_490_dividend__less__times__div,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N3 @ ( times_times_nat @ N3 @ ( divide_divide_nat @ M @ N3 ) ) ) ) ) ).

% dividend_less_times_div
thf(fact_491_dividend__less__div__times,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( divide_divide_nat @ M @ N3 ) @ N3 ) ) ) ) ).

% dividend_less_div_times
thf(fact_492_split__div,axiom,
    ! [P: nat > $o,M: nat,N3: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N3 ) )
      = ( ( ( N3 = zero_zero_nat )
         => ( P @ zero_zero_nat ) )
        & ( ( N3 != zero_zero_nat )
         => ! [I2: nat,J: nat] :
              ( ( ord_less_nat @ J @ N3 )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N3 @ I2 ) @ J ) )
               => ( P @ I2 ) ) ) ) ) ) ).

% split_div
thf(fact_493_half__gt__zero__iff,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( ord_less_real @ zero_zero_real @ A2 ) ) ).

% half_gt_zero_iff
thf(fact_494_half__gt__zero__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
      = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).

% half_gt_zero_iff
thf(fact_495_half__gt__zero,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% half_gt_zero
thf(fact_496_half__gt__zero,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% half_gt_zero
thf(fact_497_power2__less__0,axiom,
    ! [A2: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_z3403309356797280102nteger ) ).

% power2_less_0
thf(fact_498_power2__less__0,axiom,
    ! [A2: real] :
      ~ ( ord_less_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).

% power2_less_0
thf(fact_499_power2__less__0,axiom,
    ! [A2: rat] :
      ~ ( ord_less_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).

% power2_less_0
thf(fact_500_power2__less__0,axiom,
    ! [A2: int] :
      ~ ( ord_less_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).

% power2_less_0
thf(fact_501_less__2__cases__iff,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ( N3 = zero_zero_nat )
        | ( N3
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases_iff
thf(fact_502_less__2__cases,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
     => ( ( N3 = zero_zero_nat )
        | ( N3
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases
thf(fact_503_sum__power2__gt__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_z3403309356797280102nteger )
        | ( Y != zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_504_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_505_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_506_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_507_not__sum__power2__lt__zero,axiom,
    ! [X: code_integer,Y: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_z3403309356797280102nteger ) ).

% not_sum_power2_lt_zero
thf(fact_508_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_509_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_510_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_511_power2__sum,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( power_8256067586552552935nteger @ ( plus_p5714425477246183910nteger @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_512_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_513_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_514_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_515_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_516_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_517_power__odd__eq,axiom,
    ! [A2: complex,N3: nat] :
      ( ( power_power_complex @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( times_times_complex @ A2 @ ( power_power_complex @ ( power_power_complex @ A2 @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_518_power__odd__eq,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( power_8256067586552552935nteger @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_519_power__odd__eq,axiom,
    ! [A2: real,N3: nat] :
      ( ( power_power_real @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( times_times_real @ A2 @ ( power_power_real @ ( power_power_real @ A2 @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_520_power__odd__eq,axiom,
    ! [A2: rat,N3: nat] :
      ( ( power_power_rat @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( times_times_rat @ A2 @ ( power_power_rat @ ( power_power_rat @ A2 @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_521_power__odd__eq,axiom,
    ! [A2: nat,N3: nat] :
      ( ( power_power_nat @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( times_times_nat @ A2 @ ( power_power_nat @ ( power_power_nat @ A2 @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_522_power__odd__eq,axiom,
    ! [A2: int,N3: nat] :
      ( ( power_power_int @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( times_times_int @ A2 @ ( power_power_int @ ( power_power_int @ A2 @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_523_power__odd__eq,axiom,
    ! [A2: assn,N3: nat] :
      ( ( power_power_assn @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( times_times_assn @ A2 @ ( power_power_assn @ ( power_power_assn @ A2 @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_524_Suc__n__div__2__gt__zero,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% Suc_n_div_2_gt_zero
thf(fact_525_div__2__gt__zero,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N3 )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% div_2_gt_zero
thf(fact_526_power2__diff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( power_8256067586552552935nteger @ ( minus_8373710615458151222nteger @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_527_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_528_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_529_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_530_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_531_odd__power__less__zero,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) @ zero_z3403309356797280102nteger ) ) ).

% odd_power_less_zero
thf(fact_532_odd__power__less__zero,axiom,
    ! [A2: real,N3: nat] :
      ( ( ord_less_real @ A2 @ zero_zero_real )
     => ( ord_less_real @ ( power_power_real @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) @ zero_zero_real ) ) ).

% odd_power_less_zero
thf(fact_533_odd__power__less__zero,axiom,
    ! [A2: rat,N3: nat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ord_less_rat @ ( power_power_rat @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) @ zero_zero_rat ) ) ).

% odd_power_less_zero
thf(fact_534_odd__power__less__zero,axiom,
    ! [A2: int,N3: nat] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ord_less_int @ ( power_power_int @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) @ zero_zero_int ) ) ).

% odd_power_less_zero
thf(fact_535_highboundn,axiom,
    ( ( ma != mi )
   => ( ( ord_less_eq_nat @ xa @ ma )
     => ( ord_less_nat @ ( vEBT_VEBT_high @ xnew @ ( divide_divide_nat @ na @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) ) ).

% highboundn
thf(fact_536_highbound,axiom,
    ( ( ma != mi )
   => ( ( ord_less_eq_nat @ xa @ ma )
     => ( ord_less_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ na @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) ) ).

% highbound
thf(fact_537_xbound,axiom,
    ( ( ord_less_eq_nat @ mi @ xa )
   => ( ( ord_less_eq_nat @ xa @ ma )
     => ( ord_less_eq_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ na @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) ) ).

% xbound
thf(fact_538_big__assn__simp_H,axiom,
    ! [H2: nat,TreeList: list_VEBT_VEBT,Xaa: vEBT_VEBT,L2: nat,X: vEBT_VEBTi,Xb: option_nat,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ H2 @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( ( Xaa
          = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
       => ( entails
          @ ( times_times_assn
            @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Xaa @ X )
              @ ( pure_assn
                @ ( Xb
                  = ( vEBT_vebt_mint @ Xaa ) ) ) )
            @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ H2 @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ H2 @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) )
          @ ( times_times_assn
            @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ H2 @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) )
              @ ( pure_assn
                @ ( Xb
                  = ( vEBT_vebt_mint @ Xaa ) ) ) )
            @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ Xaa ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ H2 @ X ) ) ) ) ) ) ).

% big_assn_simp'
thf(fact_539_big__assn__simp,axiom,
    ! [H2: nat,TreeList: list_VEBT_VEBT,L2: nat,X: vEBT_VEBTi,Xaa: option_nat,X13: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ H2 @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
     => ( entails
        @ ( times_times_assn
          @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) @ X )
            @ ( pure_assn
              @ ( Xaa
                = ( vEBT_vebt_mint @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) ) ) )
          @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ H2 @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( insert_nat @ H2 @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) )
        @ ( times_times_assn
          @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is @ H2 @ X ) ) @ ( vEBT_vebt_assn_raw @ Summary @ X14 ) )
            @ ( pure_assn
              @ ( Xaa
                = ( vEBT_vebt_mint @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) ) ) )
          @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) @ ( list_u6098035379799741383_VEBTi @ Tree_is @ H2 @ X ) ) ) ) ) ).

% big_assn_simp
thf(fact_540_mimaxprop,axiom,
    ( ( ord_less_eq_nat @ mi @ ma )
    & ( ord_less_eq_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ) ).

% mimaxprop
thf(fact_541_atLeastLessThan__singleton,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
      = ( insert_nat @ M @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_542_nth__update__invalid,axiom,
    ! [I: nat,L2: list_VEBT_VEBT,J2: nat,X: vEBT_VEBT] :
      ( ~ ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ L2 @ J2 @ X ) @ I )
        = ( nth_VEBT_VEBT @ L2 @ I ) ) ) ).

% nth_update_invalid
thf(fact_543_nth__update__invalid,axiom,
    ! [I: nat,L2: list_real,J2: nat,X: real] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_real @ L2 ) )
     => ( ( nth_real @ ( list_update_real @ L2 @ J2 @ X ) @ I )
        = ( nth_real @ L2 @ I ) ) ) ).

% nth_update_invalid
thf(fact_544_nth__update__invalid,axiom,
    ! [I: nat,L2: list_o,J2: nat,X: $o] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_o @ L2 ) )
     => ( ( nth_o @ ( list_update_o @ L2 @ J2 @ X ) @ I )
        = ( nth_o @ L2 @ I ) ) ) ).

% nth_update_invalid
thf(fact_545_nth__update__invalid,axiom,
    ! [I: nat,L2: list_nat,J2: nat,X: nat] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_nat @ L2 ) )
     => ( ( nth_nat @ ( list_update_nat @ L2 @ J2 @ X ) @ I )
        = ( nth_nat @ L2 @ I ) ) ) ).

% nth_update_invalid
thf(fact_546_nth__update__invalid,axiom,
    ! [I: nat,L2: list_VEBT_VEBTi,J2: nat,X: vEBT_VEBTi] :
      ( ~ ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L2 ) )
     => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ L2 @ J2 @ X ) @ I )
        = ( nth_VEBT_VEBTi @ L2 @ I ) ) ) ).

% nth_update_invalid
thf(fact_547_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_548_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_real,X: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
     => ( ( nth_real @ ( list_update_real @ Xs2 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_549_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_o,X: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_550_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_551_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_552_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: vEBT_VEBT > vEBT_VEBT > assn,X: vEBT_VEBT,Xi: vEBT_VEBT,I3: set_nat,I: nat,Xs2: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ X @ Xi ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L3204528365124325536T_VEBT @ I3 @ A @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_553_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: real > vEBT_VEBTi > assn,X: real,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs2: list_real,Xsi: list_VEBT_VEBTi,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ X @ Xi ) @ ( vEBT_L7851252805511451907_VEBTi @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L7851252805511451907_VEBTi @ I3 @ A @ ( list_update_real @ Xs2 @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_554_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: real > vEBT_VEBT > assn,X: real,Xi: vEBT_VEBT,I3: set_nat,I: nat,Xs2: list_real,Xsi: list_VEBT_VEBT,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ X @ Xi ) @ ( vEBT_L3095048238742455910T_VEBT @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L3095048238742455910T_VEBT @ I3 @ A @ ( list_update_real @ Xs2 @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_555_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: $o > vEBT_VEBTi > assn,X: $o,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs2: list_o,Xsi: list_VEBT_VEBTi,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ X @ Xi ) @ ( vEBT_L6286945158656146733_VEBTi @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L6286945158656146733_VEBTi @ I3 @ A @ ( list_update_o @ Xs2 @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_556_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: $o > vEBT_VEBT > assn,X: $o,Xi: vEBT_VEBT,I3: set_nat,I: nat,Xs2: list_o,Xsi: list_VEBT_VEBT,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ X @ Xi ) @ ( vEBT_L1319876754960170684T_VEBT @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L1319876754960170684T_VEBT @ I3 @ A @ ( list_update_o @ Xs2 @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_557_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: nat > vEBT_VEBTi > assn,X: nat,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs2: list_nat,Xsi: list_VEBT_VEBTi,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ X @ Xi ) @ ( vEBT_L7489483478785760935_VEBTi @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L7489483478785760935_VEBTi @ I3 @ A @ ( list_update_nat @ Xs2 @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_558_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: nat > vEBT_VEBT > assn,X: nat,Xi: vEBT_VEBT,I3: set_nat,I: nat,Xs2: list_nat,Xsi: list_VEBT_VEBT,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ X @ Xi ) @ ( vEBT_L8511957252848910786T_VEBT @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L8511957252848910786T_VEBT @ I3 @ A @ ( list_update_nat @ Xs2 @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_559_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: vEBT_VEBTi > vEBT_VEBTi > assn,X: vEBT_VEBTi,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs2: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ X @ Xi ) @ ( vEBT_L886525131989349516_VEBTi @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L886525131989349516_VEBTi @ I3 @ A @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_560_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: vEBT_VEBTi > vEBT_VEBT > assn,X: vEBT_VEBTi,Xi: vEBT_VEBT,I3: set_nat,I: nat,Xs2: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ X @ Xi ) @ ( vEBT_L2497118539674116125T_VEBT @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L2497118539674116125T_VEBT @ I3 @ A @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_561_listI__assn__reinsert__upd,axiom,
    ! [P: assn,A: vEBT_VEBT > vEBT_VEBTi > assn,X: vEBT_VEBT,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs2: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ X @ Xi ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L1528199826722428489_VEBTi @ I3 @ A @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd
thf(fact_562_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N3 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_563_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N3 @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% mult_le_cancel2
thf(fact_564_semiring__norm_I87_J,axiom,
    ! [M: num,N3: num] :
      ( ( ( bit0 @ M )
        = ( bit0 @ N3 ) )
      = ( M = N3 ) ) ).

% semiring_norm(87)
thf(fact_565_diff__Suc__Suc,axiom,
    ! [M: nat,N3: nat] :
      ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N3 ) )
      = ( minus_minus_nat @ M @ N3 ) ) ).

% diff_Suc_Suc
thf(fact_566_Suc__diff__diff,axiom,
    ! [M: nat,N3: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N3 ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N3 ) @ K ) ) ).

% Suc_diff_diff
thf(fact_567_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% nat.inject
thf(fact_568_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_569_diff__0__eq__0,axiom,
    ! [N3: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N3 )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_570_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_571_diff__diff__left,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K )
      = ( minus_minus_nat @ I @ ( plus_plus_nat @ J2 @ K ) ) ) ).

% diff_diff_left
thf(fact_572_diff__diff__cancel,axiom,
    ! [I: nat,N3: nat] :
      ( ( ord_less_eq_nat @ I @ N3 )
     => ( ( minus_minus_nat @ N3 @ ( minus_minus_nat @ N3 @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_573_list__update__overwrite,axiom,
    ! [Xs2: list_VEBT_VEBTi,I: nat,X: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( list_u6098035379799741383_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X ) @ I @ Y )
      = ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ Y ) ) ).

% list_update_overwrite
thf(fact_574_list__update__overwrite,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ I @ Y )
      = ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ Y ) ) ).

% list_update_overwrite
thf(fact_575_semiring__norm_I85_J,axiom,
    ! [M: num] :
      ( ( bit0 @ M )
     != one ) ).

% semiring_norm(85)
thf(fact_576_semiring__norm_I83_J,axiom,
    ! [N3: num] :
      ( one
     != ( bit0 @ N3 ) ) ).

% semiring_norm(83)
thf(fact_577_lessI,axiom,
    ! [N3: nat] : ( ord_less_nat @ N3 @ ( suc @ N3 ) ) ).

% lessI
thf(fact_578_Suc__mono,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N3 ) ) ) ).

% Suc_mono
thf(fact_579_Suc__less__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N3 ) )
      = ( ord_less_nat @ M @ N3 ) ) ).

% Suc_less_eq
thf(fact_580_zero__less__diff,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N3 @ M ) )
      = ( ord_less_nat @ M @ N3 ) ) ).

% zero_less_diff
thf(fact_581_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_582_neq0__conv,axiom,
    ! [N3: nat] :
      ( ( N3 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).

% neq0_conv
thf(fact_583_less__nat__zero__code,axiom,
    ! [N3: nat] :
      ~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_584_add__Suc__right,axiom,
    ! [M: nat,N3: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N3 ) )
      = ( suc @ ( plus_plus_nat @ M @ N3 ) ) ) ).

% add_Suc_right
thf(fact_585_Suc__le__mono,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N3 ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N3 @ M ) ) ).

% Suc_le_mono
thf(fact_586_add__is__0,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( plus_plus_nat @ M @ N3 )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N3 = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_587_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_588_diff__is__0__eq_H,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( minus_minus_nat @ M @ N3 )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_589_diff__is__0__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( minus_minus_nat @ M @ N3 )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% diff_is_0_eq
thf(fact_590_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).

% bot_nat_0.extremum
thf(fact_591_le0,axiom,
    ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).

% le0
thf(fact_592_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N3 ) )
      = ( ord_less_nat @ M @ N3 ) ) ).

% nat_add_left_cancel_less
thf(fact_593_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ) ).

% Nat.diff_diff_right
thf(fact_594_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
        = ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_595_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_596_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N3 ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% nat_add_left_cancel_le
thf(fact_597_mult__is__0,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( times_times_nat @ M @ N3 )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        | ( N3 = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_598_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_599_mult__cancel1,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N3 ) )
      = ( ( M = N3 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_600_mult__cancel2,axiom,
    ! [M: nat,K: nat,N3: nat] :
      ( ( ( times_times_nat @ M @ K )
        = ( times_times_nat @ N3 @ K ) )
      = ( ( M = N3 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_601_ivl__subset,axiom,
    ! [I: rat,J2: rat,M: rat,N3: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or4029947393144176647an_rat @ I @ J2 ) @ ( set_or4029947393144176647an_rat @ M @ N3 ) )
      = ( ( ord_less_eq_rat @ J2 @ I )
        | ( ( ord_less_eq_rat @ M @ I )
          & ( ord_less_eq_rat @ J2 @ N3 ) ) ) ) ).

% ivl_subset
thf(fact_602_ivl__subset,axiom,
    ! [I: num,J2: num,M: num,N3: num] :
      ( ( ord_less_eq_set_num @ ( set_or1222409239386451017an_num @ I @ J2 ) @ ( set_or1222409239386451017an_num @ M @ N3 ) )
      = ( ( ord_less_eq_num @ J2 @ I )
        | ( ( ord_less_eq_num @ M @ I )
          & ( ord_less_eq_num @ J2 @ N3 ) ) ) ) ).

% ivl_subset
thf(fact_603_ivl__subset,axiom,
    ! [I: nat,J2: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J2 ) @ ( set_or4665077453230672383an_nat @ M @ N3 ) )
      = ( ( ord_less_eq_nat @ J2 @ I )
        | ( ( ord_less_eq_nat @ M @ I )
          & ( ord_less_eq_nat @ J2 @ N3 ) ) ) ) ).

% ivl_subset
thf(fact_604_ivl__subset,axiom,
    ! [I: int,J2: int,M: int,N3: int] :
      ( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I @ J2 ) @ ( set_or4662586982721622107an_int @ M @ N3 ) )
      = ( ( ord_less_eq_int @ J2 @ I )
        | ( ( ord_less_eq_int @ M @ I )
          & ( ord_less_eq_int @ J2 @ N3 ) ) ) ) ).

% ivl_subset
thf(fact_605_ivl__subset,axiom,
    ! [I: code_integer,J2: code_integer,M: code_integer,N3: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or8404916559141939852nteger @ I @ J2 ) @ ( set_or8404916559141939852nteger @ M @ N3 ) )
      = ( ( ord_le3102999989581377725nteger @ J2 @ I )
        | ( ( ord_le3102999989581377725nteger @ M @ I )
          & ( ord_le3102999989581377725nteger @ J2 @ N3 ) ) ) ) ).

% ivl_subset
thf(fact_606_length__list__update,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) )
      = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).

% length_list_update
thf(fact_607_length__list__update,axiom,
    ! [Xs2: list_real,I: nat,X: real] :
      ( ( size_size_list_real @ ( list_update_real @ Xs2 @ I @ X ) )
      = ( size_size_list_real @ Xs2 ) ) ).

% length_list_update
thf(fact_608_length__list__update,axiom,
    ! [Xs2: list_o,I: nat,X: $o] :
      ( ( size_size_list_o @ ( list_update_o @ Xs2 @ I @ X ) )
      = ( size_size_list_o @ Xs2 ) ) ).

% length_list_update
thf(fact_609_length__list__update,axiom,
    ! [Xs2: list_nat,I: nat,X: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs2 @ I @ X ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_list_update
thf(fact_610_length__list__update,axiom,
    ! [Xs2: list_VEBT_VEBTi,I: nat,X: vEBT_VEBTi] :
      ( ( size_s7982070591426661849_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X ) )
      = ( size_s7982070591426661849_VEBTi @ Xs2 ) ) ).

% length_list_update
thf(fact_611_list__update__id,axiom,
    ! [Xs2: list_nat,I: nat] :
      ( ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_612_list__update__id,axiom,
    ! [Xs2: list_VEBT_VEBTi,I: nat] :
      ( ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ ( nth_VEBT_VEBTi @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_613_list__update__id,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat] :
      ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( nth_VEBT_VEBT @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_614_nth__list__update__neq,axiom,
    ! [I: nat,J2: nat,Xs2: list_nat,X: nat] :
      ( ( I != J2 )
     => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ J2 )
        = ( nth_nat @ Xs2 @ J2 ) ) ) ).

% nth_list_update_neq
thf(fact_615_nth__list__update__neq,axiom,
    ! [I: nat,J2: nat,Xs2: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( I != J2 )
     => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X ) @ J2 )
        = ( nth_VEBT_VEBTi @ Xs2 @ J2 ) ) ) ).

% nth_list_update_neq
thf(fact_616_nth__list__update__neq,axiom,
    ! [I: nat,J2: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( I != J2 )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ J2 )
        = ( nth_VEBT_VEBT @ Xs2 @ J2 ) ) ) ).

% nth_list_update_neq
thf(fact_617_semiring__norm_I6_J,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N3 ) )
      = ( bit0 @ ( plus_plus_num @ M @ N3 ) ) ) ).

% semiring_norm(6)
thf(fact_618_semiring__norm_I13_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N3 ) )
      = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N3 ) ) ) ) ).

% semiring_norm(13)
thf(fact_619_semiring__norm_I12_J,axiom,
    ! [N3: num] :
      ( ( times_times_num @ one @ N3 )
      = N3 ) ).

% semiring_norm(12)
thf(fact_620_semiring__norm_I11_J,axiom,
    ! [M: num] :
      ( ( times_times_num @ M @ one )
      = M ) ).

% semiring_norm(11)
thf(fact_621_semiring__norm_I71_J,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N3 ) )
      = ( ord_less_eq_num @ M @ N3 ) ) ).

% semiring_norm(71)
thf(fact_622_semiring__norm_I78_J,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N3 ) )
      = ( ord_less_num @ M @ N3 ) ) ).

% semiring_norm(78)
thf(fact_623_semiring__norm_I68_J,axiom,
    ! [N3: num] : ( ord_less_eq_num @ one @ N3 ) ).

% semiring_norm(68)
thf(fact_624_semiring__norm_I75_J,axiom,
    ! [M: num] :
      ~ ( ord_less_num @ M @ one ) ).

% semiring_norm(75)
thf(fact_625_listlength,axiom,
    ( ( size_s6755466524823107622T_VEBT @ treeList )
    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ na @ ( divide_divide_nat @ na @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% listlength
thf(fact_626_zero__comp__diff__simps_I1_J,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B2 ) )
      = ( ord_less_eq_real @ B2 @ A2 ) ) ).

% zero_comp_diff_simps(1)
thf(fact_627_zero__comp__diff__simps_I1_J,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B2 ) )
      = ( ord_less_eq_rat @ B2 @ A2 ) ) ).

% zero_comp_diff_simps(1)
thf(fact_628_zero__comp__diff__simps_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B2 ) )
      = ( ord_less_eq_int @ B2 @ A2 ) ) ).

% zero_comp_diff_simps(1)
thf(fact_629_zero__comp__diff__simps_I2_J,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B2 ) )
      = ( ord_less_real @ B2 @ A2 ) ) ).

% zero_comp_diff_simps(2)
thf(fact_630_zero__comp__diff__simps_I2_J,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B2 ) )
      = ( ord_less_rat @ B2 @ A2 ) ) ).

% zero_comp_diff_simps(2)
thf(fact_631_zero__comp__diff__simps_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B2 ) )
      = ( ord_less_int @ B2 @ A2 ) ) ).

% zero_comp_diff_simps(2)
thf(fact_632_Suc__pred,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( suc @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) )
        = N3 ) ) ).

% Suc_pred
thf(fact_633_less__Suc0,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ N3 @ ( suc @ zero_zero_nat ) )
      = ( N3 = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_634_zero__less__Suc,axiom,
    ! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N3 ) ) ).

% zero_less_Suc
thf(fact_635_atLeastLessThan__iff,axiom,
    ! [I: real,L2: real,U: real] :
      ( ( member_real @ I @ ( set_or66887138388493659n_real @ L2 @ U ) )
      = ( ( ord_less_eq_real @ L2 @ I )
        & ( ord_less_real @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_636_atLeastLessThan__iff,axiom,
    ! [I: set_nat,L2: set_nat,U: set_nat] :
      ( ( member_set_nat @ I @ ( set_or3540276404033026485et_nat @ L2 @ U ) )
      = ( ( ord_less_eq_set_nat @ L2 @ I )
        & ( ord_less_set_nat @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_637_atLeastLessThan__iff,axiom,
    ! [I: rat,L2: rat,U: rat] :
      ( ( member_rat @ I @ ( set_or4029947393144176647an_rat @ L2 @ U ) )
      = ( ( ord_less_eq_rat @ L2 @ I )
        & ( ord_less_rat @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_638_atLeastLessThan__iff,axiom,
    ! [I: num,L2: num,U: num] :
      ( ( member_num @ I @ ( set_or1222409239386451017an_num @ L2 @ U ) )
      = ( ( ord_less_eq_num @ L2 @ I )
        & ( ord_less_num @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_639_atLeastLessThan__iff,axiom,
    ! [I: nat,L2: nat,U: nat] :
      ( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L2 @ U ) )
      = ( ( ord_less_eq_nat @ L2 @ I )
        & ( ord_less_nat @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_640_atLeastLessThan__iff,axiom,
    ! [I: int,L2: int,U: int] :
      ( ( member_int @ I @ ( set_or4662586982721622107an_int @ L2 @ U ) )
      = ( ( ord_less_eq_int @ L2 @ I )
        & ( ord_less_int @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_641_atLeastLessThan__iff,axiom,
    ! [I: code_integer,L2: code_integer,U: code_integer] :
      ( ( member_Code_integer @ I @ ( set_or8404916559141939852nteger @ L2 @ U ) )
      = ( ( ord_le3102999989581377725nteger @ L2 @ I )
        & ( ord_le6747313008572928689nteger @ I @ U ) ) ) ).

% atLeastLessThan_iff
thf(fact_642_add__gr__0,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N3 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% add_gr_0
thf(fact_643_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J2 ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_644_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) @ I )
        = ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_645_mult__eq__1__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( times_times_nat @ M @ N3 )
        = ( suc @ zero_zero_nat ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N3
          = ( suc @ zero_zero_nat ) ) ) ) ).

% mult_eq_1_iff
thf(fact_646_one__eq__mult__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( times_times_nat @ M @ N3 ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N3
          = ( suc @ zero_zero_nat ) ) ) ) ).

% one_eq_mult_iff
thf(fact_647_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N3: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N3 @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N3 ) ) ) ).

% mult_less_cancel2
thf(fact_648_nat__0__less__mult__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N3 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        & ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% nat_0_less_mult_iff
thf(fact_649_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N3 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N3 ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_650_mult__Suc__right,axiom,
    ! [M: nat,N3: nat] :
      ( ( times_times_nat @ M @ ( suc @ N3 ) )
      = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N3 ) ) ) ).

% mult_Suc_right
thf(fact_651_atLeastLessThan__empty,axiom,
    ! [B2: real,A2: real] :
      ( ( ord_less_eq_real @ B2 @ A2 )
     => ( ( set_or66887138388493659n_real @ A2 @ B2 )
        = bot_bot_set_real ) ) ).

% atLeastLessThan_empty
thf(fact_652_atLeastLessThan__empty,axiom,
    ! [B2: $o,A2: $o] :
      ( ( ord_less_eq_o @ B2 @ A2 )
     => ( ( set_or7139685690850216873Than_o @ A2 @ B2 )
        = bot_bot_set_o ) ) ).

% atLeastLessThan_empty
thf(fact_653_atLeastLessThan__empty,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( set_or3540276404033026485et_nat @ A2 @ B2 )
        = bot_bot_set_set_nat ) ) ).

% atLeastLessThan_empty
thf(fact_654_atLeastLessThan__empty,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( set_or4029947393144176647an_rat @ A2 @ B2 )
        = bot_bot_set_rat ) ) ).

% atLeastLessThan_empty
thf(fact_655_atLeastLessThan__empty,axiom,
    ! [B2: num,A2: num] :
      ( ( ord_less_eq_num @ B2 @ A2 )
     => ( ( set_or1222409239386451017an_num @ A2 @ B2 )
        = bot_bot_set_num ) ) ).

% atLeastLessThan_empty
thf(fact_656_atLeastLessThan__empty,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( set_or4665077453230672383an_nat @ A2 @ B2 )
        = bot_bot_set_nat ) ) ).

% atLeastLessThan_empty
thf(fact_657_atLeastLessThan__empty,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( set_or4662586982721622107an_int @ A2 @ B2 )
        = bot_bot_set_int ) ) ).

% atLeastLessThan_empty
thf(fact_658_atLeastLessThan__empty,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B2 @ A2 )
     => ( ( set_or8404916559141939852nteger @ A2 @ B2 )
        = bot_bo3990330152332043303nteger ) ) ).

% atLeastLessThan_empty
thf(fact_659_atLeastLessThan__empty__iff,axiom,
    ! [A2: $o,B2: $o] :
      ( ( ( set_or7139685690850216873Than_o @ A2 @ B2 )
        = bot_bot_set_o )
      = ( ~ ( ord_less_o @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_660_atLeastLessThan__empty__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ( set_or66887138388493659n_real @ A2 @ B2 )
        = bot_bot_set_real )
      = ( ~ ( ord_less_real @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_661_atLeastLessThan__empty__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( set_or4029947393144176647an_rat @ A2 @ B2 )
        = bot_bot_set_rat )
      = ( ~ ( ord_less_rat @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_662_atLeastLessThan__empty__iff,axiom,
    ! [A2: num,B2: num] :
      ( ( ( set_or1222409239386451017an_num @ A2 @ B2 )
        = bot_bot_set_num )
      = ( ~ ( ord_less_num @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_663_atLeastLessThan__empty__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A2 @ B2 )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_nat @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_664_atLeastLessThan__empty__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ( set_or4662586982721622107an_int @ A2 @ B2 )
        = bot_bot_set_int )
      = ( ~ ( ord_less_int @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_665_atLeastLessThan__empty__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( set_or8404916559141939852nteger @ A2 @ B2 )
        = bot_bo3990330152332043303nteger )
      = ( ~ ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff
thf(fact_666_atLeastLessThan__empty__iff2,axiom,
    ! [A2: $o,B2: $o] :
      ( ( bot_bot_set_o
        = ( set_or7139685690850216873Than_o @ A2 @ B2 ) )
      = ( ~ ( ord_less_o @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_667_atLeastLessThan__empty__iff2,axiom,
    ! [A2: real,B2: real] :
      ( ( bot_bot_set_real
        = ( set_or66887138388493659n_real @ A2 @ B2 ) )
      = ( ~ ( ord_less_real @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_668_atLeastLessThan__empty__iff2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( bot_bot_set_rat
        = ( set_or4029947393144176647an_rat @ A2 @ B2 ) )
      = ( ~ ( ord_less_rat @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_669_atLeastLessThan__empty__iff2,axiom,
    ! [A2: num,B2: num] :
      ( ( bot_bot_set_num
        = ( set_or1222409239386451017an_num @ A2 @ B2 ) )
      = ( ~ ( ord_less_num @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_670_atLeastLessThan__empty__iff2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( bot_bot_set_nat
        = ( set_or4665077453230672383an_nat @ A2 @ B2 ) )
      = ( ~ ( ord_less_nat @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_671_atLeastLessThan__empty__iff2,axiom,
    ! [A2: int,B2: int] :
      ( ( bot_bot_set_int
        = ( set_or4662586982721622107an_int @ A2 @ B2 ) )
      = ( ~ ( ord_less_int @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_672_atLeastLessThan__empty__iff2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( bot_bo3990330152332043303nteger
        = ( set_or8404916559141939852nteger @ A2 @ B2 ) )
      = ( ~ ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ) ).

% atLeastLessThan_empty_iff2
thf(fact_673_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ( K = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N3 ) )
          = zero_zero_nat ) )
      & ( ( K != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N3 ) )
          = ( divide_divide_nat @ M @ N3 ) ) ) ) ).

% nat_mult_div_cancel_disj
thf(fact_674_ivl__diff,axiom,
    ! [I: rat,N3: rat,M: rat] :
      ( ( ord_less_eq_rat @ I @ N3 )
     => ( ( minus_minus_set_rat @ ( set_or4029947393144176647an_rat @ I @ M ) @ ( set_or4029947393144176647an_rat @ I @ N3 ) )
        = ( set_or4029947393144176647an_rat @ N3 @ M ) ) ) ).

% ivl_diff
thf(fact_675_ivl__diff,axiom,
    ! [I: num,N3: num,M: num] :
      ( ( ord_less_eq_num @ I @ N3 )
     => ( ( minus_minus_set_num @ ( set_or1222409239386451017an_num @ I @ M ) @ ( set_or1222409239386451017an_num @ I @ N3 ) )
        = ( set_or1222409239386451017an_num @ N3 @ M ) ) ) ).

% ivl_diff
thf(fact_676_ivl__diff,axiom,
    ! [I: nat,N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ N3 )
     => ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I @ M ) @ ( set_or4665077453230672383an_nat @ I @ N3 ) )
        = ( set_or4665077453230672383an_nat @ N3 @ M ) ) ) ).

% ivl_diff
thf(fact_677_ivl__diff,axiom,
    ! [I: int,N3: int,M: int] :
      ( ( ord_less_eq_int @ I @ N3 )
     => ( ( minus_minus_set_int @ ( set_or4662586982721622107an_int @ I @ M ) @ ( set_or4662586982721622107an_int @ I @ N3 ) )
        = ( set_or4662586982721622107an_int @ N3 @ M ) ) ) ).

% ivl_diff
thf(fact_678_ivl__diff,axiom,
    ! [I: code_integer,N3: code_integer,M: code_integer] :
      ( ( ord_le3102999989581377725nteger @ I @ N3 )
     => ( ( minus_2355218937544613996nteger @ ( set_or8404916559141939852nteger @ I @ M ) @ ( set_or8404916559141939852nteger @ I @ N3 ) )
        = ( set_or8404916559141939852nteger @ N3 @ M ) ) ) ).

% ivl_diff
thf(fact_679_semiring__norm_I2_J,axiom,
    ( ( plus_plus_num @ one @ one )
    = ( bit0 @ one ) ) ).

% semiring_norm(2)
thf(fact_680_list__update__beyond,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ I )
     => ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_681_list__update__beyond,axiom,
    ! [Xs2: list_real,I: nat,X: real] :
      ( ( ord_less_eq_nat @ ( size_size_list_real @ Xs2 ) @ I )
     => ( ( list_update_real @ Xs2 @ I @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_682_list__update__beyond,axiom,
    ! [Xs2: list_o,I: nat,X: $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ I )
     => ( ( list_update_o @ Xs2 @ I @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_683_list__update__beyond,axiom,
    ! [Xs2: list_nat,I: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I )
     => ( ( list_update_nat @ Xs2 @ I @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_684_list__update__beyond,axiom,
    ! [Xs2: list_VEBT_VEBTi,I: nat,X: vEBT_VEBTi] :
      ( ( ord_less_eq_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) @ I )
     => ( ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_685_num__double,axiom,
    ! [N3: num] :
      ( ( times_times_num @ ( bit0 @ one ) @ N3 )
      = ( bit0 @ N3 ) ) ).

% num_double
thf(fact_686_power__mult__numeral,axiom,
    ! [A2: nat,M: num,N3: num] :
      ( ( power_power_nat @ ( power_power_nat @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N3 ) )
      = ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N3 ) ) ) ) ).

% power_mult_numeral
thf(fact_687_power__mult__numeral,axiom,
    ! [A2: real,M: num,N3: num] :
      ( ( power_power_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N3 ) )
      = ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N3 ) ) ) ) ).

% power_mult_numeral
thf(fact_688_power__mult__numeral,axiom,
    ! [A2: int,M: num,N3: num] :
      ( ( power_power_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N3 ) )
      = ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N3 ) ) ) ) ).

% power_mult_numeral
thf(fact_689_power__mult__numeral,axiom,
    ! [A2: complex,M: num,N3: num] :
      ( ( power_power_complex @ ( power_power_complex @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N3 ) )
      = ( power_power_complex @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N3 ) ) ) ) ).

% power_mult_numeral
thf(fact_690_power__mult__numeral,axiom,
    ! [A2: code_integer,M: num,N3: num] :
      ( ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N3 ) )
      = ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N3 ) ) ) ) ).

% power_mult_numeral
thf(fact_691_semiring__norm_I69_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).

% semiring_norm(69)
thf(fact_692_semiring__norm_I76_J,axiom,
    ! [N3: num] : ( ord_less_num @ one @ ( bit0 @ N3 ) ) ).

% semiring_norm(76)
thf(fact_693_one__le__mult__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N3 ) )
      = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
        & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N3 ) ) ) ).

% one_le_mult_iff
thf(fact_694_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I: nat] :
      ( ( P @ K )
     => ( ! [N: nat] :
            ( ( P @ ( suc @ N ) )
           => ( P @ N ) )
       => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_695_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_696_diffs0__imp__equal,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( minus_minus_nat @ M @ N3 )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N3 @ M )
          = zero_zero_nat )
       => ( M = N3 ) ) ) ).

% diffs0_imp_equal
thf(fact_697_diff__less__mono2,axiom,
    ! [M: nat,N3: nat,L2: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ( ord_less_nat @ M @ L2 )
       => ( ord_less_nat @ ( minus_minus_nat @ L2 @ N3 ) @ ( minus_minus_nat @ L2 @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_698_less__imp__diff__less,axiom,
    ! [J2: nat,K: nat,N3: nat] :
      ( ( ord_less_nat @ J2 @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J2 @ N3 ) @ K ) ) ).

% less_imp_diff_less
thf(fact_699_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N3 ) )
      = ( minus_minus_nat @ M @ N3 ) ) ).

% Nat.diff_cancel
thf(fact_700_diff__cancel2,axiom,
    ! [M: nat,K: nat,N3: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N3 @ K ) )
      = ( minus_minus_nat @ M @ N3 ) ) ).

% diff_cancel2
thf(fact_701_diff__add__inverse,axiom,
    ! [N3: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N3 @ M ) @ N3 )
      = M ) ).

% diff_add_inverse
thf(fact_702_diff__add__inverse2,axiom,
    ! [M: nat,N3: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N3 ) @ N3 )
      = M ) ).

% diff_add_inverse2
thf(fact_703_diff__le__mono2,axiom,
    ! [M: nat,N3: nat,L2: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N3 ) @ ( minus_minus_nat @ L2 @ M ) ) ) ).

% diff_le_mono2
thf(fact_704_le__diff__iff_H,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ C2 )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A2 ) @ ( minus_minus_nat @ C2 @ B2 ) )
          = ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).

% le_diff_iff'
thf(fact_705_diff__le__self,axiom,
    ! [M: nat,N3: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N3 ) @ M ) ).

% diff_le_self
thf(fact_706_diff__le__mono,axiom,
    ! [M: nat,N3: nat,L2: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N3 @ L2 ) ) ) ).

% diff_le_mono
thf(fact_707_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N3 )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N3 @ K ) )
          = ( minus_minus_nat @ M @ N3 ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_708_le__diff__iff,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N3 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N3 @ K ) )
          = ( ord_less_eq_nat @ M @ N3 ) ) ) ) ).

% le_diff_iff
thf(fact_709_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N3 )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N3 @ K ) )
          = ( M = N3 ) ) ) ) ).

% eq_diff_iff
thf(fact_710_diff__mult__distrib,axiom,
    ! [M: nat,N3: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M @ N3 ) @ K )
      = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N3 @ K ) ) ) ).

% diff_mult_distrib
thf(fact_711_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N3 ) )
      = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N3 ) ) ) ).

% diff_mult_distrib2
thf(fact_712_le__num__One__iff,axiom,
    ! [X: num] :
      ( ( ord_less_eq_num @ X @ one )
      = ( X = one ) ) ).

% le_num_One_iff
thf(fact_713_Suc__to__right,axiom,
    ! [N3: nat,M: nat] :
      ( ( ( suc @ N3 )
        = M )
     => ( N3
        = ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ).

% Suc_to_right
thf(fact_714_Suc__diff__Suc,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ N3 @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N3 ) ) )
        = ( minus_minus_nat @ M @ N3 ) ) ) ).

% Suc_diff_Suc
thf(fact_715_diff__less__Suc,axiom,
    ! [M: nat,N3: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N3 ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_716_diff__less,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N3 ) @ M ) ) ) ).

% diff_less
thf(fact_717_Suc__diff__le,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N3 )
        = ( suc @ ( minus_minus_nat @ M @ N3 ) ) ) ) ).

% Suc_diff_le
thf(fact_718_diff__add__0,axiom,
    ! [N3: nat,M: nat] :
      ( ( minus_minus_nat @ N3 @ ( plus_plus_nat @ N3 @ M ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_719_less__diff__conv,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ).

% less_diff_conv
thf(fact_720_add__diff__inverse__nat,axiom,
    ! [M: nat,N3: nat] :
      ( ~ ( ord_less_nat @ M @ N3 )
     => ( ( plus_plus_nat @ N3 @ ( minus_minus_nat @ M @ N3 ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_721_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C2 @ A2 )
       => ( ord_less_nat @ ( minus_minus_nat @ A2 @ C2 ) @ ( minus_minus_nat @ B2 @ C2 ) ) ) ) ).

% diff_less_mono
thf(fact_722_less__diff__iff,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N3 )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N3 @ K ) )
          = ( ord_less_nat @ M @ N3 ) ) ) ) ).

% less_diff_iff
thf(fact_723_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( ( minus_minus_nat @ J2 @ I )
          = K )
        = ( J2
          = ( plus_plus_nat @ K @ I ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_724_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_725_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K )
        = ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_726_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ) ).

% Nat.le_diff_conv2
thf(fact_727_le__diff__conv,axiom,
    ! [J2: nat,K: nat,I: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
      = ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I @ K ) ) ) ).

% le_diff_conv
thf(fact_728_subset__minus__empty,axiom,
    ! [A: set_real,B: set_real] :
      ( ( ord_less_eq_set_real @ A @ B )
     => ( ( minus_minus_set_real @ A @ B )
        = bot_bot_set_real ) ) ).

% subset_minus_empty
thf(fact_729_subset__minus__empty,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ord_less_eq_set_o @ A @ B )
     => ( ( minus_minus_set_o @ A @ B )
        = bot_bot_set_o ) ) ).

% subset_minus_empty
thf(fact_730_subset__minus__empty,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( minus_minus_set_int @ A @ B )
        = bot_bot_set_int ) ) ).

% subset_minus_empty
thf(fact_731_subset__minus__empty,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( minus_minus_set_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% subset_minus_empty
thf(fact_732_diff__Suc__less,axiom,
    ! [N3: nat,I: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ord_less_nat @ ( minus_minus_nat @ N3 @ ( suc @ I ) ) @ N3 ) ) ).

% diff_Suc_less
thf(fact_733_nat__diff__split,axiom,
    ! [P: nat > $o,A2: nat,B2: nat] :
      ( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
      = ( ( ( ord_less_nat @ A2 @ B2 )
         => ( P @ zero_zero_nat ) )
        & ! [D3: nat] :
            ( ( A2
              = ( plus_plus_nat @ B2 @ D3 ) )
           => ( P @ D3 ) ) ) ) ).

% nat_diff_split
thf(fact_734_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A2: nat,B2: nat] :
      ( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
      = ( ~ ( ( ( ord_less_nat @ A2 @ B2 )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D3: nat] :
                ( ( A2
                  = ( plus_plus_nat @ B2 @ D3 ) )
                & ~ ( P @ D3 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_735_less__diff__conv2,axiom,
    ! [K: nat,J2: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
        = ( ord_less_nat @ J2 @ ( plus_plus_nat @ I @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_736_div__mult2__numeral__eq,axiom,
    ! [A2: nat,K: num,L2: num] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L2 ) )
      = ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( times_times_num @ K @ L2 ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_737_div__mult2__numeral__eq,axiom,
    ! [A2: int,K: num,L2: num] :
      ( ( divide_divide_int @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L2 ) )
      = ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( times_times_num @ K @ L2 ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_738_nat__eq__add__iff1,axiom,
    ! [J2: nat,I: nat,U: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ J2 @ I )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N3 ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J2 ) @ U ) @ M )
          = N3 ) ) ) ).

% nat_eq_add_iff1
thf(fact_739_nat__eq__add__iff2,axiom,
    ! [I: nat,J2: nat,U: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N3 ) )
        = ( M
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I ) @ U ) @ N3 ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_740_nat__le__add__iff1,axiom,
    ! [J2: nat,I: nat,U: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ J2 @ I )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N3 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J2 ) @ U ) @ M ) @ N3 ) ) ) ).

% nat_le_add_iff1
thf(fact_741_nat__le__add__iff2,axiom,
    ! [I: nat,J2: nat,U: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N3 ) )
        = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I ) @ U ) @ N3 ) ) ) ) ).

% nat_le_add_iff2
thf(fact_742_nat__diff__add__eq1,axiom,
    ! [J2: nat,I: nat,U: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ J2 @ I )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N3 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J2 ) @ U ) @ M ) @ N3 ) ) ) ).

% nat_diff_add_eq1
thf(fact_743_nat__diff__add__eq2,axiom,
    ! [I: nat,J2: nat,U: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N3 ) )
        = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I ) @ U ) @ N3 ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_744_nz__le__conv__less,axiom,
    ! [K: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ K @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ K @ ( suc @ zero_zero_nat ) ) @ M ) ) ) ).

% nz_le_conv_less
thf(fact_745_nat__less__add__iff1,axiom,
    ! [J2: nat,I: nat,U: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ J2 @ I )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N3 ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J2 ) @ U ) @ M ) @ N3 ) ) ) ).

% nat_less_add_iff1
thf(fact_746_nat__less__add__iff2,axiom,
    ! [I: nat,J2: nat,U: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N3 ) )
        = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I ) @ U ) @ N3 ) ) ) ) ).

% nat_less_add_iff2
thf(fact_747_ord__eq__le__eq__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat,D2: set_nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C2 )
       => ( ( C2 = D2 )
         => ( ord_less_eq_set_nat @ A2 @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_748_ord__eq__le__eq__trans,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_rat @ B2 @ C2 )
       => ( ( C2 = D2 )
         => ( ord_less_eq_rat @ A2 @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_749_ord__eq__le__eq__trans,axiom,
    ! [A2: num,B2: num,C2: num,D2: num] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_num @ B2 @ C2 )
       => ( ( C2 = D2 )
         => ( ord_less_eq_num @ A2 @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_750_ord__eq__le__eq__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ( C2 = D2 )
         => ( ord_less_eq_nat @ A2 @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_751_ord__eq__le__eq__trans,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_int @ B2 @ C2 )
       => ( ( C2 = D2 )
         => ( ord_less_eq_int @ A2 @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_752_bex2I,axiom,
    ! [A2: code_integer,B2: code_integer,S: set_Pr4811707699266497531nteger,P: code_integer > code_integer > $o] :
      ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ A2 @ B2 ) @ S )
     => ( ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ A2 @ B2 ) @ S )
         => ( P @ A2 @ B2 ) )
       => ? [A3: code_integer,B3: code_integer] :
            ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ A3 @ B3 ) @ S )
            & ( P @ A3 @ B3 ) ) ) ) ).

% bex2I
thf(fact_753_bex2I,axiom,
    ! [A2: heap_e7401611519738050253t_unit,B2: set_nat,S: set_Pr3948176798113811640et_nat,P: heap_e7401611519738050253t_unit > set_nat > $o] :
      ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B2 ) @ S )
     => ( ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A2 @ B2 ) @ S )
         => ( P @ A2 @ B2 ) )
       => ? [A3: heap_e7401611519738050253t_unit,B3: set_nat] :
            ( ( member6260224972018164377et_nat @ ( produc7507926704131184380et_nat @ A3 @ B3 ) @ S )
            & ( P @ A3 @ B3 ) ) ) ) ).

% bex2I
thf(fact_754_bex2I,axiom,
    ! [A2: num,B2: num,S: set_Pr8218934625190621173um_num,P: num > num > $o] :
      ( ( member7279096912039735102um_num @ ( product_Pair_num_num @ A2 @ B2 ) @ S )
     => ( ( ( member7279096912039735102um_num @ ( product_Pair_num_num @ A2 @ B2 ) @ S )
         => ( P @ A2 @ B2 ) )
       => ? [A3: num,B3: num] :
            ( ( member7279096912039735102um_num @ ( product_Pair_num_num @ A3 @ B3 ) @ S )
            & ( P @ A3 @ B3 ) ) ) ) ).

% bex2I
thf(fact_755_bex2I,axiom,
    ! [A2: nat,B2: nat,S: set_Pr1261947904930325089at_nat,P: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ S )
     => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ S )
         => ( P @ A2 @ B2 ) )
       => ? [A3: nat,B3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ S )
            & ( P @ A3 @ B3 ) ) ) ) ).

% bex2I
thf(fact_756_bex2I,axiom,
    ! [A2: int,B2: int,S: set_Pr958786334691620121nt_int,P: int > int > $o] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B2 ) @ S )
     => ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A2 @ B2 ) @ S )
         => ( P @ A2 @ B2 ) )
       => ? [A3: int,B3: int] :
            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A3 @ B3 ) @ S )
            & ( P @ A3 @ B3 ) ) ) ) ).

% bex2I
thf(fact_757_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y ) )
     => ( X = Y ) ) ).

% Suc_inject
thf(fact_758_n__not__Suc__n,axiom,
    ! [N3: nat] :
      ( N3
     != ( suc @ N3 ) ) ).

% n_not_Suc_n
thf(fact_759_nat__neq__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( M != N3 )
      = ( ( ord_less_nat @ M @ N3 )
        | ( ord_less_nat @ N3 @ M ) ) ) ).

% nat_neq_iff
thf(fact_760_less__not__refl,axiom,
    ! [N3: nat] :
      ~ ( ord_less_nat @ N3 @ N3 ) ).

% less_not_refl
thf(fact_761_less__not__refl2,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ N3 @ M )
     => ( M != N3 ) ) ).

% less_not_refl2
thf(fact_762_less__not__refl3,axiom,
    ! [S2: nat,T: nat] :
      ( ( ord_less_nat @ S2 @ T )
     => ( S2 != T ) ) ).

% less_not_refl3
thf(fact_763_less__irrefl__nat,axiom,
    ! [N3: nat] :
      ~ ( ord_less_nat @ N3 @ N3 ) ).

% less_irrefl_nat
thf(fact_764_nat__less__induct,axiom,
    ! [P: nat > $o,N3: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N )
             => ( P @ M2 ) )
         => ( P @ N ) )
     => ( P @ N3 ) ) ).

% nat_less_induct
thf(fact_765_infinite__descent,axiom,
    ! [P: nat > $o,N3: nat] :
      ( ! [N: nat] :
          ( ~ ( P @ N )
         => ? [M2: nat] :
              ( ( ord_less_nat @ M2 @ N )
              & ~ ( P @ M2 ) ) )
     => ( P @ N3 ) ) ).

% infinite_descent
thf(fact_766_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_767_set__notEmptyE,axiom,
    ! [S: set_VEBT_VEBT] :
      ( ( S != bot_bo8194388402131092736T_VEBT )
     => ~ ! [X4: vEBT_VEBT] :
            ~ ( member_VEBT_VEBT @ X4 @ S ) ) ).

% set_notEmptyE
thf(fact_768_set__notEmptyE,axiom,
    ! [S: set_complex] :
      ( ( S != bot_bot_set_complex )
     => ~ ! [X4: complex] :
            ~ ( member_complex @ X4 @ S ) ) ).

% set_notEmptyE
thf(fact_769_set__notEmptyE,axiom,
    ! [S: set_real] :
      ( ( S != bot_bot_set_real )
     => ~ ! [X4: real] :
            ~ ( member_real @ X4 @ S ) ) ).

% set_notEmptyE
thf(fact_770_set__notEmptyE,axiom,
    ! [S: set_o] :
      ( ( S != bot_bot_set_o )
     => ~ ! [X4: $o] :
            ~ ( member_o @ X4 @ S ) ) ).

% set_notEmptyE
thf(fact_771_set__notEmptyE,axiom,
    ! [S: set_nat] :
      ( ( S != bot_bot_set_nat )
     => ~ ! [X4: nat] :
            ~ ( member_nat @ X4 @ S ) ) ).

% set_notEmptyE
thf(fact_772_set__notEmptyE,axiom,
    ! [S: set_int] :
      ( ( S != bot_bot_set_int )
     => ~ ! [X4: int] :
            ~ ( member_int @ X4 @ S ) ) ).

% set_notEmptyE
thf(fact_773_memb__imp__not__empty,axiom,
    ! [X: vEBT_VEBT,S: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ S )
     => ( S != bot_bo8194388402131092736T_VEBT ) ) ).

% memb_imp_not_empty
thf(fact_774_memb__imp__not__empty,axiom,
    ! [X: complex,S: set_complex] :
      ( ( member_complex @ X @ S )
     => ( S != bot_bot_set_complex ) ) ).

% memb_imp_not_empty
thf(fact_775_memb__imp__not__empty,axiom,
    ! [X: real,S: set_real] :
      ( ( member_real @ X @ S )
     => ( S != bot_bot_set_real ) ) ).

% memb_imp_not_empty
thf(fact_776_memb__imp__not__empty,axiom,
    ! [X: $o,S: set_o] :
      ( ( member_o @ X @ S )
     => ( S != bot_bot_set_o ) ) ).

% memb_imp_not_empty
thf(fact_777_memb__imp__not__empty,axiom,
    ! [X: nat,S: set_nat] :
      ( ( member_nat @ X @ S )
     => ( S != bot_bot_set_nat ) ) ).

% memb_imp_not_empty
thf(fact_778_memb__imp__not__empty,axiom,
    ! [X: int,S: set_int] :
      ( ( member_int @ X @ S )
     => ( S != bot_bot_set_int ) ) ).

% memb_imp_not_empty
thf(fact_779_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_780_size__neq__size__imp__neq,axiom,
    ! [X: list_real,Y: list_real] :
      ( ( ( size_size_list_real @ X )
       != ( size_size_list_real @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_781_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_782_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_783_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_784_size__neq__size__imp__neq,axiom,
    ! [X: list_VEBT_VEBTi,Y: list_VEBT_VEBTi] :
      ( ( ( size_s7982070591426661849_VEBTi @ X )
       != ( size_s7982070591426661849_VEBTi @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_785_bounded__Max__nat,axiom,
    ! [P: nat > $o,X: nat,M3: nat] :
      ( ( P @ X )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( ord_less_eq_nat @ X4 @ M3 ) )
       => ~ ! [M4: nat] :
              ( ( P @ M4 )
             => ~ ! [X5: nat] :
                    ( ( P @ X5 )
                   => ( ord_less_eq_nat @ X5 @ M4 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_786_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B2: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B2 ) )
       => ? [X4: nat] :
            ( ( P @ X4 )
            & ! [Y4: nat] :
                ( ( P @ Y4 )
               => ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_787_nat__le__linear,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
      | ( ord_less_eq_nat @ N3 @ M ) ) ).

% nat_le_linear
thf(fact_788_le__antisym,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( M = N3 ) ) ) ).

% le_antisym
thf(fact_789_eq__imp__le,axiom,
    ! [M: nat,N3: nat] :
      ( ( M = N3 )
     => ( ord_less_eq_nat @ M @ N3 ) ) ).

% eq_imp_le
thf(fact_790_le__trans,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( ord_less_eq_nat @ J2 @ K )
       => ( ord_less_eq_nat @ I @ K ) ) ) ).

% le_trans
thf(fact_791_le__refl,axiom,
    ! [N3: nat] : ( ord_less_eq_nat @ N3 @ N3 ) ).

% le_refl
thf(fact_792_Ex__list__of__length,axiom,
    ! [N3: nat] :
    ? [Xs3: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ Xs3 )
      = N3 ) ).

% Ex_list_of_length
thf(fact_793_Ex__list__of__length,axiom,
    ! [N3: nat] :
    ? [Xs3: list_real] :
      ( ( size_size_list_real @ Xs3 )
      = N3 ) ).

% Ex_list_of_length
thf(fact_794_Ex__list__of__length,axiom,
    ! [N3: nat] :
    ? [Xs3: list_o] :
      ( ( size_size_list_o @ Xs3 )
      = N3 ) ).

% Ex_list_of_length
thf(fact_795_Ex__list__of__length,axiom,
    ! [N3: nat] :
    ? [Xs3: list_nat] :
      ( ( size_size_list_nat @ Xs3 )
      = N3 ) ).

% Ex_list_of_length
thf(fact_796_Ex__list__of__length,axiom,
    ! [N3: nat] :
    ? [Xs3: list_VEBT_VEBTi] :
      ( ( size_s7982070591426661849_VEBTi @ Xs3 )
      = N3 ) ).

% Ex_list_of_length
thf(fact_797_neq__if__length__neq,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
       != ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_798_neq__if__length__neq,axiom,
    ! [Xs2: list_real,Ys: list_real] :
      ( ( ( size_size_list_real @ Xs2 )
       != ( size_size_list_real @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_799_neq__if__length__neq,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
       != ( size_size_list_o @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_800_neq__if__length__neq,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
       != ( size_size_list_nat @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_801_neq__if__length__neq,axiom,
    ! [Xs2: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
       != ( size_s7982070591426661849_VEBTi @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_802_list__update__swap,axiom,
    ! [I: nat,I4: nat,Xs2: list_VEBT_VEBTi,X: vEBT_VEBTi,X6: vEBT_VEBTi] :
      ( ( I != I4 )
     => ( ( list_u6098035379799741383_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X ) @ I4 @ X6 )
        = ( list_u6098035379799741383_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I4 @ X6 ) @ I @ X ) ) ) ).

% list_update_swap
thf(fact_803_list__update__swap,axiom,
    ! [I: nat,I4: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT,X6: vEBT_VEBT] :
      ( ( I != I4 )
     => ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ I4 @ X6 )
        = ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I4 @ X6 ) @ I @ X ) ) ) ).

% list_update_swap
thf(fact_804_power__diff,axiom,
    ! [A2: code_integer,N3: nat,M: nat] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( ( power_8256067586552552935nteger @ A2 @ ( minus_minus_nat @ M @ N3 ) )
          = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A2 @ M ) @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ) ) ).

% power_diff
thf(fact_805_power__diff,axiom,
    ! [A2: complex,N3: nat,M: nat] :
      ( ( A2 != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( ( power_power_complex @ A2 @ ( minus_minus_nat @ M @ N3 ) )
          = ( divide1717551699836669952omplex @ ( power_power_complex @ A2 @ M ) @ ( power_power_complex @ A2 @ N3 ) ) ) ) ) ).

% power_diff
thf(fact_806_power__diff,axiom,
    ! [A2: real,N3: nat,M: nat] :
      ( ( A2 != zero_zero_real )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( ( power_power_real @ A2 @ ( minus_minus_nat @ M @ N3 ) )
          = ( divide_divide_real @ ( power_power_real @ A2 @ M ) @ ( power_power_real @ A2 @ N3 ) ) ) ) ) ).

% power_diff
thf(fact_807_power__diff,axiom,
    ! [A2: rat,N3: nat,M: nat] :
      ( ( A2 != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( ( power_power_rat @ A2 @ ( minus_minus_nat @ M @ N3 ) )
          = ( divide_divide_rat @ ( power_power_rat @ A2 @ M ) @ ( power_power_rat @ A2 @ N3 ) ) ) ) ) ).

% power_diff
thf(fact_808_power__diff,axiom,
    ! [A2: nat,N3: nat,M: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( ( power_power_nat @ A2 @ ( minus_minus_nat @ M @ N3 ) )
          = ( divide_divide_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N3 ) ) ) ) ) ).

% power_diff
thf(fact_809_power__diff,axiom,
    ! [A2: int,N3: nat,M: nat] :
      ( ( A2 != zero_zero_int )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( ( power_power_int @ A2 @ ( minus_minus_nat @ M @ N3 ) )
          = ( divide_divide_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N3 ) ) ) ) ) ).

% power_diff
thf(fact_810_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M5: nat,N2: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M5 @ N2 )
            | ( N2 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M5 @ N2 ) @ N2 ) ) ) ) ) ).

% div_if
thf(fact_811_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N3 ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N3 ) ) ) ) ).

% diff_le_diff_pow
thf(fact_812_le__div__geq,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( ( divide_divide_nat @ M @ N3 )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N3 ) @ N3 ) ) ) ) ) ).

% le_div_geq
thf(fact_813_le__some__optE,axiom,
    ! [M: set_nat,X: option_set_nat] :
      ( ( ord_le2843612097646854710et_nat @ ( some_set_nat @ M ) @ X )
     => ~ ! [M6: set_nat] :
            ( ( X
              = ( some_set_nat @ M6 ) )
           => ~ ( ord_less_eq_set_nat @ M @ M6 ) ) ) ).

% le_some_optE
thf(fact_814_le__some__optE,axiom,
    ! [M: rat,X: option_rat] :
      ( ( ord_le2406147912482264968on_rat @ ( some_rat @ M ) @ X )
     => ~ ! [M6: rat] :
            ( ( X
              = ( some_rat @ M6 ) )
           => ~ ( ord_less_eq_rat @ M @ M6 ) ) ) ).

% le_some_optE
thf(fact_815_le__some__optE,axiom,
    ! [M: num,X: option_num] :
      ( ( ord_le6622620407824499402on_num @ ( some_num @ M ) @ X )
     => ~ ! [M6: num] :
            ( ( X
              = ( some_num @ M6 ) )
           => ~ ( ord_less_eq_num @ M @ M6 ) ) ) ).

% le_some_optE
thf(fact_816_le__some__optE,axiom,
    ! [M: nat,X: option_nat] :
      ( ( ord_le5914376470875661696on_nat @ ( some_nat @ M ) @ X )
     => ~ ! [M6: nat] :
            ( ( X
              = ( some_nat @ M6 ) )
           => ~ ( ord_less_eq_nat @ M @ M6 ) ) ) ).

% le_some_optE
thf(fact_817_le__some__optE,axiom,
    ! [M: int,X: option_int] :
      ( ( ord_le1736525451366464988on_int @ ( some_int @ M ) @ X )
     => ~ ! [M6: int] :
            ( ( X
              = ( some_int @ M6 ) )
           => ~ ( ord_less_eq_int @ M @ M6 ) ) ) ).

% le_some_optE
thf(fact_818_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_819_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_820_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_821_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_822_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_823_nat__induct,axiom,
    ! [P: nat > $o,N3: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N: nat] :
            ( ( P @ N )
           => ( P @ ( suc @ N ) ) )
       => ( P @ N3 ) ) ) ).

% nat_induct
thf(fact_824_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N3: nat] :
      ( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
     => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
       => ( ! [X4: nat,Y3: nat] :
              ( ( P @ X4 @ Y3 )
             => ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
         => ( P @ M @ N3 ) ) ) ) ).

% diff_induct
thf(fact_825_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N: nat] :
            ( ( P @ ( suc @ N ) )
           => ( P @ N ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_826_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_827_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_828_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_829_not0__implies__Suc,axiom,
    ! [N3: nat] :
      ( ( N3 != zero_zero_nat )
     => ? [M4: nat] :
          ( N3
          = ( suc @ M4 ) ) ) ).

% not0_implies_Suc
thf(fact_830_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ I @ K )
     => ( ( K
         != ( suc @ I ) )
       => ~ ! [J3: nat] :
              ( ( ord_less_nat @ I @ J3 )
             => ( K
               != ( suc @ J3 ) ) ) ) ) ).

% Nat.lessE
thf(fact_831_Suc__lessD,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N3 )
     => ( ord_less_nat @ M @ N3 ) ) ).

% Suc_lessD
thf(fact_832_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ K )
     => ~ ! [J3: nat] :
            ( ( ord_less_nat @ I @ J3 )
           => ( K
             != ( suc @ J3 ) ) ) ) ).

% Suc_lessE
thf(fact_833_Suc__lessI,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ( ( suc @ M )
         != N3 )
       => ( ord_less_nat @ ( suc @ M ) @ N3 ) ) ) ).

% Suc_lessI
thf(fact_834_less__SucE,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N3 ) )
     => ( ~ ( ord_less_nat @ M @ N3 )
       => ( M = N3 ) ) ) ).

% less_SucE
thf(fact_835_less__SucI,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ord_less_nat @ M @ ( suc @ N3 ) ) ) ).

% less_SucI
thf(fact_836_Ex__less__Suc,axiom,
    ! [N3: nat,P: nat > $o] :
      ( ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N3 ) )
            & ( P @ I2 ) ) )
      = ( ( P @ N3 )
        | ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ N3 )
            & ( P @ I2 ) ) ) ) ).

% Ex_less_Suc
thf(fact_837_less__Suc__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N3 ) )
      = ( ( ord_less_nat @ M @ N3 )
        | ( M = N3 ) ) ) ).

% less_Suc_eq
thf(fact_838_not__less__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( ~ ( ord_less_nat @ M @ N3 ) )
      = ( ord_less_nat @ N3 @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_839_Nat_OAll__less__Suc,axiom,
    ! [N3: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N3 ) )
           => ( P @ I2 ) ) )
      = ( ( P @ N3 )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ N3 )
           => ( P @ I2 ) ) ) ) ).

% Nat.All_less_Suc
thf(fact_840_Suc__less__eq2,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N3 ) @ M )
      = ( ? [M7: nat] :
            ( ( M
              = ( suc @ M7 ) )
            & ( ord_less_nat @ N3 @ M7 ) ) ) ) ).

% Suc_less_eq2
thf(fact_841_less__antisym,axiom,
    ! [N3: nat,M: nat] :
      ( ~ ( ord_less_nat @ N3 @ M )
     => ( ( ord_less_nat @ N3 @ ( suc @ M ) )
       => ( M = N3 ) ) ) ).

% less_antisym
thf(fact_842_Suc__less__SucD,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N3 ) )
     => ( ord_less_nat @ M @ N3 ) ) ).

% Suc_less_SucD
thf(fact_843_less__trans__Suc,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ( ord_less_nat @ J2 @ K )
       => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_844_less__Suc__induct,axiom,
    ! [I: nat,J2: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ! [I5: nat] : ( P @ I5 @ ( suc @ I5 ) )
       => ( ! [I5: nat,J3: nat,K2: nat] :
              ( ( ord_less_nat @ I5 @ J3 )
             => ( ( ord_less_nat @ J3 @ K2 )
               => ( ( P @ I5 @ J3 )
                 => ( ( P @ J3 @ K2 )
                   => ( P @ I5 @ K2 ) ) ) ) )
         => ( P @ I @ J2 ) ) ) ) ).

% less_Suc_induct
thf(fact_845_strict__inc__induct,axiom,
    ! [I: nat,J2: nat,P: nat > $o] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ! [I5: nat] :
            ( ( J2
              = ( suc @ I5 ) )
           => ( P @ I5 ) )
       => ( ! [I5: nat] :
              ( ( ord_less_nat @ I5 @ J2 )
             => ( ( P @ ( suc @ I5 ) )
               => ( P @ I5 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_846_not__less__less__Suc__eq,axiom,
    ! [N3: nat,M: nat] :
      ( ~ ( ord_less_nat @ N3 @ M )
     => ( ( ord_less_nat @ N3 @ ( suc @ M ) )
        = ( N3 = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_847_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_848_gr0I,axiom,
    ! [N3: nat] :
      ( ( N3 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).

% gr0I
thf(fact_849_not__gr0,axiom,
    ! [N3: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
      = ( N3 = zero_zero_nat ) ) ).

% not_gr0
thf(fact_850_not__less0,axiom,
    ! [N3: nat] :
      ~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).

% not_less0
thf(fact_851_less__zeroE,axiom,
    ! [N3: nat] :
      ~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).

% less_zeroE
thf(fact_852_gr__implies__not0,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( N3 != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_853_infinite__descent0,axiom,
    ! [P: nat > $o,N3: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( ~ ( P @ N )
             => ? [M2: nat] :
                  ( ( ord_less_nat @ M2 @ N )
                  & ~ ( P @ M2 ) ) ) )
       => ( P @ N3 ) ) ) ).

% infinite_descent0
thf(fact_854_nat__arith_Osuc1,axiom,
    ! [A: nat,K: nat,A2: nat] :
      ( ( A
        = ( plus_plus_nat @ K @ A2 ) )
     => ( ( suc @ A )
        = ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).

% nat_arith.suc1
thf(fact_855_add__Suc,axiom,
    ! [M: nat,N3: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N3 )
      = ( suc @ ( plus_plus_nat @ M @ N3 ) ) ) ).

% add_Suc
thf(fact_856_add__Suc__shift,axiom,
    ! [M: nat,N3: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N3 )
      = ( plus_plus_nat @ M @ ( suc @ N3 ) ) ) ).

% add_Suc_shift
thf(fact_857_Suc__leD,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N3 )
     => ( ord_less_eq_nat @ M @ N3 ) ) ).

% Suc_leD
thf(fact_858_le__SucE,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N3 ) )
     => ( ~ ( ord_less_eq_nat @ M @ N3 )
       => ( M
          = ( suc @ N3 ) ) ) ) ).

% le_SucE
thf(fact_859_le__SucI,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ord_less_eq_nat @ M @ ( suc @ N3 ) ) ) ).

% le_SucI
thf(fact_860_Suc__le__D,axiom,
    ! [N3: nat,M8: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N3 ) @ M8 )
     => ? [M4: nat] :
          ( M8
          = ( suc @ M4 ) ) ) ).

% Suc_le_D
thf(fact_861_le__Suc__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N3 ) )
      = ( ( ord_less_eq_nat @ M @ N3 )
        | ( M
          = ( suc @ N3 ) ) ) ) ).

% le_Suc_eq
thf(fact_862_Suc__n__not__le__n,axiom,
    ! [N3: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N3 ) @ N3 ) ).

% Suc_n_not_le_n
thf(fact_863_not__less__eq__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N3 ) )
      = ( ord_less_eq_nat @ ( suc @ N3 ) @ M ) ) ).

% not_less_eq_eq
thf(fact_864_full__nat__induct,axiom,
    ! [P: nat > $o,N3: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
             => ( P @ M2 ) )
         => ( P @ N ) )
     => ( P @ N3 ) ) ).

% full_nat_induct
thf(fact_865_nat__induct__at__least,axiom,
    ! [M: nat,N3: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( P @ M )
       => ( ! [N: nat] :
              ( ( ord_less_eq_nat @ M @ N )
             => ( ( P @ N )
               => ( P @ ( suc @ N ) ) ) )
         => ( P @ N3 ) ) ) ) ).

% nat_induct_at_least
thf(fact_866_transitive__stepwise__le,axiom,
    ! [M: nat,N3: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ! [X4: nat] : ( R @ X4 @ X4 )
       => ( ! [X4: nat,Y3: nat,Z2: nat] :
              ( ( R @ X4 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X4 @ Z2 ) ) )
         => ( ! [N: nat] : ( R @ N @ ( suc @ N ) )
           => ( R @ M @ N3 ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_867_plus__nat_Oadd__0,axiom,
    ! [N3: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N3 )
      = N3 ) ).

% plus_nat.add_0
thf(fact_868_add__eq__self__zero,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( plus_plus_nat @ M @ N3 )
        = M )
     => ( N3 = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_869_less__eq__nat_Osimps_I1_J,axiom,
    ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).

% less_eq_nat.simps(1)
thf(fact_870_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
      = ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_871_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_872_le__0__eq,axiom,
    ! [N3: nat] :
      ( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
      = ( N3 = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_873_add__lessD1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ K )
     => ( ord_less_nat @ I @ K ) ) ).

% add_lessD1
thf(fact_874_add__less__mono,axiom,
    ! [I: nat,J2: nat,K: nat,L2: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ( ord_less_nat @ K @ L2 )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ) ).

% add_less_mono
thf(fact_875_not__add__less1,axiom,
    ! [I: nat,J2: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ I ) ).

% not_add_less1
thf(fact_876_not__add__less2,axiom,
    ! [J2: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I ) @ I ) ).

% not_add_less2
thf(fact_877_add__less__mono1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).

% add_less_mono1
thf(fact_878_trans__less__add1,axiom,
    ! [I: nat,J2: nat,M: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J2 @ M ) ) ) ).

% trans_less_add1
thf(fact_879_trans__less__add2,axiom,
    ! [I: nat,J2: nat,M: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J2 ) ) ) ).

% trans_less_add2
thf(fact_880_less__add__eq__less,axiom,
    ! [K: nat,L2: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ K @ L2 )
     => ( ( ( plus_plus_nat @ M @ L2 )
          = ( plus_plus_nat @ K @ N3 ) )
       => ( ord_less_nat @ M @ N3 ) ) ) ).

% less_add_eq_less
thf(fact_881_exists__leI,axiom,
    ! [N3: nat,P: nat > $o] :
      ( ( ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ N3 )
           => ~ ( P @ N4 ) )
       => ( P @ N3 ) )
     => ? [N5: nat] :
          ( ( ord_less_eq_nat @ N5 @ N3 )
          & ( P @ N5 ) ) ) ).

% exists_leI
thf(fact_882_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M5: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M5 @ N2 )
          & ( M5 != N2 ) ) ) ) ).

% nat_less_le
thf(fact_883_less__imp__le__nat,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ord_less_eq_nat @ M @ N3 ) ) ).

% less_imp_le_nat
thf(fact_884_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N2: nat] :
          ( ( ord_less_nat @ M5 @ N2 )
          | ( M5 = N2 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_885_less__or__eq__imp__le,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( ord_less_nat @ M @ N3 )
        | ( M = N3 ) )
     => ( ord_less_eq_nat @ M @ N3 ) ) ).

% less_or_eq_imp_le
thf(fact_886_le__neq__implies__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( M != N3 )
       => ( ord_less_nat @ M @ N3 ) ) ) ).

% le_neq_implies_less
thf(fact_887_less__mono__imp__le__mono,axiom,
    ! [F2: nat > nat,I: nat,J2: nat] :
      ( ! [I5: nat,J3: nat] :
          ( ( ord_less_nat @ I5 @ J3 )
         => ( ord_less_nat @ ( F2 @ I5 ) @ ( F2 @ J3 ) ) )
     => ( ( ord_less_eq_nat @ I @ J2 )
       => ( ord_less_eq_nat @ ( F2 @ I ) @ ( F2 @ J2 ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_888_length__induct,axiom,
    ! [P: list_VEBT_VEBT > $o,Xs2: list_VEBT_VEBT] :
      ( ! [Xs3: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys2 ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_889_length__induct,axiom,
    ! [P: list_real > $o,Xs2: list_real] :
      ( ! [Xs3: list_real] :
          ( ! [Ys2: list_real] :
              ( ( ord_less_nat @ ( size_size_list_real @ Ys2 ) @ ( size_size_list_real @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_890_length__induct,axiom,
    ! [P: list_o > $o,Xs2: list_o] :
      ( ! [Xs3: list_o] :
          ( ! [Ys2: list_o] :
              ( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_891_length__induct,axiom,
    ! [P: list_nat > $o,Xs2: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys2: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_892_length__induct,axiom,
    ! [P: list_VEBT_VEBTi > $o,Xs2: list_VEBT_VEBTi] :
      ( ! [Xs3: list_VEBT_VEBTi] :
          ( ! [Ys2: list_VEBT_VEBTi] :
              ( ( ord_less_nat @ ( size_s7982070591426661849_VEBTi @ Ys2 ) @ ( size_s7982070591426661849_VEBTi @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_893_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ( times_times_nat @ ( suc @ K ) @ M )
        = ( times_times_nat @ ( suc @ K ) @ N3 ) )
      = ( M = N3 ) ) ).

% Suc_mult_cancel1
thf(fact_894_add__leE,axiom,
    ! [M: nat,K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N3 )
     => ~ ( ( ord_less_eq_nat @ M @ N3 )
         => ~ ( ord_less_eq_nat @ K @ N3 ) ) ) ).

% add_leE
thf(fact_895_le__add1,axiom,
    ! [N3: nat,M: nat] : ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ N3 @ M ) ) ).

% le_add1
thf(fact_896_le__add2,axiom,
    ! [N3: nat,M: nat] : ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ M @ N3 ) ) ).

% le_add2
thf(fact_897_add__leD1,axiom,
    ! [M: nat,K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N3 )
     => ( ord_less_eq_nat @ M @ N3 ) ) ).

% add_leD1
thf(fact_898_add__leD2,axiom,
    ! [M: nat,K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N3 )
     => ( ord_less_eq_nat @ K @ N3 ) ) ).

% add_leD2
thf(fact_899_le__Suc__ex,axiom,
    ! [K: nat,L2: nat] :
      ( ( ord_less_eq_nat @ K @ L2 )
     => ? [N: nat] :
          ( L2
          = ( plus_plus_nat @ K @ N ) ) ) ).

% le_Suc_ex
thf(fact_900_add__le__mono,axiom,
    ! [I: nat,J2: nat,K: nat,L2: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( ord_less_eq_nat @ K @ L2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ) ).

% add_le_mono
thf(fact_901_add__le__mono1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).

% add_le_mono1
thf(fact_902_trans__le__add1,axiom,
    ! [I: nat,J2: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J2 @ M ) ) ) ).

% trans_le_add1
thf(fact_903_trans__le__add2,axiom,
    ! [I: nat,J2: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J2 ) ) ) ).

% trans_le_add2
thf(fact_904_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N2: nat] :
        ? [K3: nat] :
          ( N2
          = ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_905_mult__0,axiom,
    ! [N3: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N3 )
      = zero_zero_nat ) ).

% mult_0
thf(fact_906_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N3 ) )
      = ( ( K = zero_zero_nat )
        | ( M = N3 ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_907_atLeastLessThan__subset__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or4029947393144176647an_rat @ A2 @ B2 ) @ ( set_or4029947393144176647an_rat @ C2 @ D2 ) )
     => ( ( ord_less_eq_rat @ B2 @ A2 )
        | ( ( ord_less_eq_rat @ C2 @ A2 )
          & ( ord_less_eq_rat @ B2 @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_908_atLeastLessThan__subset__iff,axiom,
    ! [A2: num,B2: num,C2: num,D2: num] :
      ( ( ord_less_eq_set_num @ ( set_or1222409239386451017an_num @ A2 @ B2 ) @ ( set_or1222409239386451017an_num @ C2 @ D2 ) )
     => ( ( ord_less_eq_num @ B2 @ A2 )
        | ( ( ord_less_eq_num @ C2 @ A2 )
          & ( ord_less_eq_num @ B2 @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_909_atLeastLessThan__subset__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A2 @ B2 ) @ ( set_or4665077453230672383an_nat @ C2 @ D2 ) )
     => ( ( ord_less_eq_nat @ B2 @ A2 )
        | ( ( ord_less_eq_nat @ C2 @ A2 )
          & ( ord_less_eq_nat @ B2 @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_910_atLeastLessThan__subset__iff,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A2 @ B2 ) @ ( set_or4662586982721622107an_int @ C2 @ D2 ) )
     => ( ( ord_less_eq_int @ B2 @ A2 )
        | ( ( ord_less_eq_int @ C2 @ A2 )
          & ( ord_less_eq_int @ B2 @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_911_atLeastLessThan__subset__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or8404916559141939852nteger @ A2 @ B2 ) @ ( set_or8404916559141939852nteger @ C2 @ D2 ) )
     => ( ( ord_le3102999989581377725nteger @ B2 @ A2 )
        | ( ( ord_le3102999989581377725nteger @ C2 @ A2 )
          & ( ord_le3102999989581377725nteger @ B2 @ D2 ) ) ) ) ).

% atLeastLessThan_subset_iff
thf(fact_912_atLeastLessThan__inj_I2_J,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( ( set_or66887138388493659n_real @ A2 @ B2 )
        = ( set_or66887138388493659n_real @ C2 @ D2 ) )
     => ( ( ord_less_real @ A2 @ B2 )
       => ( ( ord_less_real @ C2 @ D2 )
         => ( B2 = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_913_atLeastLessThan__inj_I2_J,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ( set_or4029947393144176647an_rat @ A2 @ B2 )
        = ( set_or4029947393144176647an_rat @ C2 @ D2 ) )
     => ( ( ord_less_rat @ A2 @ B2 )
       => ( ( ord_less_rat @ C2 @ D2 )
         => ( B2 = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_914_atLeastLessThan__inj_I2_J,axiom,
    ! [A2: num,B2: num,C2: num,D2: num] :
      ( ( ( set_or1222409239386451017an_num @ A2 @ B2 )
        = ( set_or1222409239386451017an_num @ C2 @ D2 ) )
     => ( ( ord_less_num @ A2 @ B2 )
       => ( ( ord_less_num @ C2 @ D2 )
         => ( B2 = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_915_atLeastLessThan__inj_I2_J,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A2 @ B2 )
        = ( set_or4665077453230672383an_nat @ C2 @ D2 ) )
     => ( ( ord_less_nat @ A2 @ B2 )
       => ( ( ord_less_nat @ C2 @ D2 )
         => ( B2 = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_916_atLeastLessThan__inj_I2_J,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ( set_or4662586982721622107an_int @ A2 @ B2 )
        = ( set_or4662586982721622107an_int @ C2 @ D2 ) )
     => ( ( ord_less_int @ A2 @ B2 )
       => ( ( ord_less_int @ C2 @ D2 )
         => ( B2 = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_917_atLeastLessThan__inj_I2_J,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ( set_or8404916559141939852nteger @ A2 @ B2 )
        = ( set_or8404916559141939852nteger @ C2 @ D2 ) )
     => ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
       => ( ( ord_le6747313008572928689nteger @ C2 @ D2 )
         => ( B2 = D2 ) ) ) ) ).

% atLeastLessThan_inj(2)
thf(fact_918_atLeastLessThan__inj_I1_J,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( ( set_or66887138388493659n_real @ A2 @ B2 )
        = ( set_or66887138388493659n_real @ C2 @ D2 ) )
     => ( ( ord_less_real @ A2 @ B2 )
       => ( ( ord_less_real @ C2 @ D2 )
         => ( A2 = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_919_atLeastLessThan__inj_I1_J,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ( set_or4029947393144176647an_rat @ A2 @ B2 )
        = ( set_or4029947393144176647an_rat @ C2 @ D2 ) )
     => ( ( ord_less_rat @ A2 @ B2 )
       => ( ( ord_less_rat @ C2 @ D2 )
         => ( A2 = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_920_atLeastLessThan__inj_I1_J,axiom,
    ! [A2: num,B2: num,C2: num,D2: num] :
      ( ( ( set_or1222409239386451017an_num @ A2 @ B2 )
        = ( set_or1222409239386451017an_num @ C2 @ D2 ) )
     => ( ( ord_less_num @ A2 @ B2 )
       => ( ( ord_less_num @ C2 @ D2 )
         => ( A2 = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_921_atLeastLessThan__inj_I1_J,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ( set_or4665077453230672383an_nat @ A2 @ B2 )
        = ( set_or4665077453230672383an_nat @ C2 @ D2 ) )
     => ( ( ord_less_nat @ A2 @ B2 )
       => ( ( ord_less_nat @ C2 @ D2 )
         => ( A2 = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_922_atLeastLessThan__inj_I1_J,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ( set_or4662586982721622107an_int @ A2 @ B2 )
        = ( set_or4662586982721622107an_int @ C2 @ D2 ) )
     => ( ( ord_less_int @ A2 @ B2 )
       => ( ( ord_less_int @ C2 @ D2 )
         => ( A2 = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_923_atLeastLessThan__inj_I1_J,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ( set_or8404916559141939852nteger @ A2 @ B2 )
        = ( set_or8404916559141939852nteger @ C2 @ D2 ) )
     => ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
       => ( ( ord_le6747313008572928689nteger @ C2 @ D2 )
         => ( A2 = C2 ) ) ) ) ).

% atLeastLessThan_inj(1)
thf(fact_924_atLeastLessThan__eq__iff,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_real @ C2 @ D2 )
       => ( ( ( set_or66887138388493659n_real @ A2 @ B2 )
            = ( set_or66887138388493659n_real @ C2 @ D2 ) )
          = ( ( A2 = C2 )
            & ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_925_atLeastLessThan__eq__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ C2 @ D2 )
       => ( ( ( set_or4029947393144176647an_rat @ A2 @ B2 )
            = ( set_or4029947393144176647an_rat @ C2 @ D2 ) )
          = ( ( A2 = C2 )
            & ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_926_atLeastLessThan__eq__iff,axiom,
    ! [A2: num,B2: num,C2: num,D2: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ( ( ord_less_num @ C2 @ D2 )
       => ( ( ( set_or1222409239386451017an_num @ A2 @ B2 )
            = ( set_or1222409239386451017an_num @ C2 @ D2 ) )
          = ( ( A2 = C2 )
            & ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_927_atLeastLessThan__eq__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ C2 @ D2 )
       => ( ( ( set_or4665077453230672383an_nat @ A2 @ B2 )
            = ( set_or4665077453230672383an_nat @ C2 @ D2 ) )
          = ( ( A2 = C2 )
            & ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_928_atLeastLessThan__eq__iff,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ C2 @ D2 )
       => ( ( ( set_or4662586982721622107an_int @ A2 @ B2 )
            = ( set_or4662586982721622107an_int @ C2 @ D2 ) )
          = ( ( A2 = C2 )
            & ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_929_atLeastLessThan__eq__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ( ord_le6747313008572928689nteger @ C2 @ D2 )
       => ( ( ( set_or8404916559141939852nteger @ A2 @ B2 )
            = ( set_or8404916559141939852nteger @ C2 @ D2 ) )
          = ( ( A2 = C2 )
            & ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_eq_iff
thf(fact_930_add__mult__distrib,axiom,
    ! [M: nat,N3: nat,K: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M @ N3 ) @ K )
      = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N3 @ K ) ) ) ).

% add_mult_distrib
thf(fact_931_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N3 ) )
      = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N3 ) ) ) ).

% add_mult_distrib2
thf(fact_932_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J2: nat,K: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ K ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J2 ) @ U ) @ K ) ) ).

% left_add_mult_distrib
thf(fact_933_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_934_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_935_mult__le__mono,axiom,
    ! [I: nat,J2: nat,K: nat,L2: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( ord_less_eq_nat @ K @ L2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J2 @ L2 ) ) ) ) ).

% mult_le_mono
thf(fact_936_mult__le__mono1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ).

% mult_le_mono1
thf(fact_937_mult__le__mono2,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J2 ) ) ) ).

% mult_le_mono2
thf(fact_938_list__assn__mono,axiom,
    ! [P: vEBT_VEBT > vEBT_VEBTi > assn,P2: vEBT_VEBT > vEBT_VEBTi > assn,L2: list_VEBT_VEBT,L3: list_VEBT_VEBTi] :
      ( ! [X4: vEBT_VEBT,X7: vEBT_VEBTi] : ( entails @ ( P @ X4 @ X7 ) @ ( P2 @ X4 @ X7 ) )
     => ( entails @ ( vEBT_L6296928887356842470_VEBTi @ P @ L2 @ L3 ) @ ( vEBT_L6296928887356842470_VEBTi @ P2 @ L2 @ L3 ) ) ) ).

% list_assn_mono
thf(fact_939_lift__Suc__mono__less,axiom,
    ! [F2: nat > real,N3: nat,N6: nat] :
      ( ! [N: nat] : ( ord_less_real @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ N3 @ N6 )
       => ( ord_less_real @ ( F2 @ N3 ) @ ( F2 @ N6 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_940_lift__Suc__mono__less,axiom,
    ! [F2: nat > rat,N3: nat,N6: nat] :
      ( ! [N: nat] : ( ord_less_rat @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ N3 @ N6 )
       => ( ord_less_rat @ ( F2 @ N3 ) @ ( F2 @ N6 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_941_lift__Suc__mono__less,axiom,
    ! [F2: nat > num,N3: nat,N6: nat] :
      ( ! [N: nat] : ( ord_less_num @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ N3 @ N6 )
       => ( ord_less_num @ ( F2 @ N3 ) @ ( F2 @ N6 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_942_lift__Suc__mono__less,axiom,
    ! [F2: nat > nat,N3: nat,N6: nat] :
      ( ! [N: nat] : ( ord_less_nat @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ N3 @ N6 )
       => ( ord_less_nat @ ( F2 @ N3 ) @ ( F2 @ N6 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_943_lift__Suc__mono__less,axiom,
    ! [F2: nat > int,N3: nat,N6: nat] :
      ( ! [N: nat] : ( ord_less_int @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ N3 @ N6 )
       => ( ord_less_int @ ( F2 @ N3 ) @ ( F2 @ N6 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_944_lift__Suc__mono__less__iff,axiom,
    ! [F2: nat > real,N3: nat,M: nat] :
      ( ! [N: nat] : ( ord_less_real @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
     => ( ( ord_less_real @ ( F2 @ N3 ) @ ( F2 @ M ) )
        = ( ord_less_nat @ N3 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_945_lift__Suc__mono__less__iff,axiom,
    ! [F2: nat > rat,N3: nat,M: nat] :
      ( ! [N: nat] : ( ord_less_rat @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
     => ( ( ord_less_rat @ ( F2 @ N3 ) @ ( F2 @ M ) )
        = ( ord_less_nat @ N3 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_946_lift__Suc__mono__less__iff,axiom,
    ! [F2: nat > num,N3: nat,M: nat] :
      ( ! [N: nat] : ( ord_less_num @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
     => ( ( ord_less_num @ ( F2 @ N3 ) @ ( F2 @ M ) )
        = ( ord_less_nat @ N3 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_947_lift__Suc__mono__less__iff,axiom,
    ! [F2: nat > nat,N3: nat,M: nat] :
      ( ! [N: nat] : ( ord_less_nat @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
     => ( ( ord_less_nat @ ( F2 @ N3 ) @ ( F2 @ M ) )
        = ( ord_less_nat @ N3 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_948_lift__Suc__mono__less__iff,axiom,
    ! [F2: nat > int,N3: nat,M: nat] :
      ( ! [N: nat] : ( ord_less_int @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
     => ( ( ord_less_int @ ( F2 @ N3 ) @ ( F2 @ M ) )
        = ( ord_less_nat @ N3 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_949_lift__Suc__antimono__le,axiom,
    ! [F2: nat > set_nat,N3: nat,N6: nat] :
      ( ! [N: nat] : ( ord_less_eq_set_nat @ ( F2 @ ( suc @ N ) ) @ ( F2 @ N ) )
     => ( ( ord_less_eq_nat @ N3 @ N6 )
       => ( ord_less_eq_set_nat @ ( F2 @ N6 ) @ ( F2 @ N3 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_950_lift__Suc__antimono__le,axiom,
    ! [F2: nat > rat,N3: nat,N6: nat] :
      ( ! [N: nat] : ( ord_less_eq_rat @ ( F2 @ ( suc @ N ) ) @ ( F2 @ N ) )
     => ( ( ord_less_eq_nat @ N3 @ N6 )
       => ( ord_less_eq_rat @ ( F2 @ N6 ) @ ( F2 @ N3 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_951_lift__Suc__antimono__le,axiom,
    ! [F2: nat > num,N3: nat,N6: nat] :
      ( ! [N: nat] : ( ord_less_eq_num @ ( F2 @ ( suc @ N ) ) @ ( F2 @ N ) )
     => ( ( ord_less_eq_nat @ N3 @ N6 )
       => ( ord_less_eq_num @ ( F2 @ N6 ) @ ( F2 @ N3 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_952_lift__Suc__antimono__le,axiom,
    ! [F2: nat > nat,N3: nat,N6: nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ ( F2 @ ( suc @ N ) ) @ ( F2 @ N ) )
     => ( ( ord_less_eq_nat @ N3 @ N6 )
       => ( ord_less_eq_nat @ ( F2 @ N6 ) @ ( F2 @ N3 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_953_lift__Suc__antimono__le,axiom,
    ! [F2: nat > int,N3: nat,N6: nat] :
      ( ! [N: nat] : ( ord_less_eq_int @ ( F2 @ ( suc @ N ) ) @ ( F2 @ N ) )
     => ( ( ord_less_eq_nat @ N3 @ N6 )
       => ( ord_less_eq_int @ ( F2 @ N6 ) @ ( F2 @ N3 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_954_lift__Suc__mono__le,axiom,
    ! [F2: nat > set_nat,N3: nat,N6: nat] :
      ( ! [N: nat] : ( ord_less_eq_set_nat @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ N3 @ N6 )
       => ( ord_less_eq_set_nat @ ( F2 @ N3 ) @ ( F2 @ N6 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_955_lift__Suc__mono__le,axiom,
    ! [F2: nat > rat,N3: nat,N6: nat] :
      ( ! [N: nat] : ( ord_less_eq_rat @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ N3 @ N6 )
       => ( ord_less_eq_rat @ ( F2 @ N3 ) @ ( F2 @ N6 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_956_lift__Suc__mono__le,axiom,
    ! [F2: nat > num,N3: nat,N6: nat] :
      ( ! [N: nat] : ( ord_less_eq_num @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ N3 @ N6 )
       => ( ord_less_eq_num @ ( F2 @ N3 ) @ ( F2 @ N6 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_957_lift__Suc__mono__le,axiom,
    ! [F2: nat > nat,N3: nat,N6: nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ N3 @ N6 )
       => ( ord_less_eq_nat @ ( F2 @ N3 ) @ ( F2 @ N6 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_958_lift__Suc__mono__le,axiom,
    ! [F2: nat > int,N3: nat,N6: nat] :
      ( ! [N: nat] : ( ord_less_eq_int @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ N3 @ N6 )
       => ( ord_less_eq_int @ ( F2 @ N3 ) @ ( F2 @ N6 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_959_Ex__less__Suc2,axiom,
    ! [N3: nat,P: nat > $o] :
      ( ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N3 ) )
            & ( P @ I2 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ N3 )
            & ( P @ ( suc @ I2 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_960_gr0__conv__Suc,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
      = ( ? [M5: nat] :
            ( N3
            = ( suc @ M5 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_961_All__less__Suc2,axiom,
    ! [N3: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N3 ) )
           => ( P @ I2 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ N3 )
           => ( P @ ( suc @ I2 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_962_gr0__implies__Suc,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ? [M4: nat] :
          ( N3
          = ( suc @ M4 ) ) ) ).

% gr0_implies_Suc
thf(fact_963_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N3 ) )
      = ( ( M = zero_zero_nat )
        | ? [J: nat] :
            ( ( M
              = ( suc @ J ) )
            & ( ord_less_nat @ J @ N3 ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_964_add__is__1,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( plus_plus_nat @ M @ N3 )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N3 = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N3
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_965_one__is__add,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N3 ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N3 = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N3
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_966_nat__compl__induct,axiom,
    ! [P: nat > $o,N3: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N )
               => ( P @ Nn ) )
           => ( P @ ( suc @ N ) ) )
       => ( P @ N3 ) ) ) ).

% nat_compl_induct
thf(fact_967_nat__compl__induct_H,axiom,
    ! [P: nat > $o,N3: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N )
               => ( P @ Nn ) )
           => ( P @ ( suc @ N ) ) )
       => ( P @ N3 ) ) ) ).

% nat_compl_induct'
thf(fact_968_less__natE,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ~ ! [Q5: nat] :
            ( N3
           != ( suc @ ( plus_plus_nat @ M @ Q5 ) ) ) ) ).

% less_natE
thf(fact_969_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_970_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_971_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M5: nat,N2: nat] :
        ? [K3: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_972_less__imp__Suc__add,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ? [K2: nat] :
          ( N3
          = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_973_nat__in__between__eq_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( ord_less_eq_nat @ A2 @ B2 )
        & ( ord_less_nat @ B2 @ ( suc @ A2 ) ) )
      = ( B2 = A2 ) ) ).

% nat_in_between_eq(2)
thf(fact_974_nat__in__between__eq_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( ord_less_nat @ A2 @ B2 )
        & ( ord_less_eq_nat @ B2 @ ( suc @ A2 ) ) )
      = ( B2
        = ( suc @ A2 ) ) ) ).

% nat_in_between_eq(1)
thf(fact_975_Suc__leI,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N3 ) ) ).

% Suc_leI
thf(fact_976_Suc__le__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N3 )
      = ( ord_less_nat @ M @ N3 ) ) ).

% Suc_le_eq
thf(fact_977_dec__induct,axiom,
    ! [I: nat,J2: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( P @ I )
       => ( ! [N: nat] :
              ( ( ord_less_eq_nat @ I @ N )
             => ( ( ord_less_nat @ N @ J2 )
               => ( ( P @ N )
                 => ( P @ ( suc @ N ) ) ) ) )
         => ( P @ J2 ) ) ) ) ).

% dec_induct
thf(fact_978_inc__induct,axiom,
    ! [I: nat,J2: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( P @ J2 )
       => ( ! [N: nat] :
              ( ( ord_less_eq_nat @ I @ N )
             => ( ( ord_less_nat @ N @ J2 )
               => ( ( P @ ( suc @ N ) )
                 => ( P @ N ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_979_Suc__le__lessD,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N3 )
     => ( ord_less_nat @ M @ N3 ) ) ).

% Suc_le_lessD
thf(fact_980_le__less__Suc__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( ord_less_nat @ N3 @ ( suc @ M ) )
        = ( N3 = M ) ) ) ).

% le_less_Suc_eq
thf(fact_981_less__Suc__eq__le,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N3 ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% less_Suc_eq_le
thf(fact_982_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_983_le__imp__less__Suc,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ord_less_nat @ M @ ( suc @ N3 ) ) ) ).

% le_imp_less_Suc
thf(fact_984_less__imp__add__positive,axiom,
    ! [I: nat,J2: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I @ K2 )
            = J2 ) ) ) ).

% less_imp_add_positive
thf(fact_985_ex__least__nat__le,axiom,
    ! [P: nat > $o,N3: nat] :
      ( ( P @ N3 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N3 )
            & ! [I6: nat] :
                ( ( ord_less_nat @ I6 @ K2 )
               => ~ ( P @ I6 ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_986_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N3 ) )
      = ( ord_less_nat @ M @ N3 ) ) ).

% Suc_mult_less_cancel1
thf(fact_987_mono__nat__linear__lb,axiom,
    ! [F2: nat > nat,M: nat,K: nat] :
      ( ! [M4: nat,N: nat] :
          ( ( ord_less_nat @ M4 @ N )
         => ( ord_less_nat @ ( F2 @ M4 ) @ ( F2 @ N ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F2 @ M ) @ K ) @ ( F2 @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_988_mult__less__mono1,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J2 @ K ) ) ) ) ).

% mult_less_mono1
thf(fact_989_mult__less__mono2,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J2 ) ) ) ) ).

% mult_less_mono2
thf(fact_990_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ( times_times_nat @ K @ M )
          = ( times_times_nat @ K @ N3 ) )
        = ( M = N3 ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_991_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N3 ) )
        = ( ord_less_nat @ M @ N3 ) ) ) ).

% nat_mult_less_cancel1
thf(fact_992_mult__Suc,axiom,
    ! [M: nat,N3: nat] :
      ( ( times_times_nat @ ( suc @ M ) @ N3 )
      = ( plus_plus_nat @ N3 @ ( times_times_nat @ M @ N3 ) ) ) ).

% mult_Suc
thf(fact_993_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N3 ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% Suc_mult_le_cancel1
thf(fact_994_mlex__bound,axiom,
    ! [A2: nat,A: nat,B2: nat,N7: nat] :
      ( ( ord_less_nat @ A2 @ A )
     => ( ( ord_less_nat @ B2 @ N7 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N7 ) @ B2 ) @ ( times_times_nat @ A @ N7 ) ) ) ) ).

% mlex_bound
thf(fact_995_mlex__fst__decrI,axiom,
    ! [A2: nat,A4: nat,B2: nat,N7: nat,B4: nat] :
      ( ( ord_less_nat @ A2 @ A4 )
     => ( ( ord_less_nat @ B2 @ N7 )
       => ( ( ord_less_nat @ B4 @ N7 )
         => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N7 ) @ B2 ) @ ( plus_plus_nat @ ( times_times_nat @ A4 @ N7 ) @ B4 ) ) ) ) ) ).

% mlex_fst_decrI
thf(fact_996_mlex__snd__decrI,axiom,
    ! [A2: nat,A4: nat,B2: nat,B4: nat,N7: nat] :
      ( ( A2 = A4 )
     => ( ( ord_less_nat @ B2 @ B4 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N7 ) @ B2 ) @ ( plus_plus_nat @ ( times_times_nat @ A4 @ N7 ) @ B4 ) ) ) ) ).

% mlex_snd_decrI
thf(fact_997_nth__equalityI,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( ( nth_VEBT_VEBT @ Xs2 @ I5 )
              = ( nth_VEBT_VEBT @ Ys @ I5 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_998_nth__equalityI,axiom,
    ! [Xs2: list_real,Ys: list_real] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_real @ Xs2 ) )
           => ( ( nth_real @ Xs2 @ I5 )
              = ( nth_real @ Ys @ I5 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_999_nth__equalityI,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_o @ Xs2 ) )
           => ( ( nth_o @ Xs2 @ I5 )
              = ( nth_o @ Ys @ I5 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_1000_nth__equalityI,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs2 ) )
           => ( ( nth_nat @ Xs2 @ I5 )
              = ( nth_nat @ Ys @ I5 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_1001_nth__equalityI,axiom,
    ! [Xs2: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
        = ( size_s7982070591426661849_VEBTi @ Ys ) )
     => ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
           => ( ( nth_VEBT_VEBTi @ Xs2 @ I5 )
              = ( nth_VEBT_VEBTi @ Ys @ I5 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_1002_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vEBT_VEBT > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X8: vEBT_VEBT] : ( P @ I2 @ X8 ) ) )
      = ( ? [Xs: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ Xs )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_VEBT_VEBT @ Xs @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1003_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > real > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X8: real] : ( P @ I2 @ X8 ) ) )
      = ( ? [Xs: list_real] :
            ( ( ( size_size_list_real @ Xs )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_real @ Xs @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1004_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > $o > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X8: $o] : ( P @ I2 @ X8 ) ) )
      = ( ? [Xs: list_o] :
            ( ( ( size_size_list_o @ Xs )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_o @ Xs @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1005_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X8: nat] : ( P @ I2 @ X8 ) ) )
      = ( ? [Xs: list_nat] :
            ( ( ( size_size_list_nat @ Xs )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1006_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vEBT_VEBTi > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X8: vEBT_VEBTi] : ( P @ I2 @ X8 ) ) )
      = ( ? [Xs: list_VEBT_VEBTi] :
            ( ( ( size_s7982070591426661849_VEBTi @ Xs )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_VEBT_VEBTi @ Xs @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1007_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_VEBT_VEBT,Z3: list_VEBT_VEBT] : Y5 = Z3 )
    = ( ^ [Xs: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( ( size_s6755466524823107622T_VEBT @ Xs )
            = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
             => ( ( nth_VEBT_VEBT @ Xs @ I2 )
                = ( nth_VEBT_VEBT @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1008_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_real,Z3: list_real] : Y5 = Z3 )
    = ( ^ [Xs: list_real,Ys3: list_real] :
          ( ( ( size_size_list_real @ Xs )
            = ( size_size_list_real @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs ) )
             => ( ( nth_real @ Xs @ I2 )
                = ( nth_real @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1009_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_o,Z3: list_o] : Y5 = Z3 )
    = ( ^ [Xs: list_o,Ys3: list_o] :
          ( ( ( size_size_list_o @ Xs )
            = ( size_size_list_o @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
             => ( ( nth_o @ Xs @ I2 )
                = ( nth_o @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1010_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_nat,Z3: list_nat] : Y5 = Z3 )
    = ( ^ [Xs: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
             => ( ( nth_nat @ Xs @ I2 )
                = ( nth_nat @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1011_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_VEBT_VEBTi,Z3: list_VEBT_VEBTi] : Y5 = Z3 )
    = ( ^ [Xs: list_VEBT_VEBTi,Ys3: list_VEBT_VEBTi] :
          ( ( ( size_s7982070591426661849_VEBTi @ Xs )
            = ( size_s7982070591426661849_VEBTi @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
             => ( ( nth_VEBT_VEBTi @ Xs @ I2 )
                = ( nth_VEBT_VEBTi @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1012_obtain__list__from__elements,axiom,
    ! [N3: nat,P: vEBT_VEBT > nat > $o] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ N3 )
         => ? [Li: vEBT_VEBT] : ( P @ Li @ I5 ) )
     => ~ ! [L4: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ L4 )
              = N3 )
           => ~ ! [I6: nat] :
                  ( ( ord_less_nat @ I6 @ N3 )
                 => ( P @ ( nth_VEBT_VEBT @ L4 @ I6 ) @ I6 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_1013_obtain__list__from__elements,axiom,
    ! [N3: nat,P: real > nat > $o] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ N3 )
         => ? [Li: real] : ( P @ Li @ I5 ) )
     => ~ ! [L4: list_real] :
            ( ( ( size_size_list_real @ L4 )
              = N3 )
           => ~ ! [I6: nat] :
                  ( ( ord_less_nat @ I6 @ N3 )
                 => ( P @ ( nth_real @ L4 @ I6 ) @ I6 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_1014_obtain__list__from__elements,axiom,
    ! [N3: nat,P: $o > nat > $o] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ N3 )
         => ? [Li: $o] : ( P @ Li @ I5 ) )
     => ~ ! [L4: list_o] :
            ( ( ( size_size_list_o @ L4 )
              = N3 )
           => ~ ! [I6: nat] :
                  ( ( ord_less_nat @ I6 @ N3 )
                 => ( P @ ( nth_o @ L4 @ I6 ) @ I6 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_1015_obtain__list__from__elements,axiom,
    ! [N3: nat,P: nat > nat > $o] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ N3 )
         => ? [Li: nat] : ( P @ Li @ I5 ) )
     => ~ ! [L4: list_nat] :
            ( ( ( size_size_list_nat @ L4 )
              = N3 )
           => ~ ! [I6: nat] :
                  ( ( ord_less_nat @ I6 @ N3 )
                 => ( P @ ( nth_nat @ L4 @ I6 ) @ I6 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_1016_obtain__list__from__elements,axiom,
    ! [N3: nat,P: vEBT_VEBTi > nat > $o] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ N3 )
         => ? [Li: vEBT_VEBTi] : ( P @ Li @ I5 ) )
     => ~ ! [L4: list_VEBT_VEBTi] :
            ( ( ( size_s7982070591426661849_VEBTi @ L4 )
              = N3 )
           => ~ ! [I6: nat] :
                  ( ( ord_less_nat @ I6 @ N3 )
                 => ( P @ ( nth_VEBT_VEBTi @ L4 @ I6 ) @ I6 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_1017_mlex__leI,axiom,
    ! [A2: nat,A4: nat,B2: nat,B4: nat,N7: nat] :
      ( ( ord_less_eq_nat @ A2 @ A4 )
     => ( ( ord_less_eq_nat @ B2 @ B4 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ A2 @ N7 ) @ B2 ) @ ( plus_plus_nat @ ( times_times_nat @ A4 @ N7 ) @ B4 ) ) ) ) ).

% mlex_leI
thf(fact_1018_insert__minus__eq,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ( X != Y )
     => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X @ A ) @ ( insert_VEBT_VEBT @ Y @ bot_bo8194388402131092736T_VEBT ) )
        = ( insert_VEBT_VEBT @ X @ ( minus_5127226145743854075T_VEBT @ A @ ( insert_VEBT_VEBT @ Y @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ).

% insert_minus_eq
thf(fact_1019_insert__minus__eq,axiom,
    ! [X: real,Y: real,A: set_real] :
      ( ( X != Y )
     => ( ( minus_minus_set_real @ ( insert_real @ X @ A ) @ ( insert_real @ Y @ bot_bot_set_real ) )
        = ( insert_real @ X @ ( minus_minus_set_real @ A @ ( insert_real @ Y @ bot_bot_set_real ) ) ) ) ) ).

% insert_minus_eq
thf(fact_1020_insert__minus__eq,axiom,
    ! [X: $o,Y: $o,A: set_o] :
      ( ( X != Y )
     => ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ ( insert_o @ Y @ bot_bot_set_o ) )
        = ( insert_o @ X @ ( minus_minus_set_o @ A @ ( insert_o @ Y @ bot_bot_set_o ) ) ) ) ) ).

% insert_minus_eq
thf(fact_1021_insert__minus__eq,axiom,
    ! [X: int,Y: int,A: set_int] :
      ( ( X != Y )
     => ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ ( insert_int @ Y @ bot_bot_set_int ) )
        = ( insert_int @ X @ ( minus_minus_set_int @ A @ ( insert_int @ Y @ bot_bot_set_int ) ) ) ) ) ).

% insert_minus_eq
thf(fact_1022_insert__minus__eq,axiom,
    ! [X: nat,Y: nat,A: set_nat] :
      ( ( X != Y )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ ( insert_nat @ Y @ bot_bot_set_nat ) )
        = ( insert_nat @ X @ ( minus_minus_set_nat @ A @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) ) ) ).

% insert_minus_eq
thf(fact_1023_set__minus__singleton__eq,axiom,
    ! [X: vEBT_VEBT,X2: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X @ X2 )
     => ( ( minus_5127226145743854075T_VEBT @ X2 @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) )
        = X2 ) ) ).

% set_minus_singleton_eq
thf(fact_1024_set__minus__singleton__eq,axiom,
    ! [X: complex,X2: set_complex] :
      ( ~ ( member_complex @ X @ X2 )
     => ( ( minus_811609699411566653omplex @ X2 @ ( insert_complex @ X @ bot_bot_set_complex ) )
        = X2 ) ) ).

% set_minus_singleton_eq
thf(fact_1025_set__minus__singleton__eq,axiom,
    ! [X: real,X2: set_real] :
      ( ~ ( member_real @ X @ X2 )
     => ( ( minus_minus_set_real @ X2 @ ( insert_real @ X @ bot_bot_set_real ) )
        = X2 ) ) ).

% set_minus_singleton_eq
thf(fact_1026_set__minus__singleton__eq,axiom,
    ! [X: $o,X2: set_o] :
      ( ~ ( member_o @ X @ X2 )
     => ( ( minus_minus_set_o @ X2 @ ( insert_o @ X @ bot_bot_set_o ) )
        = X2 ) ) ).

% set_minus_singleton_eq
thf(fact_1027_set__minus__singleton__eq,axiom,
    ! [X: int,X2: set_int] :
      ( ~ ( member_int @ X @ X2 )
     => ( ( minus_minus_set_int @ X2 @ ( insert_int @ X @ bot_bot_set_int ) )
        = X2 ) ) ).

% set_minus_singleton_eq
thf(fact_1028_set__minus__singleton__eq,axiom,
    ! [X: nat,X2: set_nat] :
      ( ~ ( member_nat @ X @ X2 )
     => ( ( minus_minus_set_nat @ X2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
        = X2 ) ) ).

% set_minus_singleton_eq
thf(fact_1029_remove__subset,axiom,
    ! [X: vEBT_VEBT,S: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ S )
     => ( ord_le3480810397992357184T_VEBT @ ( minus_5127226145743854075T_VEBT @ S @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) @ S ) ) ).

% remove_subset
thf(fact_1030_remove__subset,axiom,
    ! [X: complex,S: set_complex] :
      ( ( member_complex @ X @ S )
     => ( ord_less_set_complex @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ X @ bot_bot_set_complex ) ) @ S ) ) ).

% remove_subset
thf(fact_1031_remove__subset,axiom,
    ! [X: real,S: set_real] :
      ( ( member_real @ X @ S )
     => ( ord_less_set_real @ ( minus_minus_set_real @ S @ ( insert_real @ X @ bot_bot_set_real ) ) @ S ) ) ).

% remove_subset
thf(fact_1032_remove__subset,axiom,
    ! [X: $o,S: set_o] :
      ( ( member_o @ X @ S )
     => ( ord_less_set_o @ ( minus_minus_set_o @ S @ ( insert_o @ X @ bot_bot_set_o ) ) @ S ) ) ).

% remove_subset
thf(fact_1033_remove__subset,axiom,
    ! [X: int,S: set_int] :
      ( ( member_int @ X @ S )
     => ( ord_less_set_int @ ( minus_minus_set_int @ S @ ( insert_int @ X @ bot_bot_set_int ) ) @ S ) ) ).

% remove_subset
thf(fact_1034_remove__subset,axiom,
    ! [X: nat,S: set_nat] :
      ( ( member_nat @ X @ S )
     => ( ord_less_set_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ S ) ) ).

% remove_subset
thf(fact_1035_ex__nat__less__eq,axiom,
    ! [N3: nat,P: nat > $o] :
      ( ( ? [M5: nat] :
            ( ( ord_less_nat @ M5 @ N3 )
            & ( P @ M5 ) ) )
      = ( ? [X3: nat] :
            ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) )
            & ( P @ X3 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_1036_all__nat__less__eq,axiom,
    ! [N3: nat,P: nat > $o] :
      ( ( ! [M5: nat] :
            ( ( ord_less_nat @ M5 @ N3 )
           => ( P @ M5 ) ) )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) )
           => ( P @ X3 ) ) ) ) ).

% all_nat_less_eq
thf(fact_1037_atLeastLessThan0,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_1038_ex__least__nat__less,axiom,
    ! [P: nat > $o,N3: nat] :
      ( ( P @ N3 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N3 )
            & ! [I6: nat] :
                ( ( ord_less_eq_nat @ I6 @ K2 )
               => ~ ( P @ I6 ) )
            & ( P @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_1039_one__less__mult,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N3 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N3 ) ) ) ) ).

% one_less_mult
thf(fact_1040_n__less__m__mult__n,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N3 @ ( times_times_nat @ M @ N3 ) ) ) ) ).

% n_less_m_mult_n
thf(fact_1041_n__less__n__mult__m,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N3 @ ( times_times_nat @ N3 @ M ) ) ) ) ).

% n_less_n_mult_m
thf(fact_1042_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N3 ) )
        = ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% nat_mult_le_cancel1
thf(fact_1043_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N3 ) )
        = ( divide_divide_nat @ M @ N3 ) ) ) ).

% nat_mult_div_cancel1
thf(fact_1044_nth__list__update,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,J2: nat,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ( I = J2 )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ J2 )
            = X ) )
        & ( ( I != J2 )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ J2 )
            = ( nth_VEBT_VEBT @ Xs2 @ J2 ) ) ) ) ) ).

% nth_list_update
thf(fact_1045_nth__list__update,axiom,
    ! [I: nat,Xs2: list_real,J2: nat,X: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
     => ( ( ( I = J2 )
         => ( ( nth_real @ ( list_update_real @ Xs2 @ I @ X ) @ J2 )
            = X ) )
        & ( ( I != J2 )
         => ( ( nth_real @ ( list_update_real @ Xs2 @ I @ X ) @ J2 )
            = ( nth_real @ Xs2 @ J2 ) ) ) ) ) ).

% nth_list_update
thf(fact_1046_nth__list__update,axiom,
    ! [I: nat,Xs2: list_o,X: $o,J2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X ) @ J2 )
        = ( ( ( I = J2 )
           => X )
          & ( ( I != J2 )
           => ( nth_o @ Xs2 @ J2 ) ) ) ) ) ).

% nth_list_update
thf(fact_1047_nth__list__update,axiom,
    ! [I: nat,Xs2: list_nat,J2: nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ( I = J2 )
         => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ J2 )
            = X ) )
        & ( ( I != J2 )
         => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ J2 )
            = ( nth_nat @ Xs2 @ J2 ) ) ) ) ) ).

% nth_list_update
thf(fact_1048_nth__list__update,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,J2: nat,X: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( ( I = J2 )
         => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X ) @ J2 )
            = X ) )
        & ( ( I != J2 )
         => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X ) @ J2 )
            = ( nth_VEBT_VEBTi @ Xs2 @ J2 ) ) ) ) ) ).

% nth_list_update
thf(fact_1049_nth__list__update_H,axiom,
    ! [I: nat,J2: nat,L2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ( ( I = J2 )
          & ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) ) )
       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ X ) @ J2 )
          = X ) )
      & ( ~ ( ( I = J2 )
            & ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) ) )
       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ X ) @ J2 )
          = ( nth_VEBT_VEBT @ L2 @ J2 ) ) ) ) ).

% nth_list_update'
thf(fact_1050_nth__list__update_H,axiom,
    ! [I: nat,J2: nat,L2: list_real,X: real] :
      ( ( ( ( I = J2 )
          & ( ord_less_nat @ I @ ( size_size_list_real @ L2 ) ) )
       => ( ( nth_real @ ( list_update_real @ L2 @ I @ X ) @ J2 )
          = X ) )
      & ( ~ ( ( I = J2 )
            & ( ord_less_nat @ I @ ( size_size_list_real @ L2 ) ) )
       => ( ( nth_real @ ( list_update_real @ L2 @ I @ X ) @ J2 )
          = ( nth_real @ L2 @ J2 ) ) ) ) ).

% nth_list_update'
thf(fact_1051_nth__list__update_H,axiom,
    ! [L2: list_o,I: nat,X: $o,J2: nat] :
      ( ( nth_o @ ( list_update_o @ L2 @ I @ X ) @ J2 )
      = ( ( ( ( I = J2 )
            & ( ord_less_nat @ I @ ( size_size_list_o @ L2 ) ) )
         => X )
        & ( ~ ( ( I = J2 )
              & ( ord_less_nat @ I @ ( size_size_list_o @ L2 ) ) )
         => ( nth_o @ L2 @ J2 ) ) ) ) ).

% nth_list_update'
thf(fact_1052_nth__list__update_H,axiom,
    ! [I: nat,J2: nat,L2: list_nat,X: nat] :
      ( ( ( ( I = J2 )
          & ( ord_less_nat @ I @ ( size_size_list_nat @ L2 ) ) )
       => ( ( nth_nat @ ( list_update_nat @ L2 @ I @ X ) @ J2 )
          = X ) )
      & ( ~ ( ( I = J2 )
            & ( ord_less_nat @ I @ ( size_size_list_nat @ L2 ) ) )
       => ( ( nth_nat @ ( list_update_nat @ L2 @ I @ X ) @ J2 )
          = ( nth_nat @ L2 @ J2 ) ) ) ) ).

% nth_list_update'
thf(fact_1053_nth__list__update_H,axiom,
    ! [I: nat,J2: nat,L2: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( ( ( I = J2 )
          & ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L2 ) ) )
       => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ L2 @ I @ X ) @ J2 )
          = X ) )
      & ( ~ ( ( I = J2 )
            & ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L2 ) ) )
       => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ L2 @ I @ X ) @ J2 )
          = ( nth_VEBT_VEBTi @ L2 @ J2 ) ) ) ) ).

% nth_list_update'
thf(fact_1054_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X )
          = Xs2 )
        = ( ( nth_VEBT_VEBT @ Xs2 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_1055_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_real,X: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
     => ( ( ( list_update_real @ Xs2 @ I @ X )
          = Xs2 )
        = ( ( nth_real @ Xs2 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_1056_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_o,X: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( ( list_update_o @ Xs2 @ I @ X )
          = Xs2 )
        = ( ( nth_o @ Xs2 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_1057_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ( list_update_nat @ Xs2 @ I @ X )
          = Xs2 )
        = ( ( nth_nat @ Xs2 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_1058_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X )
          = Xs2 )
        = ( ( nth_VEBT_VEBTi @ Xs2 @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_1059_atLeast0__lessThan__Suc,axiom,
    ! [N3: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N3 ) )
      = ( insert_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) ) ).

% atLeast0_lessThan_Suc
thf(fact_1060_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs2: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A: vEBT_VEBT > vEBT_VEBT > assn,A5: vEBT_VEBT > vEBT_VEBT > assn] :
      ( ( I3 = I7 )
     => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
          = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
       => ( ( ( size_s6755466524823107622T_VEBT @ Xsi )
            = ( size_s6755466524823107622T_VEBT @ Xsi2 ) )
         => ( ! [I5: nat] :
                ( ( member_nat @ I5 @ I3 )
               => ( ( ord_less_nat @ I5 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
                 => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                      = ( size_s6755466524823107622T_VEBT @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBT @ Xs2 @ I5 )
                        = ( nth_VEBT_VEBT @ Xs4 @ I5 ) )
                      & ( ( nth_VEBT_VEBT @ Xsi @ I5 )
                        = ( nth_VEBT_VEBT @ Xsi2 @ I5 ) )
                      & ( ( A @ ( nth_VEBT_VEBT @ Xs2 @ I5 ) @ ( nth_VEBT_VEBT @ Xsi @ I5 ) )
                        = ( A5 @ ( nth_VEBT_VEBT @ Xs4 @ I5 ) @ ( nth_VEBT_VEBT @ Xsi2 @ I5 ) ) ) ) ) ) )
           => ( ( vEBT_L3204528365124325536T_VEBT @ I3 @ A @ Xs2 @ Xsi )
              = ( vEBT_L3204528365124325536T_VEBT @ I7 @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_1061_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs2: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_real,Xsi2: list_real,A: vEBT_VEBT > real > assn,A5: vEBT_VEBT > real > assn] :
      ( ( I3 = I7 )
     => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
          = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
       => ( ( ( size_size_list_real @ Xsi )
            = ( size_size_list_real @ Xsi2 ) )
         => ( ! [I5: nat] :
                ( ( member_nat @ I5 @ I3 )
               => ( ( ord_less_nat @ I5 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
                 => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                      = ( size_size_list_real @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBT @ Xs2 @ I5 )
                        = ( nth_VEBT_VEBT @ Xs4 @ I5 ) )
                      & ( ( nth_real @ Xsi @ I5 )
                        = ( nth_real @ Xsi2 @ I5 ) )
                      & ( ( A @ ( nth_VEBT_VEBT @ Xs2 @ I5 ) @ ( nth_real @ Xsi @ I5 ) )
                        = ( A5 @ ( nth_VEBT_VEBT @ Xs4 @ I5 ) @ ( nth_real @ Xsi2 @ I5 ) ) ) ) ) ) )
           => ( ( vEBT_L4281036506115550016T_real @ I3 @ A @ Xs2 @ Xsi )
              = ( vEBT_L4281036506115550016T_real @ I7 @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_1062_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs2: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_o,Xsi2: list_o,A: vEBT_VEBT > $o > assn,A5: vEBT_VEBT > $o > assn] :
      ( ( I3 = I7 )
     => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
          = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
       => ( ( ( size_size_list_o @ Xsi )
            = ( size_size_list_o @ Xsi2 ) )
         => ( ! [I5: nat] :
                ( ( member_nat @ I5 @ I3 )
               => ( ( ord_less_nat @ I5 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
                 => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                      = ( size_size_list_o @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBT @ Xs2 @ I5 )
                        = ( nth_VEBT_VEBT @ Xs4 @ I5 ) )
                      & ( ( nth_o @ Xsi @ I5 )
                        = ( nth_o @ Xsi2 @ I5 ) )
                      & ( ( A @ ( nth_VEBT_VEBT @ Xs2 @ I5 ) @ ( nth_o @ Xsi @ I5 ) )
                        = ( A5 @ ( nth_VEBT_VEBT @ Xs4 @ I5 ) @ ( nth_o @ Xsi2 @ I5 ) ) ) ) ) ) )
           => ( ( vEBT_L7058566406413635588VEBT_o @ I3 @ A @ Xs2 @ Xsi )
              = ( vEBT_L7058566406413635588VEBT_o @ I7 @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_1063_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs2: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_nat,Xsi2: list_nat,A: vEBT_VEBT > nat > assn,A5: vEBT_VEBT > nat > assn] :
      ( ( I3 = I7 )
     => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
          = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
       => ( ( ( size_size_list_nat @ Xsi )
            = ( size_size_list_nat @ Xsi2 ) )
         => ( ! [I5: nat] :
                ( ( member_nat @ I5 @ I3 )
               => ( ( ord_less_nat @ I5 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
                 => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                      = ( size_size_list_nat @ Xsi ) )
                   => ( ( ( nth_VEBT_VEBT @ Xs2 @ I5 )
                        = ( nth_VEBT_VEBT @ Xs4 @ I5 ) )
                      & ( ( nth_nat @ Xsi @ I5 )
                        = ( nth_nat @ Xsi2 @ I5 ) )
                      & ( ( A @ ( nth_VEBT_VEBT @ Xs2 @ I5 ) @ ( nth_nat @ Xsi @ I5 ) )
                        = ( A5 @ ( nth_VEBT_VEBT @ Xs4 @ I5 ) @ ( nth_nat @ Xsi2 @ I5 ) ) ) ) ) ) )
           => ( ( vEBT_L8650695023172932196BT_nat @ I3 @ A @ Xs2 @ Xsi )
              = ( vEBT_L8650695023172932196BT_nat @ I7 @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_1064_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs2: list_real,Xs4: list_real,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A: real > vEBT_VEBT > assn,A5: real > vEBT_VEBT > assn] :
      ( ( I3 = I7 )
     => ( ( ( size_size_list_real @ Xs2 )
          = ( size_size_list_real @ Xs4 ) )
       => ( ( ( size_s6755466524823107622T_VEBT @ Xsi )
            = ( size_s6755466524823107622T_VEBT @ Xsi2 ) )
         => ( ! [I5: nat] :
                ( ( member_nat @ I5 @ I3 )
               => ( ( ord_less_nat @ I5 @ ( size_size_list_real @ Xs2 ) )
                 => ( ( ( size_size_list_real @ Xs2 )
                      = ( size_s6755466524823107622T_VEBT @ Xsi ) )
                   => ( ( ( nth_real @ Xs2 @ I5 )
                        = ( nth_real @ Xs4 @ I5 ) )
                      & ( ( nth_VEBT_VEBT @ Xsi @ I5 )
                        = ( nth_VEBT_VEBT @ Xsi2 @ I5 ) )
                      & ( ( A @ ( nth_real @ Xs2 @ I5 ) @ ( nth_VEBT_VEBT @ Xsi @ I5 ) )
                        = ( A5 @ ( nth_real @ Xs4 @ I5 ) @ ( nth_VEBT_VEBT @ Xsi2 @ I5 ) ) ) ) ) ) )
           => ( ( vEBT_L3095048238742455910T_VEBT @ I3 @ A @ Xs2 @ Xsi )
              = ( vEBT_L3095048238742455910T_VEBT @ I7 @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_1065_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs2: list_real,Xs4: list_real,Xsi: list_real,Xsi2: list_real,A: real > real > assn,A5: real > real > assn] :
      ( ( I3 = I7 )
     => ( ( ( size_size_list_real @ Xs2 )
          = ( size_size_list_real @ Xs4 ) )
       => ( ( ( size_size_list_real @ Xsi )
            = ( size_size_list_real @ Xsi2 ) )
         => ( ! [I5: nat] :
                ( ( member_nat @ I5 @ I3 )
               => ( ( ord_less_nat @ I5 @ ( size_size_list_real @ Xs2 ) )
                 => ( ( ( size_size_list_real @ Xs2 )
                      = ( size_size_list_real @ Xsi ) )
                   => ( ( ( nth_real @ Xs2 @ I5 )
                        = ( nth_real @ Xs4 @ I5 ) )
                      & ( ( nth_real @ Xsi @ I5 )
                        = ( nth_real @ Xsi2 @ I5 ) )
                      & ( ( A @ ( nth_real @ Xs2 @ I5 ) @ ( nth_real @ Xsi @ I5 ) )
                        = ( A5 @ ( nth_real @ Xs4 @ I5 ) @ ( nth_real @ Xsi2 @ I5 ) ) ) ) ) ) )
           => ( ( vEBT_L5184575500739366650l_real @ I3 @ A @ Xs2 @ Xsi )
              = ( vEBT_L5184575500739366650l_real @ I7 @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_1066_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs2: list_real,Xs4: list_real,Xsi: list_o,Xsi2: list_o,A: real > $o > assn,A5: real > $o > assn] :
      ( ( I3 = I7 )
     => ( ( ( size_size_list_real @ Xs2 )
          = ( size_size_list_real @ Xs4 ) )
       => ( ( ( size_size_list_o @ Xsi )
            = ( size_size_list_o @ Xsi2 ) )
         => ( ! [I5: nat] :
                ( ( member_nat @ I5 @ I3 )
               => ( ( ord_less_nat @ I5 @ ( size_size_list_real @ Xs2 ) )
                 => ( ( ( size_size_list_real @ Xs2 )
                      = ( size_size_list_o @ Xsi ) )
                   => ( ( ( nth_real @ Xs2 @ I5 )
                        = ( nth_real @ Xs4 @ I5 ) )
                      & ( ( nth_o @ Xsi @ I5 )
                        = ( nth_o @ Xsi2 @ I5 ) )
                      & ( ( A @ ( nth_real @ Xs2 @ I5 ) @ ( nth_o @ Xsi @ I5 ) )
                        = ( A5 @ ( nth_real @ Xs4 @ I5 ) @ ( nth_o @ Xsi2 @ I5 ) ) ) ) ) ) )
           => ( ( vEBT_L7980206306069228746real_o @ I3 @ A @ Xs2 @ Xsi )
              = ( vEBT_L7980206306069228746real_o @ I7 @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_1067_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs2: list_real,Xs4: list_real,Xsi: list_nat,Xsi2: list_nat,A: real > nat > assn,A5: real > nat > assn] :
      ( ( I3 = I7 )
     => ( ( ( size_size_list_real @ Xs2 )
          = ( size_size_list_real @ Xs4 ) )
       => ( ( ( size_size_list_nat @ Xsi )
            = ( size_size_list_nat @ Xsi2 ) )
         => ( ! [I5: nat] :
                ( ( member_nat @ I5 @ I3 )
               => ( ( ord_less_nat @ I5 @ ( size_size_list_real @ Xs2 ) )
                 => ( ( ( size_size_list_real @ Xs2 )
                      = ( size_size_list_nat @ Xsi ) )
                   => ( ( ( nth_real @ Xs2 @ I5 )
                        = ( nth_real @ Xs4 @ I5 ) )
                      & ( ( nth_nat @ Xsi @ I5 )
                        = ( nth_nat @ Xsi2 @ I5 ) )
                      & ( ( A @ ( nth_real @ Xs2 @ I5 ) @ ( nth_nat @ Xsi @ I5 ) )
                        = ( A5 @ ( nth_real @ Xs4 @ I5 ) @ ( nth_nat @ Xsi2 @ I5 ) ) ) ) ) ) )
           => ( ( vEBT_L234762979517870878al_nat @ I3 @ A @ Xs2 @ Xsi )
              = ( vEBT_L234762979517870878al_nat @ I7 @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_1068_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs2: list_real,Xs4: list_real,Xsi: list_VEBT_VEBTi,Xsi2: list_VEBT_VEBTi,A: real > vEBT_VEBTi > assn,A5: real > vEBT_VEBTi > assn] :
      ( ( I3 = I7 )
     => ( ( ( size_size_list_real @ Xs2 )
          = ( size_size_list_real @ Xs4 ) )
       => ( ( ( size_s7982070591426661849_VEBTi @ Xsi )
            = ( size_s7982070591426661849_VEBTi @ Xsi2 ) )
         => ( ! [I5: nat] :
                ( ( member_nat @ I5 @ I3 )
               => ( ( ord_less_nat @ I5 @ ( size_size_list_real @ Xs2 ) )
                 => ( ( ( size_size_list_real @ Xs2 )
                      = ( size_s7982070591426661849_VEBTi @ Xsi ) )
                   => ( ( ( nth_real @ Xs2 @ I5 )
                        = ( nth_real @ Xs4 @ I5 ) )
                      & ( ( nth_VEBT_VEBTi @ Xsi @ I5 )
                        = ( nth_VEBT_VEBTi @ Xsi2 @ I5 ) )
                      & ( ( A @ ( nth_real @ Xs2 @ I5 ) @ ( nth_VEBT_VEBTi @ Xsi @ I5 ) )
                        = ( A5 @ ( nth_real @ Xs4 @ I5 ) @ ( nth_VEBT_VEBTi @ Xsi2 @ I5 ) ) ) ) ) ) )
           => ( ( vEBT_L7851252805511451907_VEBTi @ I3 @ A @ Xs2 @ Xsi )
              = ( vEBT_L7851252805511451907_VEBTi @ I7 @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_1069_listI__assn__cong,axiom,
    ! [I3: set_nat,I7: set_nat,Xs2: list_o,Xs4: list_o,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A: $o > vEBT_VEBT > assn,A5: $o > vEBT_VEBT > assn] :
      ( ( I3 = I7 )
     => ( ( ( size_size_list_o @ Xs2 )
          = ( size_size_list_o @ Xs4 ) )
       => ( ( ( size_s6755466524823107622T_VEBT @ Xsi )
            = ( size_s6755466524823107622T_VEBT @ Xsi2 ) )
         => ( ! [I5: nat] :
                ( ( member_nat @ I5 @ I3 )
               => ( ( ord_less_nat @ I5 @ ( size_size_list_o @ Xs2 ) )
                 => ( ( ( size_size_list_o @ Xs2 )
                      = ( size_s6755466524823107622T_VEBT @ Xsi ) )
                   => ( ( ( nth_o @ Xs2 @ I5 )
                        = ( nth_o @ Xs4 @ I5 ) )
                      & ( ( nth_VEBT_VEBT @ Xsi @ I5 )
                        = ( nth_VEBT_VEBT @ Xsi2 @ I5 ) )
                      & ( ( A @ ( nth_o @ Xs2 @ I5 ) @ ( nth_VEBT_VEBT @ Xsi @ I5 ) )
                        = ( A5 @ ( nth_o @ Xs4 @ I5 ) @ ( nth_VEBT_VEBT @ Xsi2 @ I5 ) ) ) ) ) ) )
           => ( ( vEBT_L1319876754960170684T_VEBT @ I3 @ A @ Xs2 @ Xsi )
              = ( vEBT_L1319876754960170684T_VEBT @ I7 @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ) ).

% listI_assn_cong
thf(fact_1070_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: vEBT_VEBT > vEBT_VEBT > assn,A5: vEBT_VEBT > vEBT_VEBT > assn,Xs2: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT] :
      ( ( I3 = I7 )
     => ( ( A = A5 )
       => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
            = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
         => ( ( ( size_s6755466524823107622T_VEBT @ Xsi )
              = ( size_s6755466524823107622T_VEBT @ Xsi2 ) )
           => ( ! [I5: nat] :
                  ( ( member_nat @ I5 @ I3 )
                 => ( ( ord_less_nat @ I5 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
                   => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                        = ( size_s6755466524823107622T_VEBT @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBT @ Xs2 @ I5 )
                          = ( nth_VEBT_VEBT @ Xs4 @ I5 ) )
                        & ( ( nth_VEBT_VEBT @ Xsi @ I5 )
                          = ( nth_VEBT_VEBT @ Xsi2 @ I5 ) ) ) ) ) )
             => ( ( vEBT_L3204528365124325536T_VEBT @ I3 @ A @ Xs2 @ Xsi )
                = ( vEBT_L3204528365124325536T_VEBT @ I7 @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_1071_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: vEBT_VEBT > real > assn,A5: vEBT_VEBT > real > assn,Xs2: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_real,Xsi2: list_real] :
      ( ( I3 = I7 )
     => ( ( A = A5 )
       => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
            = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
         => ( ( ( size_size_list_real @ Xsi )
              = ( size_size_list_real @ Xsi2 ) )
           => ( ! [I5: nat] :
                  ( ( member_nat @ I5 @ I3 )
                 => ( ( ord_less_nat @ I5 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
                   => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                        = ( size_size_list_real @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBT @ Xs2 @ I5 )
                          = ( nth_VEBT_VEBT @ Xs4 @ I5 ) )
                        & ( ( nth_real @ Xsi @ I5 )
                          = ( nth_real @ Xsi2 @ I5 ) ) ) ) ) )
             => ( ( vEBT_L4281036506115550016T_real @ I3 @ A @ Xs2 @ Xsi )
                = ( vEBT_L4281036506115550016T_real @ I7 @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_1072_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: vEBT_VEBT > $o > assn,A5: vEBT_VEBT > $o > assn,Xs2: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_o,Xsi2: list_o] :
      ( ( I3 = I7 )
     => ( ( A = A5 )
       => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
            = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
         => ( ( ( size_size_list_o @ Xsi )
              = ( size_size_list_o @ Xsi2 ) )
           => ( ! [I5: nat] :
                  ( ( member_nat @ I5 @ I3 )
                 => ( ( ord_less_nat @ I5 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
                   => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                        = ( size_size_list_o @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBT @ Xs2 @ I5 )
                          = ( nth_VEBT_VEBT @ Xs4 @ I5 ) )
                        & ( ( nth_o @ Xsi @ I5 )
                          = ( nth_o @ Xsi2 @ I5 ) ) ) ) ) )
             => ( ( vEBT_L7058566406413635588VEBT_o @ I3 @ A @ Xs2 @ Xsi )
                = ( vEBT_L7058566406413635588VEBT_o @ I7 @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_1073_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: vEBT_VEBT > nat > assn,A5: vEBT_VEBT > nat > assn,Xs2: list_VEBT_VEBT,Xs4: list_VEBT_VEBT,Xsi: list_nat,Xsi2: list_nat] :
      ( ( I3 = I7 )
     => ( ( A = A5 )
       => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
            = ( size_s6755466524823107622T_VEBT @ Xs4 ) )
         => ( ( ( size_size_list_nat @ Xsi )
              = ( size_size_list_nat @ Xsi2 ) )
           => ( ! [I5: nat] :
                  ( ( member_nat @ I5 @ I3 )
                 => ( ( ord_less_nat @ I5 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
                   => ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                        = ( size_size_list_nat @ Xsi ) )
                     => ( ( ( nth_VEBT_VEBT @ Xs2 @ I5 )
                          = ( nth_VEBT_VEBT @ Xs4 @ I5 ) )
                        & ( ( nth_nat @ Xsi @ I5 )
                          = ( nth_nat @ Xsi2 @ I5 ) ) ) ) ) )
             => ( ( vEBT_L8650695023172932196BT_nat @ I3 @ A @ Xs2 @ Xsi )
                = ( vEBT_L8650695023172932196BT_nat @ I7 @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_1074_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: real > vEBT_VEBT > assn,A5: real > vEBT_VEBT > assn,Xs2: list_real,Xs4: list_real,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT] :
      ( ( I3 = I7 )
     => ( ( A = A5 )
       => ( ( ( size_size_list_real @ Xs2 )
            = ( size_size_list_real @ Xs4 ) )
         => ( ( ( size_s6755466524823107622T_VEBT @ Xsi )
              = ( size_s6755466524823107622T_VEBT @ Xsi2 ) )
           => ( ! [I5: nat] :
                  ( ( member_nat @ I5 @ I3 )
                 => ( ( ord_less_nat @ I5 @ ( size_size_list_real @ Xs2 ) )
                   => ( ( ( size_size_list_real @ Xs2 )
                        = ( size_s6755466524823107622T_VEBT @ Xsi ) )
                     => ( ( ( nth_real @ Xs2 @ I5 )
                          = ( nth_real @ Xs4 @ I5 ) )
                        & ( ( nth_VEBT_VEBT @ Xsi @ I5 )
                          = ( nth_VEBT_VEBT @ Xsi2 @ I5 ) ) ) ) ) )
             => ( ( vEBT_L3095048238742455910T_VEBT @ I3 @ A @ Xs2 @ Xsi )
                = ( vEBT_L3095048238742455910T_VEBT @ I7 @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_1075_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: real > real > assn,A5: real > real > assn,Xs2: list_real,Xs4: list_real,Xsi: list_real,Xsi2: list_real] :
      ( ( I3 = I7 )
     => ( ( A = A5 )
       => ( ( ( size_size_list_real @ Xs2 )
            = ( size_size_list_real @ Xs4 ) )
         => ( ( ( size_size_list_real @ Xsi )
              = ( size_size_list_real @ Xsi2 ) )
           => ( ! [I5: nat] :
                  ( ( member_nat @ I5 @ I3 )
                 => ( ( ord_less_nat @ I5 @ ( size_size_list_real @ Xs2 ) )
                   => ( ( ( size_size_list_real @ Xs2 )
                        = ( size_size_list_real @ Xsi ) )
                     => ( ( ( nth_real @ Xs2 @ I5 )
                          = ( nth_real @ Xs4 @ I5 ) )
                        & ( ( nth_real @ Xsi @ I5 )
                          = ( nth_real @ Xsi2 @ I5 ) ) ) ) ) )
             => ( ( vEBT_L5184575500739366650l_real @ I3 @ A @ Xs2 @ Xsi )
                = ( vEBT_L5184575500739366650l_real @ I7 @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_1076_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: real > $o > assn,A5: real > $o > assn,Xs2: list_real,Xs4: list_real,Xsi: list_o,Xsi2: list_o] :
      ( ( I3 = I7 )
     => ( ( A = A5 )
       => ( ( ( size_size_list_real @ Xs2 )
            = ( size_size_list_real @ Xs4 ) )
         => ( ( ( size_size_list_o @ Xsi )
              = ( size_size_list_o @ Xsi2 ) )
           => ( ! [I5: nat] :
                  ( ( member_nat @ I5 @ I3 )
                 => ( ( ord_less_nat @ I5 @ ( size_size_list_real @ Xs2 ) )
                   => ( ( ( size_size_list_real @ Xs2 )
                        = ( size_size_list_o @ Xsi ) )
                     => ( ( ( nth_real @ Xs2 @ I5 )
                          = ( nth_real @ Xs4 @ I5 ) )
                        & ( ( nth_o @ Xsi @ I5 )
                          = ( nth_o @ Xsi2 @ I5 ) ) ) ) ) )
             => ( ( vEBT_L7980206306069228746real_o @ I3 @ A @ Xs2 @ Xsi )
                = ( vEBT_L7980206306069228746real_o @ I7 @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_1077_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: real > nat > assn,A5: real > nat > assn,Xs2: list_real,Xs4: list_real,Xsi: list_nat,Xsi2: list_nat] :
      ( ( I3 = I7 )
     => ( ( A = A5 )
       => ( ( ( size_size_list_real @ Xs2 )
            = ( size_size_list_real @ Xs4 ) )
         => ( ( ( size_size_list_nat @ Xsi )
              = ( size_size_list_nat @ Xsi2 ) )
           => ( ! [I5: nat] :
                  ( ( member_nat @ I5 @ I3 )
                 => ( ( ord_less_nat @ I5 @ ( size_size_list_real @ Xs2 ) )
                   => ( ( ( size_size_list_real @ Xs2 )
                        = ( size_size_list_nat @ Xsi ) )
                     => ( ( ( nth_real @ Xs2 @ I5 )
                          = ( nth_real @ Xs4 @ I5 ) )
                        & ( ( nth_nat @ Xsi @ I5 )
                          = ( nth_nat @ Xsi2 @ I5 ) ) ) ) ) )
             => ( ( vEBT_L234762979517870878al_nat @ I3 @ A @ Xs2 @ Xsi )
                = ( vEBT_L234762979517870878al_nat @ I7 @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_1078_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: real > vEBT_VEBTi > assn,A5: real > vEBT_VEBTi > assn,Xs2: list_real,Xs4: list_real,Xsi: list_VEBT_VEBTi,Xsi2: list_VEBT_VEBTi] :
      ( ( I3 = I7 )
     => ( ( A = A5 )
       => ( ( ( size_size_list_real @ Xs2 )
            = ( size_size_list_real @ Xs4 ) )
         => ( ( ( size_s7982070591426661849_VEBTi @ Xsi )
              = ( size_s7982070591426661849_VEBTi @ Xsi2 ) )
           => ( ! [I5: nat] :
                  ( ( member_nat @ I5 @ I3 )
                 => ( ( ord_less_nat @ I5 @ ( size_size_list_real @ Xs2 ) )
                   => ( ( ( size_size_list_real @ Xs2 )
                        = ( size_s7982070591426661849_VEBTi @ Xsi ) )
                     => ( ( ( nth_real @ Xs2 @ I5 )
                          = ( nth_real @ Xs4 @ I5 ) )
                        & ( ( nth_VEBT_VEBTi @ Xsi @ I5 )
                          = ( nth_VEBT_VEBTi @ Xsi2 @ I5 ) ) ) ) ) )
             => ( ( vEBT_L7851252805511451907_VEBTi @ I3 @ A @ Xs2 @ Xsi )
                = ( vEBT_L7851252805511451907_VEBTi @ I7 @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_1079_listI__assn__weak__cong,axiom,
    ! [I3: set_nat,I7: set_nat,A: $o > vEBT_VEBT > assn,A5: $o > vEBT_VEBT > assn,Xs2: list_o,Xs4: list_o,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT] :
      ( ( I3 = I7 )
     => ( ( A = A5 )
       => ( ( ( size_size_list_o @ Xs2 )
            = ( size_size_list_o @ Xs4 ) )
         => ( ( ( size_s6755466524823107622T_VEBT @ Xsi )
              = ( size_s6755466524823107622T_VEBT @ Xsi2 ) )
           => ( ! [I5: nat] :
                  ( ( member_nat @ I5 @ I3 )
                 => ( ( ord_less_nat @ I5 @ ( size_size_list_o @ Xs2 ) )
                   => ( ( ( size_size_list_o @ Xs2 )
                        = ( size_s6755466524823107622T_VEBT @ Xsi ) )
                     => ( ( ( nth_o @ Xs2 @ I5 )
                          = ( nth_o @ Xs4 @ I5 ) )
                        & ( ( nth_VEBT_VEBT @ Xsi @ I5 )
                          = ( nth_VEBT_VEBT @ Xsi2 @ I5 ) ) ) ) ) )
             => ( ( vEBT_L1319876754960170684T_VEBT @ I3 @ A @ Xs2 @ Xsi )
                = ( vEBT_L1319876754960170684T_VEBT @ I7 @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ) ) ).

% listI_assn_weak_cong
thf(fact_1080_subst__not__in,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_VEBT_VEBT,A: vEBT_VEBT > vEBT_VEBT > assn,X1: vEBT_VEBT,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( vEBT_L3204528365124325536T_VEBT @ I3 @ A @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X22 ) )
          = ( vEBT_L3204528365124325536T_VEBT @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_1081_subst__not__in,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_real,A: real > vEBT_VEBTi > assn,X1: real,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( vEBT_L7851252805511451907_VEBTi @ I3 @ A @ ( list_update_real @ Xs2 @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X22 ) )
          = ( vEBT_L7851252805511451907_VEBTi @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_1082_subst__not__in,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_real,A: real > vEBT_VEBT > assn,X1: real,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( vEBT_L3095048238742455910T_VEBT @ I3 @ A @ ( list_update_real @ Xs2 @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X22 ) )
          = ( vEBT_L3095048238742455910T_VEBT @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_1083_subst__not__in,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_o,A: $o > vEBT_VEBTi > assn,X1: $o,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( vEBT_L6286945158656146733_VEBTi @ I3 @ A @ ( list_update_o @ Xs2 @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X22 ) )
          = ( vEBT_L6286945158656146733_VEBTi @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_1084_subst__not__in,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_o,A: $o > vEBT_VEBT > assn,X1: $o,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( vEBT_L1319876754960170684T_VEBT @ I3 @ A @ ( list_update_o @ Xs2 @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X22 ) )
          = ( vEBT_L1319876754960170684T_VEBT @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_1085_subst__not__in,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_nat,A: nat > vEBT_VEBTi > assn,X1: nat,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( vEBT_L7489483478785760935_VEBTi @ I3 @ A @ ( list_update_nat @ Xs2 @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X22 ) )
          = ( vEBT_L7489483478785760935_VEBTi @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_1086_subst__not__in,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_nat,A: nat > vEBT_VEBT > assn,X1: nat,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( vEBT_L8511957252848910786T_VEBT @ I3 @ A @ ( list_update_nat @ Xs2 @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X22 ) )
          = ( vEBT_L8511957252848910786T_VEBT @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_1087_subst__not__in,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_VEBT_VEBTi,A: vEBT_VEBTi > vEBT_VEBTi > assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( vEBT_L886525131989349516_VEBTi @ I3 @ A @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X22 ) )
          = ( vEBT_L886525131989349516_VEBTi @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_1088_subst__not__in,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_VEBT_VEBTi,A: vEBT_VEBTi > vEBT_VEBT > assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( vEBT_L2497118539674116125T_VEBT @ I3 @ A @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X22 ) )
          = ( vEBT_L2497118539674116125T_VEBT @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_1089_subst__not__in,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_VEBT_VEBT,A: vEBT_VEBT > vEBT_VEBTi > assn,X1: vEBT_VEBT,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( vEBT_L1528199826722428489_VEBTi @ I3 @ A @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X22 ) )
          = ( vEBT_L1528199826722428489_VEBTi @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ).

% subst_not_in
thf(fact_1090_atLeastLessThanSuc,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( ord_less_eq_nat @ M @ N3 )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N3 ) )
          = ( insert_nat @ N3 @ ( set_or4665077453230672383an_nat @ M @ N3 ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N3 )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N3 ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
thf(fact_1091_listI__assn__conv,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT,A: vEBT_VEBT > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi] :
      ( ( N3
        = ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( vEBT_L1528199826722428489_VEBTi @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) @ A @ Xs2 @ Xsi )
        = ( vEBT_L6296928887356842470_VEBTi @ A @ Xs2 @ Xsi ) ) ) ).

% listI_assn_conv
thf(fact_1092_list__assn__conv__idx,axiom,
    ( vEBT_L6296928887356842470_VEBTi
    = ( ^ [A6: vEBT_VEBT > vEBT_VEBTi > assn,Xs: list_VEBT_VEBT] : ( vEBT_L1528199826722428489_VEBTi @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) @ A6 @ Xs ) ) ) ).

% list_assn_conv_idx
thf(fact_1093_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_VEBT_VEBT,A: vEBT_VEBT > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( vEBT_L3204528365124325536T_VEBT @ ( insert_nat @ I @ I3 ) @ A @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3204528365124325536T_VEBT @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_1094_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_VEBT_VEBT,A: vEBT_VEBT > nat > assn,Xsi: list_nat] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( vEBT_L8650695023172932196BT_nat @ ( insert_nat @ I @ I3 ) @ A @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8650695023172932196BT_nat @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_1095_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_real,A: real > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( vEBT_L3095048238742455910T_VEBT @ ( insert_nat @ I @ I3 ) @ A @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_real @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3095048238742455910T_VEBT @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_1096_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_real,A: real > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( vEBT_L7851252805511451907_VEBTi @ ( insert_nat @ I @ I3 ) @ A @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_real @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L7851252805511451907_VEBTi @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_1097_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_real,A: real > nat > assn,Xsi: list_nat] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( vEBT_L234762979517870878al_nat @ ( insert_nat @ I @ I3 ) @ A @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_real @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L234762979517870878al_nat @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_1098_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_o,A: $o > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( vEBT_L1319876754960170684T_VEBT @ ( insert_nat @ I @ I3 ) @ A @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_o @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L1319876754960170684T_VEBT @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_1099_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_o,A: $o > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( vEBT_L6286945158656146733_VEBTi @ ( insert_nat @ I @ I3 ) @ A @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_o @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L6286945158656146733_VEBTi @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_1100_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_o,A: $o > nat > assn,Xsi: list_nat] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( vEBT_L2281750874075065672_o_nat @ ( insert_nat @ I @ I3 ) @ A @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_o @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L2281750874075065672_o_nat @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_1101_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_nat,A: nat > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( vEBT_L8511957252848910786T_VEBT @ ( insert_nat @ I @ I3 ) @ A @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_nat @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L8511957252848910786T_VEBT @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_1102_listI__assn__insert,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_nat,A: nat > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( vEBT_L7489483478785760935_VEBTi @ ( insert_nat @ I @ I3 ) @ A @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_nat @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L7489483478785760935_VEBTi @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_insert
thf(fact_1103_listI__assn__conv_H,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT,A: vEBT_VEBT > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi,F: assn] :
      ( ( N3
        = ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( times_times_assn @ ( vEBT_L1528199826722428489_VEBTi @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) @ A @ Xs2 @ Xsi ) @ F )
        = ( times_times_assn @ ( vEBT_L6296928887356842470_VEBTi @ A @ Xs2 @ Xsi ) @ F ) ) ) ).

% listI_assn_conv'
thf(fact_1104_listI__assn__subst,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_VEBT_VEBT,A: vEBT_VEBT > vEBT_VEBT > assn,X1: vEBT_VEBT,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( vEBT_L3204528365124325536T_VEBT @ ( insert_nat @ I @ I3 ) @ A @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X22 ) )
          = ( times_times_assn @ ( A @ X1 @ X22 ) @ ( vEBT_L3204528365124325536T_VEBT @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_1105_listI__assn__subst,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_real,A: real > vEBT_VEBTi > assn,X1: real,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( vEBT_L7851252805511451907_VEBTi @ ( insert_nat @ I @ I3 ) @ A @ ( list_update_real @ Xs2 @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X22 ) )
          = ( times_times_assn @ ( A @ X1 @ X22 ) @ ( vEBT_L7851252805511451907_VEBTi @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_1106_listI__assn__subst,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_real,A: real > vEBT_VEBT > assn,X1: real,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( vEBT_L3095048238742455910T_VEBT @ ( insert_nat @ I @ I3 ) @ A @ ( list_update_real @ Xs2 @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X22 ) )
          = ( times_times_assn @ ( A @ X1 @ X22 ) @ ( vEBT_L3095048238742455910T_VEBT @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_1107_listI__assn__subst,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_o,A: $o > vEBT_VEBTi > assn,X1: $o,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( vEBT_L6286945158656146733_VEBTi @ ( insert_nat @ I @ I3 ) @ A @ ( list_update_o @ Xs2 @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X22 ) )
          = ( times_times_assn @ ( A @ X1 @ X22 ) @ ( vEBT_L6286945158656146733_VEBTi @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_1108_listI__assn__subst,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_o,A: $o > vEBT_VEBT > assn,X1: $o,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( vEBT_L1319876754960170684T_VEBT @ ( insert_nat @ I @ I3 ) @ A @ ( list_update_o @ Xs2 @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X22 ) )
          = ( times_times_assn @ ( A @ X1 @ X22 ) @ ( vEBT_L1319876754960170684T_VEBT @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_1109_listI__assn__subst,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_nat,A: nat > vEBT_VEBTi > assn,X1: nat,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( vEBT_L7489483478785760935_VEBTi @ ( insert_nat @ I @ I3 ) @ A @ ( list_update_nat @ Xs2 @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X22 ) )
          = ( times_times_assn @ ( A @ X1 @ X22 ) @ ( vEBT_L7489483478785760935_VEBTi @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_1110_listI__assn__subst,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_nat,A: nat > vEBT_VEBT > assn,X1: nat,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( vEBT_L8511957252848910786T_VEBT @ ( insert_nat @ I @ I3 ) @ A @ ( list_update_nat @ Xs2 @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X22 ) )
          = ( times_times_assn @ ( A @ X1 @ X22 ) @ ( vEBT_L8511957252848910786T_VEBT @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_1111_listI__assn__subst,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_VEBT_VEBTi,A: vEBT_VEBTi > vEBT_VEBTi > assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( vEBT_L886525131989349516_VEBTi @ ( insert_nat @ I @ I3 ) @ A @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X22 ) )
          = ( times_times_assn @ ( A @ X1 @ X22 ) @ ( vEBT_L886525131989349516_VEBTi @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_1112_listI__assn__subst,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_VEBT_VEBTi,A: vEBT_VEBTi > vEBT_VEBT > assn,X1: vEBT_VEBTi,Xsi: list_VEBT_VEBT,X22: vEBT_VEBT] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( vEBT_L2497118539674116125T_VEBT @ ( insert_nat @ I @ I3 ) @ A @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X1 ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ X22 ) )
          = ( times_times_assn @ ( A @ X1 @ X22 ) @ ( vEBT_L2497118539674116125T_VEBT @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_1113_listI__assn__subst,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_VEBT_VEBT,A: vEBT_VEBT > vEBT_VEBTi > assn,X1: vEBT_VEBT,Xsi: list_VEBT_VEBTi,X22: vEBT_VEBTi] :
      ( ~ ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( vEBT_L1528199826722428489_VEBTi @ ( insert_nat @ I @ I3 ) @ A @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X1 ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ X22 ) )
          = ( times_times_assn @ ( A @ X1 @ X22 ) @ ( vEBT_L1528199826722428489_VEBTi @ I3 @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_subst
thf(fact_1114_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_VEBT_VEBT,A: vEBT_VEBT > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( vEBT_L3204528365124325536T_VEBT @ I3 @ A @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_1115_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_VEBT_VEBT,A: vEBT_VEBT > nat > assn,Xsi: list_nat] :
      ( ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( vEBT_L8650695023172932196BT_nat @ I3 @ A @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8650695023172932196BT_nat @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_1116_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_real,A: real > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( vEBT_L3095048238742455910T_VEBT @ I3 @ A @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_real @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3095048238742455910T_VEBT @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_1117_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_real,A: real > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi] :
      ( ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( vEBT_L7851252805511451907_VEBTi @ I3 @ A @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_real @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L7851252805511451907_VEBTi @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_1118_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_real,A: real > nat > assn,Xsi: list_nat] :
      ( ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( vEBT_L234762979517870878al_nat @ I3 @ A @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_real @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L234762979517870878al_nat @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_1119_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_o,A: $o > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( vEBT_L1319876754960170684T_VEBT @ I3 @ A @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_o @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L1319876754960170684T_VEBT @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_1120_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_o,A: $o > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi] :
      ( ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( vEBT_L6286945158656146733_VEBTi @ I3 @ A @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_o @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L6286945158656146733_VEBTi @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_1121_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_o,A: $o > nat > assn,Xsi: list_nat] :
      ( ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( vEBT_L2281750874075065672_o_nat @ I3 @ A @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_o @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L2281750874075065672_o_nat @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_1122_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_nat,A: nat > vEBT_VEBT > assn,Xsi: list_VEBT_VEBT] :
      ( ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( vEBT_L8511957252848910786T_VEBT @ I3 @ A @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_nat @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L8511957252848910786T_VEBT @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_1123_listI__assn__extract,axiom,
    ! [I: nat,I3: set_nat,Xs2: list_nat,A: nat > vEBT_VEBTi > assn,Xsi: list_VEBT_VEBTi] :
      ( ( member_nat @ I @ I3 )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( vEBT_L7489483478785760935_VEBTi @ I3 @ A @ Xs2 @ Xsi )
          = ( times_times_assn @ ( A @ ( nth_nat @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L7489483478785760935_VEBTi @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) ) ) ) ).

% listI_assn_extract
thf(fact_1124_listI__assn__reinsert,axiom,
    ! [P: assn,A: vEBT_VEBT > vEBT_VEBT > assn,Xs2: list_VEBT_VEBT,I: nat,Xsi: list_VEBT_VEBT,I3: set_nat,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L3204528365124325536T_VEBT @ I3 @ A @ Xs2 @ Xsi ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_1125_listI__assn__reinsert,axiom,
    ! [P: assn,A: vEBT_VEBT > nat > assn,Xs2: list_VEBT_VEBT,I: nat,Xsi: list_nat,I3: set_nat,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8650695023172932196BT_nat @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L8650695023172932196BT_nat @ I3 @ A @ Xs2 @ Xsi ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_1126_listI__assn__reinsert,axiom,
    ! [P: assn,A: real > vEBT_VEBT > assn,Xs2: list_real,I: nat,Xsi: list_VEBT_VEBT,I3: set_nat,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_real @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3095048238742455910T_VEBT @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L3095048238742455910T_VEBT @ I3 @ A @ Xs2 @ Xsi ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_1127_listI__assn__reinsert,axiom,
    ! [P: assn,A: real > vEBT_VEBTi > assn,Xs2: list_real,I: nat,Xsi: list_VEBT_VEBTi,I3: set_nat,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_real @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L7851252805511451907_VEBTi @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L7851252805511451907_VEBTi @ I3 @ A @ Xs2 @ Xsi ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_1128_listI__assn__reinsert,axiom,
    ! [P: assn,A: real > nat > assn,Xs2: list_real,I: nat,Xsi: list_nat,I3: set_nat,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_real @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L234762979517870878al_nat @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L234762979517870878al_nat @ I3 @ A @ Xs2 @ Xsi ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_1129_listI__assn__reinsert,axiom,
    ! [P: assn,A: $o > vEBT_VEBT > assn,Xs2: list_o,I: nat,Xsi: list_VEBT_VEBT,I3: set_nat,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_o @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L1319876754960170684T_VEBT @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L1319876754960170684T_VEBT @ I3 @ A @ Xs2 @ Xsi ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_1130_listI__assn__reinsert,axiom,
    ! [P: assn,A: $o > vEBT_VEBTi > assn,Xs2: list_o,I: nat,Xsi: list_VEBT_VEBTi,I3: set_nat,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_o @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L6286945158656146733_VEBTi @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L6286945158656146733_VEBTi @ I3 @ A @ Xs2 @ Xsi ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_1131_listI__assn__reinsert,axiom,
    ! [P: assn,A: $o > nat > assn,Xs2: list_o,I: nat,Xsi: list_nat,I3: set_nat,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_o @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L2281750874075065672_o_nat @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L2281750874075065672_o_nat @ I3 @ A @ Xs2 @ Xsi ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_1132_listI__assn__reinsert,axiom,
    ! [P: assn,A: nat > vEBT_VEBT > assn,Xs2: list_nat,I: nat,Xsi: list_VEBT_VEBT,I3: set_nat,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_nat @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L8511957252848910786T_VEBT @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L8511957252848910786T_VEBT @ I3 @ A @ Xs2 @ Xsi ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_1133_listI__assn__reinsert,axiom,
    ! [P: assn,A: nat > vEBT_VEBTi > assn,Xs2: list_nat,I: nat,Xsi: list_VEBT_VEBTi,I3: set_nat,F: assn,Q: assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_nat @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L7489483478785760935_VEBTi @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( entails @ ( times_times_assn @ ( vEBT_L7489483478785760935_VEBTi @ I3 @ A @ Xs2 @ Xsi ) @ F ) @ Q )
           => ( entails @ P @ Q ) ) ) ) ) ).

% listI_assn_reinsert
thf(fact_1134_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_1135_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_1136_nat__div__eq__Suc__0__iff,axiom,
    ! [N3: nat,M: nat] :
      ( ( ( divide_divide_nat @ N3 @ M )
        = ( suc @ zero_zero_nat ) )
      = ( ( ord_less_eq_nat @ M @ N3 )
        & ( ord_less_nat @ N3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% nat_div_eq_Suc_0_iff
thf(fact_1137_ent__pure__pre__iff,axiom,
    ! [P: assn,B2: $o,Q: assn] :
      ( ( entails @ ( times_times_assn @ P @ ( pure_assn @ B2 ) ) @ Q )
      = ( B2
       => ( entails @ P @ Q ) ) ) ).

% ent_pure_pre_iff
thf(fact_1138_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N3: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N3 @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
thf(fact_1139_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N3: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N3 @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
thf(fact_1140_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_1141_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_1142_insert__Diff__single,axiom,
    ! [A2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ( insert_VEBT_VEBT @ A2 @ ( minus_5127226145743854075T_VEBT @ A @ ( insert_VEBT_VEBT @ A2 @ bot_bo8194388402131092736T_VEBT ) ) )
      = ( insert_VEBT_VEBT @ A2 @ A ) ) ).

% insert_Diff_single
thf(fact_1143_insert__Diff__single,axiom,
    ! [A2: real,A: set_real] :
      ( ( insert_real @ A2 @ ( minus_minus_set_real @ A @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
      = ( insert_real @ A2 @ A ) ) ).

% insert_Diff_single
thf(fact_1144_insert__Diff__single,axiom,
    ! [A2: $o,A: set_o] :
      ( ( insert_o @ A2 @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
      = ( insert_o @ A2 @ A ) ) ).

% insert_Diff_single
thf(fact_1145_insert__Diff__single,axiom,
    ! [A2: int,A: set_int] :
      ( ( insert_int @ A2 @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
      = ( insert_int @ A2 @ A ) ) ).

% insert_Diff_single
thf(fact_1146_insert__Diff__single,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
      = ( insert_nat @ A2 @ A ) ) ).

% insert_Diff_single
thf(fact_1147_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( ( C2 = zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C2 @ A2 ) @ ( times_times_complex @ C2 @ B2 ) )
          = zero_zero_complex ) )
      & ( ( C2 != zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C2 @ A2 ) @ ( times_times_complex @ C2 @ B2 ) )
          = ( divide1717551699836669952omplex @ A2 @ B2 ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_1148_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ( C2 = zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
          = zero_zero_real ) )
      & ( ( C2 != zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
          = ( divide_divide_real @ A2 @ B2 ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_1149_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ( C2 = zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
          = zero_zero_rat ) )
      & ( ( C2 != zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
          = ( divide_divide_rat @ A2 @ B2 ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_1150_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C2 @ A2 ) @ ( times_times_complex @ C2 @ B2 ) )
        = ( divide1717551699836669952omplex @ A2 @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_1151_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
        = ( divide_divide_real @ A2 @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_1152_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
        = ( divide_divide_rat @ A2 @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_1153_nonzero__mult__div__cancel__left,axiom,
    ! [A2: complex,B2: complex] :
      ( ( A2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A2 @ B2 ) @ A2 )
        = B2 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1154_nonzero__mult__div__cancel__left,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A2 @ B2 ) @ A2 )
        = B2 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1155_nonzero__mult__div__cancel__left,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A2 @ B2 ) @ A2 )
        = B2 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1156_nonzero__mult__div__cancel__left,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B2 ) @ A2 )
        = B2 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1157_nonzero__mult__div__cancel__left,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A2 @ B2 ) @ A2 )
        = B2 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_1158_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C2 @ A2 ) @ ( times_times_complex @ B2 @ C2 ) )
        = ( divide1717551699836669952omplex @ A2 @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_1159_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ B2 @ C2 ) )
        = ( divide_divide_real @ A2 @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_1160_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ B2 @ C2 ) )
        = ( divide_divide_rat @ A2 @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_1161_empty__Collect__eq,axiom,
    ! [P: complex > $o] :
      ( ( bot_bot_set_complex
        = ( collect_complex @ P ) )
      = ( ! [X3: complex] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_1162_empty__Collect__eq,axiom,
    ! [P: product_prod_int_int > $o] :
      ( ( bot_bo1796632182523588997nt_int
        = ( collec213857154873943460nt_int @ P ) )
      = ( ! [X3: product_prod_int_int] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_1163_empty__Collect__eq,axiom,
    ! [P: real > $o] :
      ( ( bot_bot_set_real
        = ( collect_real @ P ) )
      = ( ! [X3: real] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_1164_empty__Collect__eq,axiom,
    ! [P: $o > $o] :
      ( ( bot_bot_set_o
        = ( collect_o @ P ) )
      = ( ! [X3: $o] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_1165_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X3: nat] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_1166_empty__Collect__eq,axiom,
    ! [P: int > $o] :
      ( ( bot_bot_set_int
        = ( collect_int @ P ) )
      = ( ! [X3: int] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_1167_Collect__empty__eq,axiom,
    ! [P: complex > $o] :
      ( ( ( collect_complex @ P )
        = bot_bot_set_complex )
      = ( ! [X3: complex] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_1168_Collect__empty__eq,axiom,
    ! [P: product_prod_int_int > $o] :
      ( ( ( collec213857154873943460nt_int @ P )
        = bot_bo1796632182523588997nt_int )
      = ( ! [X3: product_prod_int_int] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_1169_Collect__empty__eq,axiom,
    ! [P: real > $o] :
      ( ( ( collect_real @ P )
        = bot_bot_set_real )
      = ( ! [X3: real] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_1170_Collect__empty__eq,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( ! [X3: $o] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_1171_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X3: nat] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_1172_Collect__empty__eq,axiom,
    ! [P: int > $o] :
      ( ( ( collect_int @ P )
        = bot_bot_set_int )
      = ( ! [X3: int] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_1173_all__not__in__conv,axiom,
    ! [A: set_VEBT_VEBT] :
      ( ( ! [X3: vEBT_VEBT] :
            ~ ( member_VEBT_VEBT @ X3 @ A ) )
      = ( A = bot_bo8194388402131092736T_VEBT ) ) ).

% all_not_in_conv
thf(fact_1174_all__not__in__conv,axiom,
    ! [A: set_complex] :
      ( ( ! [X3: complex] :
            ~ ( member_complex @ X3 @ A ) )
      = ( A = bot_bot_set_complex ) ) ).

% all_not_in_conv
thf(fact_1175_all__not__in__conv,axiom,
    ! [A: set_real] :
      ( ( ! [X3: real] :
            ~ ( member_real @ X3 @ A ) )
      = ( A = bot_bot_set_real ) ) ).

% all_not_in_conv
thf(fact_1176_all__not__in__conv,axiom,
    ! [A: set_o] :
      ( ( ! [X3: $o] :
            ~ ( member_o @ X3 @ A ) )
      = ( A = bot_bot_set_o ) ) ).

% all_not_in_conv
thf(fact_1177_all__not__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ! [X3: nat] :
            ~ ( member_nat @ X3 @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_1178_all__not__in__conv,axiom,
    ! [A: set_int] :
      ( ( ! [X3: int] :
            ~ ( member_int @ X3 @ A ) )
      = ( A = bot_bot_set_int ) ) ).

% all_not_in_conv
thf(fact_1179_empty__iff,axiom,
    ! [C2: vEBT_VEBT] :
      ~ ( member_VEBT_VEBT @ C2 @ bot_bo8194388402131092736T_VEBT ) ).

% empty_iff
thf(fact_1180_empty__iff,axiom,
    ! [C2: complex] :
      ~ ( member_complex @ C2 @ bot_bot_set_complex ) ).

% empty_iff
thf(fact_1181_empty__iff,axiom,
    ! [C2: real] :
      ~ ( member_real @ C2 @ bot_bot_set_real ) ).

% empty_iff
thf(fact_1182_empty__iff,axiom,
    ! [C2: $o] :
      ~ ( member_o @ C2 @ bot_bot_set_o ) ).

% empty_iff
thf(fact_1183_empty__iff,axiom,
    ! [C2: nat] :
      ~ ( member_nat @ C2 @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_1184_empty__iff,axiom,
    ! [C2: int] :
      ~ ( member_int @ C2 @ bot_bot_set_int ) ).

% empty_iff
thf(fact_1185_subsetI,axiom,
    ! [A: set_real,B: set_real] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A )
         => ( member_real @ X4 @ B ) )
     => ( ord_less_eq_set_real @ A @ B ) ) ).

% subsetI
thf(fact_1186_subsetI,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ A )
         => ( member_VEBT_VEBT @ X4 @ B ) )
     => ( ord_le4337996190870823476T_VEBT @ A @ B ) ) ).

% subsetI
thf(fact_1187_subsetI,axiom,
    ! [A: set_int,B: set_int] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A )
         => ( member_int @ X4 @ B ) )
     => ( ord_less_eq_set_int @ A @ B ) ) ).

% subsetI
thf(fact_1188_subsetI,axiom,
    ! [A: set_complex,B: set_complex] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ A )
         => ( member_complex @ X4 @ B ) )
     => ( ord_le211207098394363844omplex @ A @ B ) ) ).

% subsetI
thf(fact_1189_subsetI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A )
         => ( member_nat @ X4 @ B ) )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% subsetI
thf(fact_1190_subset__antisym,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ A )
       => ( A = B ) ) ) ).

% subset_antisym
thf(fact_1191_insert__absorb2,axiom,
    ! [X: nat,A: set_nat] :
      ( ( insert_nat @ X @ ( insert_nat @ X @ A ) )
      = ( insert_nat @ X @ A ) ) ).

% insert_absorb2
thf(fact_1192_insert__absorb2,axiom,
    ! [X: int,A: set_int] :
      ( ( insert_int @ X @ ( insert_int @ X @ A ) )
      = ( insert_int @ X @ A ) ) ).

% insert_absorb2
thf(fact_1193_insert__absorb2,axiom,
    ! [X: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ( insert_VEBT_VEBT @ X @ ( insert_VEBT_VEBT @ X @ A ) )
      = ( insert_VEBT_VEBT @ X @ A ) ) ).

% insert_absorb2
thf(fact_1194_insert__absorb2,axiom,
    ! [X: real,A: set_real] :
      ( ( insert_real @ X @ ( insert_real @ X @ A ) )
      = ( insert_real @ X @ A ) ) ).

% insert_absorb2
thf(fact_1195_insert__absorb2,axiom,
    ! [X: $o,A: set_o] :
      ( ( insert_o @ X @ ( insert_o @ X @ A ) )
      = ( insert_o @ X @ A ) ) ).

% insert_absorb2
thf(fact_1196_insert__iff,axiom,
    ! [A2: $o,B2: $o,A: set_o] :
      ( ( member_o @ A2 @ ( insert_o @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member_o @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_1197_insert__iff,axiom,
    ! [A2: nat,B2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ ( insert_nat @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member_nat @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_1198_insert__iff,axiom,
    ! [A2: real,B2: real,A: set_real] :
      ( ( member_real @ A2 @ ( insert_real @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member_real @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_1199_insert__iff,axiom,
    ! [A2: vEBT_VEBT,B2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ A2 @ ( insert_VEBT_VEBT @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member_VEBT_VEBT @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_1200_insert__iff,axiom,
    ! [A2: int,B2: int,A: set_int] :
      ( ( member_int @ A2 @ ( insert_int @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member_int @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_1201_insert__iff,axiom,
    ! [A2: complex,B2: complex,A: set_complex] :
      ( ( member_complex @ A2 @ ( insert_complex @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member_complex @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_1202_insertCI,axiom,
    ! [A2: $o,B: set_o,B2: $o] :
      ( ( ~ ( member_o @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member_o @ A2 @ ( insert_o @ B2 @ B ) ) ) ).

% insertCI
thf(fact_1203_insertCI,axiom,
    ! [A2: nat,B: set_nat,B2: nat] :
      ( ( ~ ( member_nat @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).

% insertCI
thf(fact_1204_insertCI,axiom,
    ! [A2: real,B: set_real,B2: real] :
      ( ( ~ ( member_real @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member_real @ A2 @ ( insert_real @ B2 @ B ) ) ) ).

% insertCI
thf(fact_1205_insertCI,axiom,
    ! [A2: vEBT_VEBT,B: set_VEBT_VEBT,B2: vEBT_VEBT] :
      ( ( ~ ( member_VEBT_VEBT @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member_VEBT_VEBT @ A2 @ ( insert_VEBT_VEBT @ B2 @ B ) ) ) ).

% insertCI
thf(fact_1206_insertCI,axiom,
    ! [A2: int,B: set_int,B2: int] :
      ( ( ~ ( member_int @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member_int @ A2 @ ( insert_int @ B2 @ B ) ) ) ).

% insertCI
thf(fact_1207_insertCI,axiom,
    ! [A2: complex,B: set_complex,B2: complex] :
      ( ( ~ ( member_complex @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member_complex @ A2 @ ( insert_complex @ B2 @ B ) ) ) ).

% insertCI
thf(fact_1208_Diff__idemp,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B ) @ B )
      = ( minus_minus_set_nat @ A @ B ) ) ).

% Diff_idemp
thf(fact_1209_Diff__iff,axiom,
    ! [C2: real,A: set_real,B: set_real] :
      ( ( member_real @ C2 @ ( minus_minus_set_real @ A @ B ) )
      = ( ( member_real @ C2 @ A )
        & ~ ( member_real @ C2 @ B ) ) ) ).

% Diff_iff
thf(fact_1210_Diff__iff,axiom,
    ! [C2: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ C2 @ ( minus_5127226145743854075T_VEBT @ A @ B ) )
      = ( ( member_VEBT_VEBT @ C2 @ A )
        & ~ ( member_VEBT_VEBT @ C2 @ B ) ) ) ).

% Diff_iff
thf(fact_1211_Diff__iff,axiom,
    ! [C2: int,A: set_int,B: set_int] :
      ( ( member_int @ C2 @ ( minus_minus_set_int @ A @ B ) )
      = ( ( member_int @ C2 @ A )
        & ~ ( member_int @ C2 @ B ) ) ) ).

% Diff_iff
thf(fact_1212_Diff__iff,axiom,
    ! [C2: complex,A: set_complex,B: set_complex] :
      ( ( member_complex @ C2 @ ( minus_811609699411566653omplex @ A @ B ) )
      = ( ( member_complex @ C2 @ A )
        & ~ ( member_complex @ C2 @ B ) ) ) ).

% Diff_iff
thf(fact_1213_Diff__iff,axiom,
    ! [C2: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) )
      = ( ( member_nat @ C2 @ A )
        & ~ ( member_nat @ C2 @ B ) ) ) ).

% Diff_iff
thf(fact_1214_DiffI,axiom,
    ! [C2: real,A: set_real,B: set_real] :
      ( ( member_real @ C2 @ A )
     => ( ~ ( member_real @ C2 @ B )
       => ( member_real @ C2 @ ( minus_minus_set_real @ A @ B ) ) ) ) ).

% DiffI
thf(fact_1215_DiffI,axiom,
    ! [C2: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ C2 @ A )
     => ( ~ ( member_VEBT_VEBT @ C2 @ B )
       => ( member_VEBT_VEBT @ C2 @ ( minus_5127226145743854075T_VEBT @ A @ B ) ) ) ) ).

% DiffI
thf(fact_1216_DiffI,axiom,
    ! [C2: int,A: set_int,B: set_int] :
      ( ( member_int @ C2 @ A )
     => ( ~ ( member_int @ C2 @ B )
       => ( member_int @ C2 @ ( minus_minus_set_int @ A @ B ) ) ) ) ).

% DiffI
thf(fact_1217_DiffI,axiom,
    ! [C2: complex,A: set_complex,B: set_complex] :
      ( ( member_complex @ C2 @ A )
     => ( ~ ( member_complex @ C2 @ B )
       => ( member_complex @ C2 @ ( minus_811609699411566653omplex @ A @ B ) ) ) ) ).

% DiffI
thf(fact_1218_DiffI,axiom,
    ! [C2: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C2 @ A )
     => ( ~ ( member_nat @ C2 @ B )
       => ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).

% DiffI
thf(fact_1219_pure__assn__eq__conv,axiom,
    ! [P: $o,Q: $o] :
      ( ( ( pure_assn @ P )
        = ( pure_assn @ Q ) )
      = ( P = Q ) ) ).

% pure_assn_eq_conv
thf(fact_1220_mult__cancel__right,axiom,
    ! [A2: complex,C2: complex,B2: complex] :
      ( ( ( times_times_complex @ A2 @ C2 )
        = ( times_times_complex @ B2 @ C2 ) )
      = ( ( C2 = zero_zero_complex )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
thf(fact_1221_mult__cancel__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( ( times_times_real @ A2 @ C2 )
        = ( times_times_real @ B2 @ C2 ) )
      = ( ( C2 = zero_zero_real )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
thf(fact_1222_mult__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ( times_times_rat @ A2 @ C2 )
        = ( times_times_rat @ B2 @ C2 ) )
      = ( ( C2 = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
thf(fact_1223_mult__cancel__right,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( ( times_times_nat @ A2 @ C2 )
        = ( times_times_nat @ B2 @ C2 ) )
      = ( ( C2 = zero_zero_nat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
thf(fact_1224_mult__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ( times_times_int @ A2 @ C2 )
        = ( times_times_int @ B2 @ C2 ) )
      = ( ( C2 = zero_zero_int )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_right
thf(fact_1225_mult__cancel__left,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( ( times_times_complex @ C2 @ A2 )
        = ( times_times_complex @ C2 @ B2 ) )
      = ( ( C2 = zero_zero_complex )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
thf(fact_1226_mult__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ( times_times_real @ C2 @ A2 )
        = ( times_times_real @ C2 @ B2 ) )
      = ( ( C2 = zero_zero_real )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
thf(fact_1227_mult__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ( times_times_rat @ C2 @ A2 )
        = ( times_times_rat @ C2 @ B2 ) )
      = ( ( C2 = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
thf(fact_1228_mult__cancel__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ( times_times_nat @ C2 @ A2 )
        = ( times_times_nat @ C2 @ B2 ) )
      = ( ( C2 = zero_zero_nat )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
thf(fact_1229_mult__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ( times_times_int @ C2 @ A2 )
        = ( times_times_int @ C2 @ B2 ) )
      = ( ( C2 = zero_zero_int )
        | ( A2 = B2 ) ) ) ).

% mult_cancel_left
thf(fact_1230_mult__eq__0__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ( times_times_complex @ A2 @ B2 )
        = zero_zero_complex )
      = ( ( A2 = zero_zero_complex )
        | ( B2 = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
thf(fact_1231_mult__eq__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ( times_times_real @ A2 @ B2 )
        = zero_zero_real )
      = ( ( A2 = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% mult_eq_0_iff
thf(fact_1232_mult__eq__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( times_times_rat @ A2 @ B2 )
        = zero_zero_rat )
      = ( ( A2 = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
thf(fact_1233_mult__eq__0__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( times_times_nat @ A2 @ B2 )
        = zero_zero_nat )
      = ( ( A2 = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
thf(fact_1234_mult__eq__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ( times_times_int @ A2 @ B2 )
        = zero_zero_int )
      = ( ( A2 = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% mult_eq_0_iff
thf(fact_1235_mult__zero__right,axiom,
    ! [A2: complex] :
      ( ( times_times_complex @ A2 @ zero_zero_complex )
      = zero_zero_complex ) ).

% mult_zero_right
thf(fact_1236_mult__zero__right,axiom,
    ! [A2: real] :
      ( ( times_times_real @ A2 @ zero_zero_real )
      = zero_zero_real ) ).

% mult_zero_right
thf(fact_1237_mult__zero__right,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ A2 @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_1238_mult__zero__right,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ A2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_1239_mult__zero__right,axiom,
    ! [A2: int] :
      ( ( times_times_int @ A2 @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_1240_mult__zero__left,axiom,
    ! [A2: complex] :
      ( ( times_times_complex @ zero_zero_complex @ A2 )
      = zero_zero_complex ) ).

% mult_zero_left
thf(fact_1241_mult__zero__left,axiom,
    ! [A2: real] :
      ( ( times_times_real @ zero_zero_real @ A2 )
      = zero_zero_real ) ).

% mult_zero_left
thf(fact_1242_mult__zero__left,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A2 )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_1243_mult__zero__left,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_1244_mult__zero__left,axiom,
    ! [A2: int] :
      ( ( times_times_int @ zero_zero_int @ A2 )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_1245_division__ring__divide__zero,axiom,
    ! [A2: complex] :
      ( ( divide1717551699836669952omplex @ A2 @ zero_zero_complex )
      = zero_zero_complex ) ).

% division_ring_divide_zero
thf(fact_1246_division__ring__divide__zero,axiom,
    ! [A2: real] :
      ( ( divide_divide_real @ A2 @ zero_zero_real )
      = zero_zero_real ) ).

% division_ring_divide_zero
thf(fact_1247_division__ring__divide__zero,axiom,
    ! [A2: rat] :
      ( ( divide_divide_rat @ A2 @ zero_zero_rat )
      = zero_zero_rat ) ).

% division_ring_divide_zero
thf(fact_1248_divide__cancel__right,axiom,
    ! [A2: complex,C2: complex,B2: complex] :
      ( ( ( divide1717551699836669952omplex @ A2 @ C2 )
        = ( divide1717551699836669952omplex @ B2 @ C2 ) )
      = ( ( C2 = zero_zero_complex )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_right
thf(fact_1249_divide__cancel__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( ( divide_divide_real @ A2 @ C2 )
        = ( divide_divide_real @ B2 @ C2 ) )
      = ( ( C2 = zero_zero_real )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_right
thf(fact_1250_divide__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ( divide_divide_rat @ A2 @ C2 )
        = ( divide_divide_rat @ B2 @ C2 ) )
      = ( ( C2 = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_right
thf(fact_1251_divide__cancel__left,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( ( divide1717551699836669952omplex @ C2 @ A2 )
        = ( divide1717551699836669952omplex @ C2 @ B2 ) )
      = ( ( C2 = zero_zero_complex )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_left
thf(fact_1252_divide__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ( divide_divide_real @ C2 @ A2 )
        = ( divide_divide_real @ C2 @ B2 ) )
      = ( ( C2 = zero_zero_real )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_left
thf(fact_1253_divide__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ( divide_divide_rat @ C2 @ A2 )
        = ( divide_divide_rat @ C2 @ B2 ) )
      = ( ( C2 = zero_zero_rat )
        | ( A2 = B2 ) ) ) ).

% divide_cancel_left
thf(fact_1254_div__by__0,axiom,
    ! [A2: complex] :
      ( ( divide1717551699836669952omplex @ A2 @ zero_zero_complex )
      = zero_zero_complex ) ).

% div_by_0
thf(fact_1255_div__by__0,axiom,
    ! [A2: real] :
      ( ( divide_divide_real @ A2 @ zero_zero_real )
      = zero_zero_real ) ).

% div_by_0
thf(fact_1256_div__by__0,axiom,
    ! [A2: rat] :
      ( ( divide_divide_rat @ A2 @ zero_zero_rat )
      = zero_zero_rat ) ).

% div_by_0
thf(fact_1257_div__by__0,axiom,
    ! [A2: nat] :
      ( ( divide_divide_nat @ A2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% div_by_0
thf(fact_1258_div__by__0,axiom,
    ! [A2: int] :
      ( ( divide_divide_int @ A2 @ zero_zero_int )
      = zero_zero_int ) ).

% div_by_0
thf(fact_1259_divide__eq__0__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ( divide1717551699836669952omplex @ A2 @ B2 )
        = zero_zero_complex )
      = ( ( A2 = zero_zero_complex )
        | ( B2 = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
thf(fact_1260_divide__eq__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ( divide_divide_real @ A2 @ B2 )
        = zero_zero_real )
      = ( ( A2 = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% divide_eq_0_iff
thf(fact_1261_divide__eq__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( divide_divide_rat @ A2 @ B2 )
        = zero_zero_rat )
      = ( ( A2 = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
thf(fact_1262_div__0,axiom,
    ! [A2: complex] :
      ( ( divide1717551699836669952omplex @ zero_zero_complex @ A2 )
      = zero_zero_complex ) ).

% div_0
thf(fact_1263_div__0,axiom,
    ! [A2: real] :
      ( ( divide_divide_real @ zero_zero_real @ A2 )
      = zero_zero_real ) ).

% div_0
thf(fact_1264_div__0,axiom,
    ! [A2: rat] :
      ( ( divide_divide_rat @ zero_zero_rat @ A2 )
      = zero_zero_rat ) ).

% div_0
thf(fact_1265_div__0,axiom,
    ! [A2: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% div_0
thf(fact_1266_div__0,axiom,
    ! [A2: int] :
      ( ( divide_divide_int @ zero_zero_int @ A2 )
      = zero_zero_int ) ).

% div_0
thf(fact_1267_times__divide__eq__right,axiom,
    ! [A2: complex,B2: complex,C2: complex] :
      ( ( times_times_complex @ A2 @ ( divide1717551699836669952omplex @ B2 @ C2 ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ A2 @ B2 ) @ C2 ) ) ).

% times_divide_eq_right
thf(fact_1268_times__divide__eq__right,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( times_times_real @ A2 @ ( divide_divide_real @ B2 @ C2 ) )
      = ( divide_divide_real @ ( times_times_real @ A2 @ B2 ) @ C2 ) ) ).

% times_divide_eq_right
thf(fact_1269_times__divide__eq__right,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( times_times_rat @ A2 @ ( divide_divide_rat @ B2 @ C2 ) )
      = ( divide_divide_rat @ ( times_times_rat @ A2 @ B2 ) @ C2 ) ) ).

% times_divide_eq_right
thf(fact_1270_divide__divide__eq__right,axiom,
    ! [A2: complex,B2: complex,C2: complex] :
      ( ( divide1717551699836669952omplex @ A2 @ ( divide1717551699836669952omplex @ B2 @ C2 ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ A2 @ C2 ) @ B2 ) ) ).

% divide_divide_eq_right
thf(fact_1271_divide__divide__eq__right,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( divide_divide_real @ A2 @ ( divide_divide_real @ B2 @ C2 ) )
      = ( divide_divide_real @ ( times_times_real @ A2 @ C2 ) @ B2 ) ) ).

% divide_divide_eq_right
thf(fact_1272_divide__divide__eq__right,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( divide_divide_rat @ A2 @ ( divide_divide_rat @ B2 @ C2 ) )
      = ( divide_divide_rat @ ( times_times_rat @ A2 @ C2 ) @ B2 ) ) ).

% divide_divide_eq_right
thf(fact_1273_divide__divide__eq__left,axiom,
    ! [A2: complex,B2: complex,C2: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A2 @ B2 ) @ C2 )
      = ( divide1717551699836669952omplex @ A2 @ ( times_times_complex @ B2 @ C2 ) ) ) ).

% divide_divide_eq_left
thf(fact_1274_divide__divide__eq__left,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ A2 @ B2 ) @ C2 )
      = ( divide_divide_real @ A2 @ ( times_times_real @ B2 @ C2 ) ) ) ).

% divide_divide_eq_left
thf(fact_1275_divide__divide__eq__left,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A2 @ B2 ) @ C2 )
      = ( divide_divide_rat @ A2 @ ( times_times_rat @ B2 @ C2 ) ) ) ).

% divide_divide_eq_left
thf(fact_1276_times__divide__eq__left,axiom,
    ! [B2: complex,C2: complex,A2: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ B2 @ C2 ) @ A2 )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ B2 @ A2 ) @ C2 ) ) ).

% times_divide_eq_left
thf(fact_1277_times__divide__eq__left,axiom,
    ! [B2: real,C2: real,A2: real] :
      ( ( times_times_real @ ( divide_divide_real @ B2 @ C2 ) @ A2 )
      = ( divide_divide_real @ ( times_times_real @ B2 @ A2 ) @ C2 ) ) ).

% times_divide_eq_left
thf(fact_1278_times__divide__eq__left,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ B2 @ C2 ) @ A2 )
      = ( divide_divide_rat @ ( times_times_rat @ B2 @ A2 ) @ C2 ) ) ).

% times_divide_eq_left
thf(fact_1279_empty__subsetI,axiom,
    ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).

% empty_subsetI
thf(fact_1280_empty__subsetI,axiom,
    ! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).

% empty_subsetI
thf(fact_1281_empty__subsetI,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).

% empty_subsetI
thf(fact_1282_empty__subsetI,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% empty_subsetI
thf(fact_1283_subset__empty,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
      = ( A = bot_bot_set_real ) ) ).

% subset_empty
thf(fact_1284_subset__empty,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
      = ( A = bot_bot_set_o ) ) ).

% subset_empty
thf(fact_1285_subset__empty,axiom,
    ! [A: set_int] :
      ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
      = ( A = bot_bot_set_int ) ) ).

% subset_empty
thf(fact_1286_subset__empty,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
      = ( A = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_1287_singletonI,axiom,
    ! [A2: vEBT_VEBT] : ( member_VEBT_VEBT @ A2 @ ( insert_VEBT_VEBT @ A2 @ bot_bo8194388402131092736T_VEBT ) ) ).

% singletonI
thf(fact_1288_singletonI,axiom,
    ! [A2: complex] : ( member_complex @ A2 @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ).

% singletonI
thf(fact_1289_singletonI,axiom,
    ! [A2: real] : ( member_real @ A2 @ ( insert_real @ A2 @ bot_bot_set_real ) ) ).

% singletonI
thf(fact_1290_singletonI,axiom,
    ! [A2: $o] : ( member_o @ A2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) ).

% singletonI
thf(fact_1291_singletonI,axiom,
    ! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_1292_singletonI,axiom,
    ! [A2: int] : ( member_int @ A2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) ).

% singletonI
thf(fact_1293_insert__subset,axiom,
    ! [X: $o,A: set_o,B: set_o] :
      ( ( ord_less_eq_set_o @ ( insert_o @ X @ A ) @ B )
      = ( ( member_o @ X @ B )
        & ( ord_less_eq_set_o @ A @ B ) ) ) ).

% insert_subset
thf(fact_1294_insert__subset,axiom,
    ! [X: real,A: set_real,B: set_real] :
      ( ( ord_less_eq_set_real @ ( insert_real @ X @ A ) @ B )
      = ( ( member_real @ X @ B )
        & ( ord_less_eq_set_real @ A @ B ) ) ) ).

% insert_subset
thf(fact_1295_insert__subset,axiom,
    ! [X: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( insert_VEBT_VEBT @ X @ A ) @ B )
      = ( ( member_VEBT_VEBT @ X @ B )
        & ( ord_le4337996190870823476T_VEBT @ A @ B ) ) ) ).

% insert_subset
thf(fact_1296_insert__subset,axiom,
    ! [X: int,A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ ( insert_int @ X @ A ) @ B )
      = ( ( member_int @ X @ B )
        & ( ord_less_eq_set_int @ A @ B ) ) ) ).

% insert_subset
thf(fact_1297_insert__subset,axiom,
    ! [X: complex,A: set_complex,B: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( insert_complex @ X @ A ) @ B )
      = ( ( member_complex @ X @ B )
        & ( ord_le211207098394363844omplex @ A @ B ) ) ) ).

% insert_subset
thf(fact_1298_insert__subset,axiom,
    ! [X: nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A ) @ B )
      = ( ( member_nat @ X @ B )
        & ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% insert_subset
thf(fact_1299_Diff__cancel,axiom,
    ! [A: set_real] :
      ( ( minus_minus_set_real @ A @ A )
      = bot_bot_set_real ) ).

% Diff_cancel
thf(fact_1300_Diff__cancel,axiom,
    ! [A: set_o] :
      ( ( minus_minus_set_o @ A @ A )
      = bot_bot_set_o ) ).

% Diff_cancel
thf(fact_1301_Diff__cancel,axiom,
    ! [A: set_int] :
      ( ( minus_minus_set_int @ A @ A )
      = bot_bot_set_int ) ).

% Diff_cancel
thf(fact_1302_Diff__cancel,axiom,
    ! [A: set_nat] :
      ( ( minus_minus_set_nat @ A @ A )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_1303_empty__Diff,axiom,
    ! [A: set_real] :
      ( ( minus_minus_set_real @ bot_bot_set_real @ A )
      = bot_bot_set_real ) ).

% empty_Diff
thf(fact_1304_empty__Diff,axiom,
    ! [A: set_o] :
      ( ( minus_minus_set_o @ bot_bot_set_o @ A )
      = bot_bot_set_o ) ).

% empty_Diff
thf(fact_1305_empty__Diff,axiom,
    ! [A: set_int] :
      ( ( minus_minus_set_int @ bot_bot_set_int @ A )
      = bot_bot_set_int ) ).

% empty_Diff
thf(fact_1306_empty__Diff,axiom,
    ! [A: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_1307_Diff__empty,axiom,
    ! [A: set_real] :
      ( ( minus_minus_set_real @ A @ bot_bot_set_real )
      = A ) ).

% Diff_empty
thf(fact_1308_Diff__empty,axiom,
    ! [A: set_o] :
      ( ( minus_minus_set_o @ A @ bot_bot_set_o )
      = A ) ).

% Diff_empty
thf(fact_1309_Diff__empty,axiom,
    ! [A: set_int] :
      ( ( minus_minus_set_int @ A @ bot_bot_set_int )
      = A ) ).

% Diff_empty
thf(fact_1310_Diff__empty,axiom,
    ! [A: set_nat] :
      ( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
      = A ) ).

% Diff_empty
thf(fact_1311_insert__Diff1,axiom,
    ! [X: $o,B: set_o,A: set_o] :
      ( ( member_o @ X @ B )
     => ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ B )
        = ( minus_minus_set_o @ A @ B ) ) ) ).

% insert_Diff1
thf(fact_1312_insert__Diff1,axiom,
    ! [X: real,B: set_real,A: set_real] :
      ( ( member_real @ X @ B )
     => ( ( minus_minus_set_real @ ( insert_real @ X @ A ) @ B )
        = ( minus_minus_set_real @ A @ B ) ) ) ).

% insert_Diff1
thf(fact_1313_insert__Diff1,axiom,
    ! [X: vEBT_VEBT,B: set_VEBT_VEBT,A: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ B )
     => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X @ A ) @ B )
        = ( minus_5127226145743854075T_VEBT @ A @ B ) ) ) ).

% insert_Diff1
thf(fact_1314_insert__Diff1,axiom,
    ! [X: int,B: set_int,A: set_int] :
      ( ( member_int @ X @ B )
     => ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ B )
        = ( minus_minus_set_int @ A @ B ) ) ) ).

% insert_Diff1
thf(fact_1315_insert__Diff1,axiom,
    ! [X: complex,B: set_complex,A: set_complex] :
      ( ( member_complex @ X @ B )
     => ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A ) @ B )
        = ( minus_811609699411566653omplex @ A @ B ) ) ) ).

% insert_Diff1
thf(fact_1316_insert__Diff1,axiom,
    ! [X: nat,B: set_nat,A: set_nat] :
      ( ( member_nat @ X @ B )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B )
        = ( minus_minus_set_nat @ A @ B ) ) ) ).

% insert_Diff1
thf(fact_1317_Diff__insert0,axiom,
    ! [X: $o,A: set_o,B: set_o] :
      ( ~ ( member_o @ X @ A )
     => ( ( minus_minus_set_o @ A @ ( insert_o @ X @ B ) )
        = ( minus_minus_set_o @ A @ B ) ) ) ).

% Diff_insert0
thf(fact_1318_Diff__insert0,axiom,
    ! [X: real,A: set_real,B: set_real] :
      ( ~ ( member_real @ X @ A )
     => ( ( minus_minus_set_real @ A @ ( insert_real @ X @ B ) )
        = ( minus_minus_set_real @ A @ B ) ) ) ).

% Diff_insert0
thf(fact_1319_Diff__insert0,axiom,
    ! [X: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X @ A )
     => ( ( minus_5127226145743854075T_VEBT @ A @ ( insert_VEBT_VEBT @ X @ B ) )
        = ( minus_5127226145743854075T_VEBT @ A @ B ) ) ) ).

% Diff_insert0
thf(fact_1320_Diff__insert0,axiom,
    ! [X: int,A: set_int,B: set_int] :
      ( ~ ( member_int @ X @ A )
     => ( ( minus_minus_set_int @ A @ ( insert_int @ X @ B ) )
        = ( minus_minus_set_int @ A @ B ) ) ) ).

% Diff_insert0
thf(fact_1321_Diff__insert0,axiom,
    ! [X: complex,A: set_complex,B: set_complex] :
      ( ~ ( member_complex @ X @ A )
     => ( ( minus_811609699411566653omplex @ A @ ( insert_complex @ X @ B ) )
        = ( minus_811609699411566653omplex @ A @ B ) ) ) ).

% Diff_insert0
thf(fact_1322_Diff__insert0,axiom,
    ! [X: nat,A: set_nat,B: set_nat] :
      ( ~ ( member_nat @ X @ A )
     => ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ B ) )
        = ( minus_minus_set_nat @ A @ B ) ) ) ).

% Diff_insert0
thf(fact_1323_psubsetI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_nat @ A @ B ) ) ) ).

% psubsetI
thf(fact_1324_merge__pure__star,axiom,
    ! [A2: $o,B2: $o] :
      ( ( times_times_assn @ ( pure_assn @ A2 ) @ ( pure_assn @ B2 ) )
      = ( pure_assn
        @ ( A2
          & B2 ) ) ) ).

% merge_pure_star
thf(fact_1325_le__add__diff__inverse2,axiom,
    ! [B2: real,A2: real] :
      ( ( ord_less_eq_real @ B2 @ A2 )
     => ( ( plus_plus_real @ ( minus_minus_real @ A2 @ B2 ) @ B2 )
        = A2 ) ) ).

% le_add_diff_inverse2
thf(fact_1326_le__add__diff__inverse2,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B2 ) @ B2 )
        = A2 ) ) ).

% le_add_diff_inverse2
thf(fact_1327_le__add__diff__inverse2,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ B2 )
        = A2 ) ) ).

% le_add_diff_inverse2
thf(fact_1328_le__add__diff__inverse2,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ B2 )
        = A2 ) ) ).

% le_add_diff_inverse2
thf(fact_1329_le__add__diff__inverse,axiom,
    ! [B2: real,A2: real] :
      ( ( ord_less_eq_real @ B2 @ A2 )
     => ( ( plus_plus_real @ B2 @ ( minus_minus_real @ A2 @ B2 ) )
        = A2 ) ) ).

% le_add_diff_inverse
thf(fact_1330_le__add__diff__inverse,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( plus_plus_rat @ B2 @ ( minus_minus_rat @ A2 @ B2 ) )
        = A2 ) ) ).

% le_add_diff_inverse
thf(fact_1331_le__add__diff__inverse,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A2 @ B2 ) )
        = A2 ) ) ).

% le_add_diff_inverse
thf(fact_1332_le__add__diff__inverse,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A2 @ B2 ) )
        = A2 ) ) ).

% le_add_diff_inverse
thf(fact_1333_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A2 @ C2 ) @ ( times_times_complex @ C2 @ B2 ) )
        = ( divide1717551699836669952omplex @ A2 @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_1334_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ C2 @ B2 ) )
        = ( divide_divide_real @ A2 @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_1335_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ C2 @ B2 ) )
        = ( divide_divide_rat @ A2 @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_1336_nonzero__mult__div__cancel__right,axiom,
    ! [B2: complex,A2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A2 @ B2 ) @ B2 )
        = A2 ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1337_nonzero__mult__div__cancel__right,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A2 @ B2 ) @ B2 )
        = A2 ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1338_nonzero__mult__div__cancel__right,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A2 @ B2 ) @ B2 )
        = A2 ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1339_nonzero__mult__div__cancel__right,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B2 ) @ B2 )
        = A2 ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1340_nonzero__mult__div__cancel__right,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A2 @ B2 ) @ B2 )
        = A2 ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_1341_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A2 @ C2 ) @ ( times_times_complex @ B2 @ C2 ) )
        = ( divide1717551699836669952omplex @ A2 @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_1342_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
        = ( divide_divide_real @ A2 @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_1343_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) )
        = ( divide_divide_rat @ A2 @ B2 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_1344_singleton__insert__inj__eq_H,axiom,
    ! [A2: vEBT_VEBT,A: set_VEBT_VEBT,B2: vEBT_VEBT] :
      ( ( ( insert_VEBT_VEBT @ A2 @ A )
        = ( insert_VEBT_VEBT @ B2 @ bot_bo8194388402131092736T_VEBT ) )
      = ( ( A2 = B2 )
        & ( ord_le4337996190870823476T_VEBT @ A @ ( insert_VEBT_VEBT @ B2 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1345_singleton__insert__inj__eq_H,axiom,
    ! [A2: real,A: set_real,B2: real] :
      ( ( ( insert_real @ A2 @ A )
        = ( insert_real @ B2 @ bot_bot_set_real ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_real @ A @ ( insert_real @ B2 @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1346_singleton__insert__inj__eq_H,axiom,
    ! [A2: $o,A: set_o,B2: $o] :
      ( ( ( insert_o @ A2 @ A )
        = ( insert_o @ B2 @ bot_bot_set_o ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_o @ A @ ( insert_o @ B2 @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1347_singleton__insert__inj__eq_H,axiom,
    ! [A2: int,A: set_int,B2: int] :
      ( ( ( insert_int @ A2 @ A )
        = ( insert_int @ B2 @ bot_bot_set_int ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_int @ A @ ( insert_int @ B2 @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1348_singleton__insert__inj__eq_H,axiom,
    ! [A2: nat,A: set_nat,B2: nat] :
      ( ( ( insert_nat @ A2 @ A )
        = ( insert_nat @ B2 @ bot_bot_set_nat ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1349_singleton__insert__inj__eq,axiom,
    ! [B2: vEBT_VEBT,A2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ( ( insert_VEBT_VEBT @ B2 @ bot_bo8194388402131092736T_VEBT )
        = ( insert_VEBT_VEBT @ A2 @ A ) )
      = ( ( A2 = B2 )
        & ( ord_le4337996190870823476T_VEBT @ A @ ( insert_VEBT_VEBT @ B2 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1350_singleton__insert__inj__eq,axiom,
    ! [B2: real,A2: real,A: set_real] :
      ( ( ( insert_real @ B2 @ bot_bot_set_real )
        = ( insert_real @ A2 @ A ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_real @ A @ ( insert_real @ B2 @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1351_singleton__insert__inj__eq,axiom,
    ! [B2: $o,A2: $o,A: set_o] :
      ( ( ( insert_o @ B2 @ bot_bot_set_o )
        = ( insert_o @ A2 @ A ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_o @ A @ ( insert_o @ B2 @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1352_singleton__insert__inj__eq,axiom,
    ! [B2: int,A2: int,A: set_int] :
      ( ( ( insert_int @ B2 @ bot_bot_set_int )
        = ( insert_int @ A2 @ A ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_int @ A @ ( insert_int @ B2 @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1353_singleton__insert__inj__eq,axiom,
    ! [B2: nat,A2: nat,A: set_nat] :
      ( ( ( insert_nat @ B2 @ bot_bot_set_nat )
        = ( insert_nat @ A2 @ A ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1354_Diff__eq__empty__iff,axiom,
    ! [A: set_real,B: set_real] :
      ( ( ( minus_minus_set_real @ A @ B )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ A @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_1355_Diff__eq__empty__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ( minus_minus_set_o @ A @ B )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ A @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_1356_Diff__eq__empty__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( minus_minus_set_int @ A @ B )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_1357_Diff__eq__empty__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( minus_minus_set_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A @ B ) ) ).

% Diff_eq_empty_iff
thf(fact_1358_in__mono,axiom,
    ! [A: set_real,B: set_real,X: real] :
      ( ( ord_less_eq_set_real @ A @ B )
     => ( ( member_real @ X @ A )
       => ( member_real @ X @ B ) ) ) ).

% in_mono
thf(fact_1359_in__mono,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A @ B )
     => ( ( member_VEBT_VEBT @ X @ A )
       => ( member_VEBT_VEBT @ X @ B ) ) ) ).

% in_mono
thf(fact_1360_in__mono,axiom,
    ! [A: set_int,B: set_int,X: int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( member_int @ X @ A )
       => ( member_int @ X @ B ) ) ) ).

% in_mono
thf(fact_1361_in__mono,axiom,
    ! [A: set_complex,B: set_complex,X: complex] :
      ( ( ord_le211207098394363844omplex @ A @ B )
     => ( ( member_complex @ X @ A )
       => ( member_complex @ X @ B ) ) ) ).

% in_mono
thf(fact_1362_in__mono,axiom,
    ! [A: set_nat,B: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( member_nat @ X @ A )
       => ( member_nat @ X @ B ) ) ) ).

% in_mono
thf(fact_1363_subsetD,axiom,
    ! [A: set_real,B: set_real,C2: real] :
      ( ( ord_less_eq_set_real @ A @ B )
     => ( ( member_real @ C2 @ A )
       => ( member_real @ C2 @ B ) ) ) ).

% subsetD
thf(fact_1364_subsetD,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT,C2: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A @ B )
     => ( ( member_VEBT_VEBT @ C2 @ A )
       => ( member_VEBT_VEBT @ C2 @ B ) ) ) ).

% subsetD
thf(fact_1365_subsetD,axiom,
    ! [A: set_int,B: set_int,C2: int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( member_int @ C2 @ A )
       => ( member_int @ C2 @ B ) ) ) ).

% subsetD
thf(fact_1366_subsetD,axiom,
    ! [A: set_complex,B: set_complex,C2: complex] :
      ( ( ord_le211207098394363844omplex @ A @ B )
     => ( ( member_complex @ C2 @ A )
       => ( member_complex @ C2 @ B ) ) ) ).

% subsetD
thf(fact_1367_subsetD,axiom,
    ! [A: set_nat,B: set_nat,C2: nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( member_nat @ C2 @ A )
       => ( member_nat @ C2 @ B ) ) ) ).

% subsetD
thf(fact_1368_psubsetD,axiom,
    ! [A: set_nat,B: set_nat,C2: nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ( member_nat @ C2 @ A )
       => ( member_nat @ C2 @ B ) ) ) ).

% psubsetD
thf(fact_1369_psubsetD,axiom,
    ! [A: set_real,B: set_real,C2: real] :
      ( ( ord_less_set_real @ A @ B )
     => ( ( member_real @ C2 @ A )
       => ( member_real @ C2 @ B ) ) ) ).

% psubsetD
thf(fact_1370_psubsetD,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT,C2: vEBT_VEBT] :
      ( ( ord_le3480810397992357184T_VEBT @ A @ B )
     => ( ( member_VEBT_VEBT @ C2 @ A )
       => ( member_VEBT_VEBT @ C2 @ B ) ) ) ).

% psubsetD
thf(fact_1371_psubsetD,axiom,
    ! [A: set_int,B: set_int,C2: int] :
      ( ( ord_less_set_int @ A @ B )
     => ( ( member_int @ C2 @ A )
       => ( member_int @ C2 @ B ) ) ) ).

% psubsetD
thf(fact_1372_psubsetD,axiom,
    ! [A: set_complex,B: set_complex,C2: complex] :
      ( ( ord_less_set_complex @ A @ B )
     => ( ( member_complex @ C2 @ A )
       => ( member_complex @ C2 @ B ) ) ) ).

% psubsetD
thf(fact_1373_psubsetE,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ~ ( ( ord_less_eq_set_nat @ A @ B )
         => ( ord_less_eq_set_nat @ B @ A ) ) ) ).

% psubsetE
thf(fact_1374_equalityE,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
     => ~ ( ( ord_less_eq_set_nat @ A @ B )
         => ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).

% equalityE
thf(fact_1375_subset__eq,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B5: set_real] :
        ! [X3: real] :
          ( ( member_real @ X3 @ A6 )
         => ( member_real @ X3 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_1376_subset__eq,axiom,
    ( ord_le4337996190870823476T_VEBT
    = ( ^ [A6: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
        ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A6 )
         => ( member_VEBT_VEBT @ X3 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_1377_subset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B5: set_int] :
        ! [X3: int] :
          ( ( member_int @ X3 @ A6 )
         => ( member_int @ X3 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_1378_subset__eq,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A6: set_complex,B5: set_complex] :
        ! [X3: complex] :
          ( ( member_complex @ X3 @ A6 )
         => ( member_complex @ X3 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_1379_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B5: set_nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ A6 )
         => ( member_nat @ X3 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_1380_equalityD1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% equalityD1
thf(fact_1381_Set_OequalityD2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = B )
     => ( ord_less_eq_set_nat @ B @ A ) ) ).

% Set.equalityD2
thf(fact_1382_psubset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A6: set_nat,B5: set_nat] :
          ( ( ord_less_eq_set_nat @ A6 @ B5 )
          & ( A6 != B5 ) ) ) ) ).

% psubset_eq
thf(fact_1383_subset__iff,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B5: set_real] :
        ! [T2: real] :
          ( ( member_real @ T2 @ A6 )
         => ( member_real @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_1384_subset__iff,axiom,
    ( ord_le4337996190870823476T_VEBT
    = ( ^ [A6: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
        ! [T2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ T2 @ A6 )
         => ( member_VEBT_VEBT @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_1385_subset__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B5: set_int] :
        ! [T2: int] :
          ( ( member_int @ T2 @ A6 )
         => ( member_int @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_1386_subset__iff,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A6: set_complex,B5: set_complex] :
        ! [T2: complex] :
          ( ( member_complex @ T2 @ A6 )
         => ( member_complex @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_1387_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B5: set_nat] :
        ! [T2: nat] :
          ( ( member_nat @ T2 @ A6 )
         => ( member_nat @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_1388_subset__refl,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% subset_refl
thf(fact_1389_Collect__mono,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X4: int] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).

% Collect_mono
thf(fact_1390_Collect__mono,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ! [X4: complex] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).

% Collect_mono
thf(fact_1391_Collect__mono,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ! [X4: product_prod_int_int] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) ) ) ).

% Collect_mono
thf(fact_1392_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X4: nat] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_1393_subset__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% subset_trans
thf(fact_1394_set__eq__subset,axiom,
    ( ( ^ [Y5: set_nat,Z3: set_nat] : Y5 = Z3 )
    = ( ^ [A6: set_nat,B5: set_nat] :
          ( ( ord_less_eq_set_nat @ A6 @ B5 )
          & ( ord_less_eq_set_nat @ B5 @ A6 ) ) ) ) ).

% set_eq_subset
thf(fact_1395_Collect__mono__iff,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
      = ( ! [X3: int] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1396_Collect__mono__iff,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
      = ( ! [X3: complex] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1397_Collect__mono__iff,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) )
      = ( ! [X3: product_prod_int_int] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1398_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1399_psubset__imp__subset,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% psubset_imp_subset
thf(fact_1400_subset__Collect__conv,axiom,
    ! [S: set_int,P: int > $o] :
      ( ( ord_less_eq_set_int @ S @ ( collect_int @ P ) )
      = ( ! [X3: int] :
            ( ( member_int @ X3 @ S )
           => ( P @ X3 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1401_subset__Collect__conv,axiom,
    ! [S: set_complex,P: complex > $o] :
      ( ( ord_le211207098394363844omplex @ S @ ( collect_complex @ P ) )
      = ( ! [X3: complex] :
            ( ( member_complex @ X3 @ S )
           => ( P @ X3 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1402_subset__Collect__conv,axiom,
    ! [S: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
      ( ( ord_le2843351958646193337nt_int @ S @ ( collec213857154873943460nt_int @ P ) )
      = ( ! [X3: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X3 @ S )
           => ( P @ X3 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1403_subset__Collect__conv,axiom,
    ! [S: set_nat,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ S @ ( collect_nat @ P ) )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ S )
           => ( P @ X3 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1404_psubset__subset__trans,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 ) ) ) ).

% psubset_subset_trans
thf(fact_1405_subset__not__subset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A6: set_nat,B5: set_nat] :
          ( ( ord_less_eq_set_nat @ A6 @ B5 )
          & ~ ( ord_less_eq_set_nat @ B5 @ A6 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_1406_subset__psubset__trans,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 ) ) ) ).

% subset_psubset_trans
thf(fact_1407_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B5: set_nat] :
          ( ( ord_less_set_nat @ A6 @ B5 )
          | ( A6 = B5 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_1408_diff__commute,axiom,
    ! [I: nat,J2: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J2 ) ) ).

% diff_commute
thf(fact_1409_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc4471711990508489141at_nat] :
      ~ ! [F3: nat > nat > nat,A3: nat,B3: nat,Acc: nat] :
          ( X
         != ( produc3209952032786966637at_nat @ F3 @ ( produc487386426758144856at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_1410_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_1411_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_1412_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_1413_linordered__field__no__ub,axiom,
    ! [X5: real] :
    ? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_1414_linordered__field__no__ub,axiom,
    ! [X5: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X5 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_1415_linordered__field__no__lb,axiom,
    ! [X5: real] :
    ? [Y3: real] : ( ord_less_real @ Y3 @ X5 ) ).

% linordered_field_no_lb
thf(fact_1416_linordered__field__no__lb,axiom,
    ! [X5: rat] :
    ? [Y3: rat] : ( ord_less_rat @ Y3 @ X5 ) ).

% linordered_field_no_lb
thf(fact_1417_realpow__pos__nth2,axiom,
    ! [A2: real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ? [R2: real] :
          ( ( ord_less_real @ zero_zero_real @ R2 )
          & ( ( power_power_real @ R2 @ ( suc @ N3 ) )
            = A2 ) ) ) ).

% realpow_pos_nth2
thf(fact_1418_not__psubset__empty,axiom,
    ! [A: set_real] :
      ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).

% not_psubset_empty
thf(fact_1419_not__psubset__empty,axiom,
    ! [A: set_o] :
      ~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).

% not_psubset_empty
thf(fact_1420_not__psubset__empty,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).

% not_psubset_empty
thf(fact_1421_not__psubset__empty,axiom,
    ! [A: set_int] :
      ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).

% not_psubset_empty
thf(fact_1422_ex__in__conv,axiom,
    ! [A: set_VEBT_VEBT] :
      ( ( ? [X3: vEBT_VEBT] : ( member_VEBT_VEBT @ X3 @ A ) )
      = ( A != bot_bo8194388402131092736T_VEBT ) ) ).

% ex_in_conv
thf(fact_1423_ex__in__conv,axiom,
    ! [A: set_complex] :
      ( ( ? [X3: complex] : ( member_complex @ X3 @ A ) )
      = ( A != bot_bot_set_complex ) ) ).

% ex_in_conv
thf(fact_1424_ex__in__conv,axiom,
    ! [A: set_real] :
      ( ( ? [X3: real] : ( member_real @ X3 @ A ) )
      = ( A != bot_bot_set_real ) ) ).

% ex_in_conv
thf(fact_1425_ex__in__conv,axiom,
    ! [A: set_o] :
      ( ( ? [X3: $o] : ( member_o @ X3 @ A ) )
      = ( A != bot_bot_set_o ) ) ).

% ex_in_conv
thf(fact_1426_ex__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ? [X3: nat] : ( member_nat @ X3 @ A ) )
      = ( A != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_1427_ex__in__conv,axiom,
    ! [A: set_int] :
      ( ( ? [X3: int] : ( member_int @ X3 @ A ) )
      = ( A != bot_bot_set_int ) ) ).

% ex_in_conv
thf(fact_1428_equals0I,axiom,
    ! [A: set_VEBT_VEBT] :
      ( ! [Y3: vEBT_VEBT] :
          ~ ( member_VEBT_VEBT @ Y3 @ A )
     => ( A = bot_bo8194388402131092736T_VEBT ) ) ).

% equals0I
thf(fact_1429_equals0I,axiom,
    ! [A: set_complex] :
      ( ! [Y3: complex] :
          ~ ( member_complex @ Y3 @ A )
     => ( A = bot_bot_set_complex ) ) ).

% equals0I
thf(fact_1430_equals0I,axiom,
    ! [A: set_real] :
      ( ! [Y3: real] :
          ~ ( member_real @ Y3 @ A )
     => ( A = bot_bot_set_real ) ) ).

% equals0I
thf(fact_1431_equals0I,axiom,
    ! [A: set_o] :
      ( ! [Y3: $o] :
          ~ ( member_o @ Y3 @ A )
     => ( A = bot_bot_set_o ) ) ).

% equals0I
thf(fact_1432_equals0I,axiom,
    ! [A: set_nat] :
      ( ! [Y3: nat] :
          ~ ( member_nat @ Y3 @ A )
     => ( A = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_1433_equals0I,axiom,
    ! [A: set_int] :
      ( ! [Y3: int] :
          ~ ( member_int @ Y3 @ A )
     => ( A = bot_bot_set_int ) ) ).

% equals0I
thf(fact_1434_equals0D,axiom,
    ! [A: set_VEBT_VEBT,A2: vEBT_VEBT] :
      ( ( A = bot_bo8194388402131092736T_VEBT )
     => ~ ( member_VEBT_VEBT @ A2 @ A ) ) ).

% equals0D
thf(fact_1435_equals0D,axiom,
    ! [A: set_complex,A2: complex] :
      ( ( A = bot_bot_set_complex )
     => ~ ( member_complex @ A2 @ A ) ) ).

% equals0D
thf(fact_1436_equals0D,axiom,
    ! [A: set_real,A2: real] :
      ( ( A = bot_bot_set_real )
     => ~ ( member_real @ A2 @ A ) ) ).

% equals0D
thf(fact_1437_equals0D,axiom,
    ! [A: set_o,A2: $o] :
      ( ( A = bot_bot_set_o )
     => ~ ( member_o @ A2 @ A ) ) ).

% equals0D
thf(fact_1438_equals0D,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( A = bot_bot_set_nat )
     => ~ ( member_nat @ A2 @ A ) ) ).

% equals0D
thf(fact_1439_equals0D,axiom,
    ! [A: set_int,A2: int] :
      ( ( A = bot_bot_set_int )
     => ~ ( member_int @ A2 @ A ) ) ).

% equals0D
thf(fact_1440_emptyE,axiom,
    ! [A2: vEBT_VEBT] :
      ~ ( member_VEBT_VEBT @ A2 @ bot_bo8194388402131092736T_VEBT ) ).

% emptyE
thf(fact_1441_emptyE,axiom,
    ! [A2: complex] :
      ~ ( member_complex @ A2 @ bot_bot_set_complex ) ).

% emptyE
thf(fact_1442_emptyE,axiom,
    ! [A2: real] :
      ~ ( member_real @ A2 @ bot_bot_set_real ) ).

% emptyE
thf(fact_1443_emptyE,axiom,
    ! [A2: $o] :
      ~ ( member_o @ A2 @ bot_bot_set_o ) ).

% emptyE
thf(fact_1444_emptyE,axiom,
    ! [A2: nat] :
      ~ ( member_nat @ A2 @ bot_bot_set_nat ) ).

% emptyE
thf(fact_1445_emptyE,axiom,
    ! [A2: int] :
      ~ ( member_int @ A2 @ bot_bot_set_int ) ).

% emptyE
thf(fact_1446_mk__disjoint__insert,axiom,
    ! [A2: $o,A: set_o] :
      ( ( member_o @ A2 @ A )
     => ? [B6: set_o] :
          ( ( A
            = ( insert_o @ A2 @ B6 ) )
          & ~ ( member_o @ A2 @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_1447_mk__disjoint__insert,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ A )
     => ? [B6: set_nat] :
          ( ( A
            = ( insert_nat @ A2 @ B6 ) )
          & ~ ( member_nat @ A2 @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_1448_mk__disjoint__insert,axiom,
    ! [A2: real,A: set_real] :
      ( ( member_real @ A2 @ A )
     => ? [B6: set_real] :
          ( ( A
            = ( insert_real @ A2 @ B6 ) )
          & ~ ( member_real @ A2 @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_1449_mk__disjoint__insert,axiom,
    ! [A2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ A2 @ A )
     => ? [B6: set_VEBT_VEBT] :
          ( ( A
            = ( insert_VEBT_VEBT @ A2 @ B6 ) )
          & ~ ( member_VEBT_VEBT @ A2 @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_1450_mk__disjoint__insert,axiom,
    ! [A2: int,A: set_int] :
      ( ( member_int @ A2 @ A )
     => ? [B6: set_int] :
          ( ( A
            = ( insert_int @ A2 @ B6 ) )
          & ~ ( member_int @ A2 @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_1451_mk__disjoint__insert,axiom,
    ! [A2: complex,A: set_complex] :
      ( ( member_complex @ A2 @ A )
     => ? [B6: set_complex] :
          ( ( A
            = ( insert_complex @ A2 @ B6 ) )
          & ~ ( member_complex @ A2 @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_1452_insert__commute,axiom,
    ! [X: nat,Y: nat,A: set_nat] :
      ( ( insert_nat @ X @ ( insert_nat @ Y @ A ) )
      = ( insert_nat @ Y @ ( insert_nat @ X @ A ) ) ) ).

% insert_commute
thf(fact_1453_insert__commute,axiom,
    ! [X: int,Y: int,A: set_int] :
      ( ( insert_int @ X @ ( insert_int @ Y @ A ) )
      = ( insert_int @ Y @ ( insert_int @ X @ A ) ) ) ).

% insert_commute
thf(fact_1454_insert__commute,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ( insert_VEBT_VEBT @ X @ ( insert_VEBT_VEBT @ Y @ A ) )
      = ( insert_VEBT_VEBT @ Y @ ( insert_VEBT_VEBT @ X @ A ) ) ) ).

% insert_commute
thf(fact_1455_insert__commute,axiom,
    ! [X: real,Y: real,A: set_real] :
      ( ( insert_real @ X @ ( insert_real @ Y @ A ) )
      = ( insert_real @ Y @ ( insert_real @ X @ A ) ) ) ).

% insert_commute
thf(fact_1456_insert__commute,axiom,
    ! [X: $o,Y: $o,A: set_o] :
      ( ( insert_o @ X @ ( insert_o @ Y @ A ) )
      = ( insert_o @ Y @ ( insert_o @ X @ A ) ) ) ).

% insert_commute
thf(fact_1457_insert__eq__iff,axiom,
    ! [A2: $o,A: set_o,B2: $o,B: set_o] :
      ( ~ ( member_o @ A2 @ A )
     => ( ~ ( member_o @ B2 @ B )
       => ( ( ( insert_o @ A2 @ A )
            = ( insert_o @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 = ~ B2 )
             => ? [C3: set_o] :
                  ( ( A
                    = ( insert_o @ B2 @ C3 ) )
                  & ~ ( member_o @ B2 @ C3 )
                  & ( B
                    = ( insert_o @ A2 @ C3 ) )
                  & ~ ( member_o @ A2 @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1458_insert__eq__iff,axiom,
    ! [A2: nat,A: set_nat,B2: nat,B: set_nat] :
      ( ~ ( member_nat @ A2 @ A )
     => ( ~ ( member_nat @ B2 @ B )
       => ( ( ( insert_nat @ A2 @ A )
            = ( insert_nat @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C3: set_nat] :
                  ( ( A
                    = ( insert_nat @ B2 @ C3 ) )
                  & ~ ( member_nat @ B2 @ C3 )
                  & ( B
                    = ( insert_nat @ A2 @ C3 ) )
                  & ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1459_insert__eq__iff,axiom,
    ! [A2: real,A: set_real,B2: real,B: set_real] :
      ( ~ ( member_real @ A2 @ A )
     => ( ~ ( member_real @ B2 @ B )
       => ( ( ( insert_real @ A2 @ A )
            = ( insert_real @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C3: set_real] :
                  ( ( A
                    = ( insert_real @ B2 @ C3 ) )
                  & ~ ( member_real @ B2 @ C3 )
                  & ( B
                    = ( insert_real @ A2 @ C3 ) )
                  & ~ ( member_real @ A2 @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1460_insert__eq__iff,axiom,
    ! [A2: vEBT_VEBT,A: set_VEBT_VEBT,B2: vEBT_VEBT,B: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ A2 @ A )
     => ( ~ ( member_VEBT_VEBT @ B2 @ B )
       => ( ( ( insert_VEBT_VEBT @ A2 @ A )
            = ( insert_VEBT_VEBT @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C3: set_VEBT_VEBT] :
                  ( ( A
                    = ( insert_VEBT_VEBT @ B2 @ C3 ) )
                  & ~ ( member_VEBT_VEBT @ B2 @ C3 )
                  & ( B
                    = ( insert_VEBT_VEBT @ A2 @ C3 ) )
                  & ~ ( member_VEBT_VEBT @ A2 @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1461_insert__eq__iff,axiom,
    ! [A2: int,A: set_int,B2: int,B: set_int] :
      ( ~ ( member_int @ A2 @ A )
     => ( ~ ( member_int @ B2 @ B )
       => ( ( ( insert_int @ A2 @ A )
            = ( insert_int @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C3: set_int] :
                  ( ( A
                    = ( insert_int @ B2 @ C3 ) )
                  & ~ ( member_int @ B2 @ C3 )
                  & ( B
                    = ( insert_int @ A2 @ C3 ) )
                  & ~ ( member_int @ A2 @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1462_insert__eq__iff,axiom,
    ! [A2: complex,A: set_complex,B2: complex,B: set_complex] :
      ( ~ ( member_complex @ A2 @ A )
     => ( ~ ( member_complex @ B2 @ B )
       => ( ( ( insert_complex @ A2 @ A )
            = ( insert_complex @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C3: set_complex] :
                  ( ( A
                    = ( insert_complex @ B2 @ C3 ) )
                  & ~ ( member_complex @ B2 @ C3 )
                  & ( B
                    = ( insert_complex @ A2 @ C3 ) )
                  & ~ ( member_complex @ A2 @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1463_insert__absorb,axiom,
    ! [A2: $o,A: set_o] :
      ( ( member_o @ A2 @ A )
     => ( ( insert_o @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_1464_insert__absorb,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ A )
     => ( ( insert_nat @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_1465_insert__absorb,axiom,
    ! [A2: real,A: set_real] :
      ( ( member_real @ A2 @ A )
     => ( ( insert_real @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_1466_insert__absorb,axiom,
    ! [A2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ A2 @ A )
     => ( ( insert_VEBT_VEBT @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_1467_insert__absorb,axiom,
    ! [A2: int,A: set_int] :
      ( ( member_int @ A2 @ A )
     => ( ( insert_int @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_1468_insert__absorb,axiom,
    ! [A2: complex,A: set_complex] :
      ( ( member_complex @ A2 @ A )
     => ( ( insert_complex @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_1469_insert__ident,axiom,
    ! [X: $o,A: set_o,B: set_o] :
      ( ~ ( member_o @ X @ A )
     => ( ~ ( member_o @ X @ B )
       => ( ( ( insert_o @ X @ A )
            = ( insert_o @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_1470_insert__ident,axiom,
    ! [X: nat,A: set_nat,B: set_nat] :
      ( ~ ( member_nat @ X @ A )
     => ( ~ ( member_nat @ X @ B )
       => ( ( ( insert_nat @ X @ A )
            = ( insert_nat @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_1471_insert__ident,axiom,
    ! [X: real,A: set_real,B: set_real] :
      ( ~ ( member_real @ X @ A )
     => ( ~ ( member_real @ X @ B )
       => ( ( ( insert_real @ X @ A )
            = ( insert_real @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_1472_insert__ident,axiom,
    ! [X: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X @ A )
     => ( ~ ( member_VEBT_VEBT @ X @ B )
       => ( ( ( insert_VEBT_VEBT @ X @ A )
            = ( insert_VEBT_VEBT @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_1473_insert__ident,axiom,
    ! [X: int,A: set_int,B: set_int] :
      ( ~ ( member_int @ X @ A )
     => ( ~ ( member_int @ X @ B )
       => ( ( ( insert_int @ X @ A )
            = ( insert_int @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_1474_insert__ident,axiom,
    ! [X: complex,A: set_complex,B: set_complex] :
      ( ~ ( member_complex @ X @ A )
     => ( ~ ( member_complex @ X @ B )
       => ( ( ( insert_complex @ X @ A )
            = ( insert_complex @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_1475_Set_Oset__insert,axiom,
    ! [X: $o,A: set_o] :
      ( ( member_o @ X @ A )
     => ~ ! [B6: set_o] :
            ( ( A
              = ( insert_o @ X @ B6 ) )
           => ( member_o @ X @ B6 ) ) ) ).

% Set.set_insert
thf(fact_1476_Set_Oset__insert,axiom,
    ! [X: nat,A: set_nat] :
      ( ( member_nat @ X @ A )
     => ~ ! [B6: set_nat] :
            ( ( A
              = ( insert_nat @ X @ B6 ) )
           => ( member_nat @ X @ B6 ) ) ) ).

% Set.set_insert
thf(fact_1477_Set_Oset__insert,axiom,
    ! [X: real,A: set_real] :
      ( ( member_real @ X @ A )
     => ~ ! [B6: set_real] :
            ( ( A
              = ( insert_real @ X @ B6 ) )
           => ( member_real @ X @ B6 ) ) ) ).

% Set.set_insert
thf(fact_1478_Set_Oset__insert,axiom,
    ! [X: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ A )
     => ~ ! [B6: set_VEBT_VEBT] :
            ( ( A
              = ( insert_VEBT_VEBT @ X @ B6 ) )
           => ( member_VEBT_VEBT @ X @ B6 ) ) ) ).

% Set.set_insert
thf(fact_1479_Set_Oset__insert,axiom,
    ! [X: int,A: set_int] :
      ( ( member_int @ X @ A )
     => ~ ! [B6: set_int] :
            ( ( A
              = ( insert_int @ X @ B6 ) )
           => ( member_int @ X @ B6 ) ) ) ).

% Set.set_insert
thf(fact_1480_Set_Oset__insert,axiom,
    ! [X: complex,A: set_complex] :
      ( ( member_complex @ X @ A )
     => ~ ! [B6: set_complex] :
            ( ( A
              = ( insert_complex @ X @ B6 ) )
           => ( member_complex @ X @ B6 ) ) ) ).

% Set.set_insert
thf(fact_1481_insertI2,axiom,
    ! [A2: $o,B: set_o,B2: $o] :
      ( ( member_o @ A2 @ B )
     => ( member_o @ A2 @ ( insert_o @ B2 @ B ) ) ) ).

% insertI2
thf(fact_1482_insertI2,axiom,
    ! [A2: nat,B: set_nat,B2: nat] :
      ( ( member_nat @ A2 @ B )
     => ( member_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).

% insertI2
thf(fact_1483_insertI2,axiom,
    ! [A2: real,B: set_real,B2: real] :
      ( ( member_real @ A2 @ B )
     => ( member_real @ A2 @ ( insert_real @ B2 @ B ) ) ) ).

% insertI2
thf(fact_1484_insertI2,axiom,
    ! [A2: vEBT_VEBT,B: set_VEBT_VEBT,B2: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ A2 @ B )
     => ( member_VEBT_VEBT @ A2 @ ( insert_VEBT_VEBT @ B2 @ B ) ) ) ).

% insertI2
thf(fact_1485_insertI2,axiom,
    ! [A2: int,B: set_int,B2: int] :
      ( ( member_int @ A2 @ B )
     => ( member_int @ A2 @ ( insert_int @ B2 @ B ) ) ) ).

% insertI2
thf(fact_1486_insertI2,axiom,
    ! [A2: complex,B: set_complex,B2: complex] :
      ( ( member_complex @ A2 @ B )
     => ( member_complex @ A2 @ ( insert_complex @ B2 @ B ) ) ) ).

% insertI2
thf(fact_1487_insertI1,axiom,
    ! [A2: $o,B: set_o] : ( member_o @ A2 @ ( insert_o @ A2 @ B ) ) ).

% insertI1
thf(fact_1488_insertI1,axiom,
    ! [A2: nat,B: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B ) ) ).

% insertI1
thf(fact_1489_insertI1,axiom,
    ! [A2: real,B: set_real] : ( member_real @ A2 @ ( insert_real @ A2 @ B ) ) ).

% insertI1
thf(fact_1490_insertI1,axiom,
    ! [A2: vEBT_VEBT,B: set_VEBT_VEBT] : ( member_VEBT_VEBT @ A2 @ ( insert_VEBT_VEBT @ A2 @ B ) ) ).

% insertI1
thf(fact_1491_insertI1,axiom,
    ! [A2: int,B: set_int] : ( member_int @ A2 @ ( insert_int @ A2 @ B ) ) ).

% insertI1
thf(fact_1492_insertI1,axiom,
    ! [A2: complex,B: set_complex] : ( member_complex @ A2 @ ( insert_complex @ A2 @ B ) ) ).

% insertI1
thf(fact_1493_insertE,axiom,
    ! [A2: $o,B2: $o,A: set_o] :
      ( ( member_o @ A2 @ ( insert_o @ B2 @ A ) )
     => ( ( A2 = ~ B2 )
       => ( member_o @ A2 @ A ) ) ) ).

% insertE
thf(fact_1494_insertE,axiom,
    ! [A2: nat,B2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ ( insert_nat @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member_nat @ A2 @ A ) ) ) ).

% insertE
thf(fact_1495_insertE,axiom,
    ! [A2: real,B2: real,A: set_real] :
      ( ( member_real @ A2 @ ( insert_real @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member_real @ A2 @ A ) ) ) ).

% insertE
thf(fact_1496_insertE,axiom,
    ! [A2: vEBT_VEBT,B2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ A2 @ ( insert_VEBT_VEBT @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member_VEBT_VEBT @ A2 @ A ) ) ) ).

% insertE
thf(fact_1497_insertE,axiom,
    ! [A2: int,B2: int,A: set_int] :
      ( ( member_int @ A2 @ ( insert_int @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member_int @ A2 @ A ) ) ) ).

% insertE
thf(fact_1498_insertE,axiom,
    ! [A2: complex,B2: complex,A: set_complex] :
      ( ( member_complex @ A2 @ ( insert_complex @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member_complex @ A2 @ A ) ) ) ).

% insertE
thf(fact_1499_subset__insertI2,axiom,
    ! [A: set_int,B: set_int,B2: int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ord_less_eq_set_int @ A @ ( insert_int @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_1500_subset__insertI2,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT,B2: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A @ B )
     => ( ord_le4337996190870823476T_VEBT @ A @ ( insert_VEBT_VEBT @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_1501_subset__insertI2,axiom,
    ! [A: set_real,B: set_real,B2: real] :
      ( ( ord_less_eq_set_real @ A @ B )
     => ( ord_less_eq_set_real @ A @ ( insert_real @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_1502_subset__insertI2,axiom,
    ! [A: set_o,B: set_o,B2: $o] :
      ( ( ord_less_eq_set_o @ A @ B )
     => ( ord_less_eq_set_o @ A @ ( insert_o @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_1503_subset__insertI2,axiom,
    ! [A: set_nat,B: set_nat,B2: nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_1504_subset__insertI,axiom,
    ! [B: set_int,A2: int] : ( ord_less_eq_set_int @ B @ ( insert_int @ A2 @ B ) ) ).

% subset_insertI
thf(fact_1505_subset__insertI,axiom,
    ! [B: set_VEBT_VEBT,A2: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ B @ ( insert_VEBT_VEBT @ A2 @ B ) ) ).

% subset_insertI
thf(fact_1506_subset__insertI,axiom,
    ! [B: set_real,A2: real] : ( ord_less_eq_set_real @ B @ ( insert_real @ A2 @ B ) ) ).

% subset_insertI
thf(fact_1507_subset__insertI,axiom,
    ! [B: set_o,A2: $o] : ( ord_less_eq_set_o @ B @ ( insert_o @ A2 @ B ) ) ).

% subset_insertI
thf(fact_1508_subset__insertI,axiom,
    ! [B: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat @ A2 @ B ) ) ).

% subset_insertI
thf(fact_1509_subset__insert,axiom,
    ! [X: $o,A: set_o,B: set_o] :
      ( ~ ( member_o @ X @ A )
     => ( ( ord_less_eq_set_o @ A @ ( insert_o @ X @ B ) )
        = ( ord_less_eq_set_o @ A @ B ) ) ) ).

% subset_insert
thf(fact_1510_subset__insert,axiom,
    ! [X: real,A: set_real,B: set_real] :
      ( ~ ( member_real @ X @ A )
     => ( ( ord_less_eq_set_real @ A @ ( insert_real @ X @ B ) )
        = ( ord_less_eq_set_real @ A @ B ) ) ) ).

% subset_insert
thf(fact_1511_subset__insert,axiom,
    ! [X: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X @ A )
     => ( ( ord_le4337996190870823476T_VEBT @ A @ ( insert_VEBT_VEBT @ X @ B ) )
        = ( ord_le4337996190870823476T_VEBT @ A @ B ) ) ) ).

% subset_insert
thf(fact_1512_subset__insert,axiom,
    ! [X: int,A: set_int,B: set_int] :
      ( ~ ( member_int @ X @ A )
     => ( ( ord_less_eq_set_int @ A @ ( insert_int @ X @ B ) )
        = ( ord_less_eq_set_int @ A @ B ) ) ) ).

% subset_insert
thf(fact_1513_subset__insert,axiom,
    ! [X: complex,A: set_complex,B: set_complex] :
      ( ~ ( member_complex @ X @ A )
     => ( ( ord_le211207098394363844omplex @ A @ ( insert_complex @ X @ B ) )
        = ( ord_le211207098394363844omplex @ A @ B ) ) ) ).

% subset_insert
thf(fact_1514_subset__insert,axiom,
    ! [X: nat,A: set_nat,B: set_nat] :
      ( ~ ( member_nat @ X @ A )
     => ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B ) )
        = ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% subset_insert
thf(fact_1515_insert__mono,axiom,
    ! [C: set_int,D: set_int,A2: int] :
      ( ( ord_less_eq_set_int @ C @ D )
     => ( ord_less_eq_set_int @ ( insert_int @ A2 @ C ) @ ( insert_int @ A2 @ D ) ) ) ).

% insert_mono
thf(fact_1516_insert__mono,axiom,
    ! [C: set_VEBT_VEBT,D: set_VEBT_VEBT,A2: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ C @ D )
     => ( ord_le4337996190870823476T_VEBT @ ( insert_VEBT_VEBT @ A2 @ C ) @ ( insert_VEBT_VEBT @ A2 @ D ) ) ) ).

% insert_mono
thf(fact_1517_insert__mono,axiom,
    ! [C: set_real,D: set_real,A2: real] :
      ( ( ord_less_eq_set_real @ C @ D )
     => ( ord_less_eq_set_real @ ( insert_real @ A2 @ C ) @ ( insert_real @ A2 @ D ) ) ) ).

% insert_mono
thf(fact_1518_insert__mono,axiom,
    ! [C: set_o,D: set_o,A2: $o] :
      ( ( ord_less_eq_set_o @ C @ D )
     => ( ord_less_eq_set_o @ ( insert_o @ A2 @ C ) @ ( insert_o @ A2 @ D ) ) ) ).

% insert_mono
thf(fact_1519_insert__mono,axiom,
    ! [C: set_nat,D: set_nat,A2: nat] :
      ( ( ord_less_eq_set_nat @ C @ D )
     => ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ C ) @ ( insert_nat @ A2 @ D ) ) ) ).

% insert_mono
thf(fact_1520_psubset__imp__ex__mem,axiom,
    ! [A: set_real,B: set_real] :
      ( ( ord_less_set_real @ A @ B )
     => ? [B3: real] : ( member_real @ B3 @ ( minus_minus_set_real @ B @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1521_psubset__imp__ex__mem,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ( ord_le3480810397992357184T_VEBT @ A @ B )
     => ? [B3: vEBT_VEBT] : ( member_VEBT_VEBT @ B3 @ ( minus_5127226145743854075T_VEBT @ B @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1522_psubset__imp__ex__mem,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_set_int @ A @ B )
     => ? [B3: int] : ( member_int @ B3 @ ( minus_minus_set_int @ B @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1523_psubset__imp__ex__mem,axiom,
    ! [A: set_complex,B: set_complex] :
      ( ( ord_less_set_complex @ A @ B )
     => ? [B3: complex] : ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1524_psubset__imp__ex__mem,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1525_DiffD2,axiom,
    ! [C2: real,A: set_real,B: set_real] :
      ( ( member_real @ C2 @ ( minus_minus_set_real @ A @ B ) )
     => ~ ( member_real @ C2 @ B ) ) ).

% DiffD2
thf(fact_1526_DiffD2,axiom,
    ! [C2: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ C2 @ ( minus_5127226145743854075T_VEBT @ A @ B ) )
     => ~ ( member_VEBT_VEBT @ C2 @ B ) ) ).

% DiffD2
thf(fact_1527_DiffD2,axiom,
    ! [C2: int,A: set_int,B: set_int] :
      ( ( member_int @ C2 @ ( minus_minus_set_int @ A @ B ) )
     => ~ ( member_int @ C2 @ B ) ) ).

% DiffD2
thf(fact_1528_DiffD2,axiom,
    ! [C2: complex,A: set_complex,B: set_complex] :
      ( ( member_complex @ C2 @ ( minus_811609699411566653omplex @ A @ B ) )
     => ~ ( member_complex @ C2 @ B ) ) ).

% DiffD2
thf(fact_1529_DiffD2,axiom,
    ! [C2: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) )
     => ~ ( member_nat @ C2 @ B ) ) ).

% DiffD2
thf(fact_1530_DiffD1,axiom,
    ! [C2: real,A: set_real,B: set_real] :
      ( ( member_real @ C2 @ ( minus_minus_set_real @ A @ B ) )
     => ( member_real @ C2 @ A ) ) ).

% DiffD1
thf(fact_1531_DiffD1,axiom,
    ! [C2: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ C2 @ ( minus_5127226145743854075T_VEBT @ A @ B ) )
     => ( member_VEBT_VEBT @ C2 @ A ) ) ).

% DiffD1
thf(fact_1532_DiffD1,axiom,
    ! [C2: int,A: set_int,B: set_int] :
      ( ( member_int @ C2 @ ( minus_minus_set_int @ A @ B ) )
     => ( member_int @ C2 @ A ) ) ).

% DiffD1
thf(fact_1533_DiffD1,axiom,
    ! [C2: complex,A: set_complex,B: set_complex] :
      ( ( member_complex @ C2 @ ( minus_811609699411566653omplex @ A @ B ) )
     => ( member_complex @ C2 @ A ) ) ).

% DiffD1
thf(fact_1534_DiffD1,axiom,
    ! [C2: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) )
     => ( member_nat @ C2 @ A ) ) ).

% DiffD1
thf(fact_1535_DiffE,axiom,
    ! [C2: real,A: set_real,B: set_real] :
      ( ( member_real @ C2 @ ( minus_minus_set_real @ A @ B ) )
     => ~ ( ( member_real @ C2 @ A )
         => ( member_real @ C2 @ B ) ) ) ).

% DiffE
thf(fact_1536_DiffE,axiom,
    ! [C2: vEBT_VEBT,A: set_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ C2 @ ( minus_5127226145743854075T_VEBT @ A @ B ) )
     => ~ ( ( member_VEBT_VEBT @ C2 @ A )
         => ( member_VEBT_VEBT @ C2 @ B ) ) ) ).

% DiffE
thf(fact_1537_DiffE,axiom,
    ! [C2: int,A: set_int,B: set_int] :
      ( ( member_int @ C2 @ ( minus_minus_set_int @ A @ B ) )
     => ~ ( ( member_int @ C2 @ A )
         => ( member_int @ C2 @ B ) ) ) ).

% DiffE
thf(fact_1538_DiffE,axiom,
    ! [C2: complex,A: set_complex,B: set_complex] :
      ( ( member_complex @ C2 @ ( minus_811609699411566653omplex @ A @ B ) )
     => ~ ( ( member_complex @ C2 @ A )
         => ( member_complex @ C2 @ B ) ) ) ).

% DiffE
thf(fact_1539_DiffE,axiom,
    ! [C2: nat,A: set_nat,B: set_nat] :
      ( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B ) )
     => ~ ( ( member_nat @ C2 @ A )
         => ( member_nat @ C2 @ B ) ) ) ).

% DiffE
thf(fact_1540_double__diff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C @ A ) )
          = A ) ) ) ).

% double_diff
thf(fact_1541_Diff__subset,axiom,
    ! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ A ) ).

% Diff_subset
thf(fact_1542_Diff__mono,axiom,
    ! [A: set_nat,C: set_nat,D: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C )
     => ( ( ord_less_eq_set_nat @ D @ B )
       => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ C @ D ) ) ) ) ).

% Diff_mono
thf(fact_1543_assn__times__assoc,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( ( times_times_assn @ ( times_times_assn @ P @ Q ) @ R )
      = ( times_times_assn @ P @ ( times_times_assn @ Q @ R ) ) ) ).

% assn_times_assoc
thf(fact_1544_assn__times__comm,axiom,
    ( times_times_assn
    = ( ^ [P3: assn,Q6: assn] : ( times_times_assn @ Q6 @ P3 ) ) ) ).

% assn_times_comm
thf(fact_1545_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_1546_ent__trans,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( ( entails @ P @ Q )
     => ( ( entails @ Q @ R )
       => ( entails @ P @ R ) ) ) ).

% ent_trans
thf(fact_1547_ent__refl,axiom,
    ! [P: assn] : ( entails @ P @ P ) ).

% ent_refl
thf(fact_1548_ent__iffI,axiom,
    ! [A: assn,B: assn] :
      ( ( entails @ A @ B )
     => ( ( entails @ B @ A )
       => ( A = B ) ) ) ).

% ent_iffI
thf(fact_1549_mult__right__cancel,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( ( times_times_complex @ A2 @ C2 )
          = ( times_times_complex @ B2 @ C2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_right_cancel
thf(fact_1550_mult__right__cancel,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( ( times_times_real @ A2 @ C2 )
          = ( times_times_real @ B2 @ C2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_right_cancel
thf(fact_1551_mult__right__cancel,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( ( times_times_rat @ A2 @ C2 )
          = ( times_times_rat @ B2 @ C2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_right_cancel
thf(fact_1552_mult__right__cancel,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( ( times_times_nat @ A2 @ C2 )
          = ( times_times_nat @ B2 @ C2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_right_cancel
thf(fact_1553_mult__right__cancel,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( C2 != zero_zero_int )
     => ( ( ( times_times_int @ A2 @ C2 )
          = ( times_times_int @ B2 @ C2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_right_cancel
thf(fact_1554_mult__left__cancel,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( ( times_times_complex @ C2 @ A2 )
          = ( times_times_complex @ C2 @ B2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_left_cancel
thf(fact_1555_mult__left__cancel,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( ( times_times_real @ C2 @ A2 )
          = ( times_times_real @ C2 @ B2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_left_cancel
thf(fact_1556_mult__left__cancel,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( ( times_times_rat @ C2 @ A2 )
          = ( times_times_rat @ C2 @ B2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_left_cancel
thf(fact_1557_mult__left__cancel,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( ( times_times_nat @ C2 @ A2 )
          = ( times_times_nat @ C2 @ B2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_left_cancel
thf(fact_1558_mult__left__cancel,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( C2 != zero_zero_int )
     => ( ( ( times_times_int @ C2 @ A2 )
          = ( times_times_int @ C2 @ B2 ) )
        = ( A2 = B2 ) ) ) ).

% mult_left_cancel
thf(fact_1559_no__zero__divisors,axiom,
    ! [A2: complex,B2: complex] :
      ( ( A2 != zero_zero_complex )
     => ( ( B2 != zero_zero_complex )
       => ( ( times_times_complex @ A2 @ B2 )
         != zero_zero_complex ) ) ) ).

% no_zero_divisors
thf(fact_1560_no__zero__divisors,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( ( times_times_real @ A2 @ B2 )
         != zero_zero_real ) ) ) ).

% no_zero_divisors
thf(fact_1561_no__zero__divisors,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( ( times_times_rat @ A2 @ B2 )
         != zero_zero_rat ) ) ) ).

% no_zero_divisors
thf(fact_1562_no__zero__divisors,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( B2 != zero_zero_nat )
       => ( ( times_times_nat @ A2 @ B2 )
         != zero_zero_nat ) ) ) ).

% no_zero_divisors
thf(fact_1563_no__zero__divisors,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( B2 != zero_zero_int )
       => ( ( times_times_int @ A2 @ B2 )
         != zero_zero_int ) ) ) ).

% no_zero_divisors
thf(fact_1564_divisors__zero,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ( times_times_complex @ A2 @ B2 )
        = zero_zero_complex )
     => ( ( A2 = zero_zero_complex )
        | ( B2 = zero_zero_complex ) ) ) ).

% divisors_zero
thf(fact_1565_divisors__zero,axiom,
    ! [A2: real,B2: real] :
      ( ( ( times_times_real @ A2 @ B2 )
        = zero_zero_real )
     => ( ( A2 = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% divisors_zero
thf(fact_1566_divisors__zero,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( times_times_rat @ A2 @ B2 )
        = zero_zero_rat )
     => ( ( A2 = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% divisors_zero
thf(fact_1567_divisors__zero,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( times_times_nat @ A2 @ B2 )
        = zero_zero_nat )
     => ( ( A2 = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% divisors_zero
thf(fact_1568_divisors__zero,axiom,
    ! [A2: int,B2: int] :
      ( ( ( times_times_int @ A2 @ B2 )
        = zero_zero_int )
     => ( ( A2 = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% divisors_zero
thf(fact_1569_mult__not__zero,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ( times_times_complex @ A2 @ B2 )
       != zero_zero_complex )
     => ( ( A2 != zero_zero_complex )
        & ( B2 != zero_zero_complex ) ) ) ).

% mult_not_zero
thf(fact_1570_mult__not__zero,axiom,
    ! [A2: real,B2: real] :
      ( ( ( times_times_real @ A2 @ B2 )
       != zero_zero_real )
     => ( ( A2 != zero_zero_real )
        & ( B2 != zero_zero_real ) ) ) ).

% mult_not_zero
thf(fact_1571_mult__not__zero,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( times_times_rat @ A2 @ B2 )
       != zero_zero_rat )
     => ( ( A2 != zero_zero_rat )
        & ( B2 != zero_zero_rat ) ) ) ).

% mult_not_zero
thf(fact_1572_mult__not__zero,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( times_times_nat @ A2 @ B2 )
       != zero_zero_nat )
     => ( ( A2 != zero_zero_nat )
        & ( B2 != zero_zero_nat ) ) ) ).

% mult_not_zero
thf(fact_1573_mult__not__zero,axiom,
    ! [A2: int,B2: int] :
      ( ( ( times_times_int @ A2 @ B2 )
       != zero_zero_int )
     => ( ( A2 != zero_zero_int )
        & ( B2 != zero_zero_int ) ) ) ).

% mult_not_zero
thf(fact_1574_num_Osize_I5_J,axiom,
    ! [X22: num] :
      ( ( size_size_num @ ( bit0 @ X22 ) )
      = ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(5)
thf(fact_1575_combine__common__factor,axiom,
    ! [A2: real,E: real,B2: real,C2: real] :
      ( ( plus_plus_real @ ( times_times_real @ A2 @ E ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ C2 ) )
      = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A2 @ B2 ) @ E ) @ C2 ) ) ).

% combine_common_factor
thf(fact_1576_combine__common__factor,axiom,
    ! [A2: rat,E: rat,B2: rat,C2: rat] :
      ( ( plus_plus_rat @ ( times_times_rat @ A2 @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ C2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A2 @ B2 ) @ E ) @ C2 ) ) ).

% combine_common_factor
thf(fact_1577_combine__common__factor,axiom,
    ! [A2: nat,E: nat,B2: nat,C2: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A2 @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B2 @ E ) @ C2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ E ) @ C2 ) ) ).

% combine_common_factor
thf(fact_1578_combine__common__factor,axiom,
    ! [A2: int,E: int,B2: int,C2: int] :
      ( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ C2 ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ E ) @ C2 ) ) ).

% combine_common_factor
thf(fact_1579_distrib__right,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( times_times_real @ ( plus_plus_real @ A2 @ B2 ) @ C2 )
      = ( plus_plus_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ).

% distrib_right
thf(fact_1580_distrib__right,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C2 )
      = ( plus_plus_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) ) ) ).

% distrib_right
thf(fact_1581_distrib__right,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 )
      = ( plus_plus_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) ) ) ).

% distrib_right
thf(fact_1582_distrib__right,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 )
      = ( plus_plus_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ).

% distrib_right
thf(fact_1583_distrib__left,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( times_times_real @ A2 @ ( plus_plus_real @ B2 @ C2 ) )
      = ( plus_plus_real @ ( times_times_real @ A2 @ B2 ) @ ( times_times_real @ A2 @ C2 ) ) ) ).

% distrib_left
thf(fact_1584_distrib__left,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( times_times_rat @ A2 @ ( plus_plus_rat @ B2 @ C2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ A2 @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) ) ).

% distrib_left
thf(fact_1585_distrib__left,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( times_times_nat @ A2 @ ( plus_plus_nat @ B2 @ C2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C2 ) ) ) ).

% distrib_left
thf(fact_1586_distrib__left,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( times_times_int @ A2 @ ( plus_plus_int @ B2 @ C2 ) )
      = ( plus_plus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C2 ) ) ) ).

% distrib_left
thf(fact_1587_comm__semiring__class_Odistrib,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( times_times_real @ ( plus_plus_real @ A2 @ B2 ) @ C2 )
      = ( plus_plus_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ).

% comm_semiring_class.distrib
thf(fact_1588_comm__semiring__class_Odistrib,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C2 )
      = ( plus_plus_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) ) ) ).

% comm_semiring_class.distrib
thf(fact_1589_comm__semiring__class_Odistrib,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 )
      = ( plus_plus_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) ) ) ).

% comm_semiring_class.distrib
thf(fact_1590_comm__semiring__class_Odistrib,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 )
      = ( plus_plus_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ).

% comm_semiring_class.distrib
thf(fact_1591_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( times_times_real @ A2 @ ( plus_plus_real @ B2 @ C2 ) )
      = ( plus_plus_real @ ( times_times_real @ A2 @ B2 ) @ ( times_times_real @ A2 @ C2 ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_1592_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( times_times_rat @ A2 @ ( plus_plus_rat @ B2 @ C2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ A2 @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_1593_ring__class_Oring__distribs_I1_J,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( times_times_int @ A2 @ ( plus_plus_int @ B2 @ C2 ) )
      = ( plus_plus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C2 ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_1594_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( times_times_real @ ( plus_plus_real @ A2 @ B2 ) @ C2 )
      = ( plus_plus_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_1595_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C2 )
      = ( plus_plus_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_1596_ring__class_Oring__distribs_I2_J,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 )
      = ( plus_plus_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_1597_right__diff__distrib_H,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( times_times_real @ A2 @ ( minus_minus_real @ B2 @ C2 ) )
      = ( minus_minus_real @ ( times_times_real @ A2 @ B2 ) @ ( times_times_real @ A2 @ C2 ) ) ) ).

% right_diff_distrib'
thf(fact_1598_right__diff__distrib_H,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( times_times_rat @ A2 @ ( minus_minus_rat @ B2 @ C2 ) )
      = ( minus_minus_rat @ ( times_times_rat @ A2 @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) ) ).

% right_diff_distrib'
thf(fact_1599_right__diff__distrib_H,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( times_times_nat @ A2 @ ( minus_minus_nat @ B2 @ C2 ) )
      = ( minus_minus_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C2 ) ) ) ).

% right_diff_distrib'
thf(fact_1600_right__diff__distrib_H,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( times_times_int @ A2 @ ( minus_minus_int @ B2 @ C2 ) )
      = ( minus_minus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C2 ) ) ) ).

% right_diff_distrib'
thf(fact_1601_left__diff__distrib_H,axiom,
    ! [B2: real,C2: real,A2: real] :
      ( ( times_times_real @ ( minus_minus_real @ B2 @ C2 ) @ A2 )
      = ( minus_minus_real @ ( times_times_real @ B2 @ A2 ) @ ( times_times_real @ C2 @ A2 ) ) ) ).

% left_diff_distrib'
thf(fact_1602_left__diff__distrib_H,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ B2 @ C2 ) @ A2 )
      = ( minus_minus_rat @ ( times_times_rat @ B2 @ A2 ) @ ( times_times_rat @ C2 @ A2 ) ) ) ).

% left_diff_distrib'
thf(fact_1603_left__diff__distrib_H,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ B2 @ C2 ) @ A2 )
      = ( minus_minus_nat @ ( times_times_nat @ B2 @ A2 ) @ ( times_times_nat @ C2 @ A2 ) ) ) ).

% left_diff_distrib'
thf(fact_1604_left__diff__distrib_H,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( times_times_int @ ( minus_minus_int @ B2 @ C2 ) @ A2 )
      = ( minus_minus_int @ ( times_times_int @ B2 @ A2 ) @ ( times_times_int @ C2 @ A2 ) ) ) ).

% left_diff_distrib'
thf(fact_1605_right__diff__distrib,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( times_times_real @ A2 @ ( minus_minus_real @ B2 @ C2 ) )
      = ( minus_minus_real @ ( times_times_real @ A2 @ B2 ) @ ( times_times_real @ A2 @ C2 ) ) ) ).

% right_diff_distrib
thf(fact_1606_right__diff__distrib,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( times_times_rat @ A2 @ ( minus_minus_rat @ B2 @ C2 ) )
      = ( minus_minus_rat @ ( times_times_rat @ A2 @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) ) ).

% right_diff_distrib
thf(fact_1607_right__diff__distrib,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( times_times_int @ A2 @ ( minus_minus_int @ B2 @ C2 ) )
      = ( minus_minus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C2 ) ) ) ).

% right_diff_distrib
thf(fact_1608_left__diff__distrib,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( times_times_real @ ( minus_minus_real @ A2 @ B2 ) @ C2 )
      = ( minus_minus_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ).

% left_diff_distrib
thf(fact_1609_left__diff__distrib,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C2 )
      = ( minus_minus_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) ) ) ).

% left_diff_distrib
thf(fact_1610_left__diff__distrib,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ C2 )
      = ( minus_minus_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ).

% left_diff_distrib
thf(fact_1611_add__divide__distrib,axiom,
    ! [A2: complex,B2: complex,C2: complex] :
      ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A2 @ B2 ) @ C2 )
      = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A2 @ C2 ) @ ( divide1717551699836669952omplex @ B2 @ C2 ) ) ) ).

% add_divide_distrib
thf(fact_1612_add__divide__distrib,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( divide_divide_real @ ( plus_plus_real @ A2 @ B2 ) @ C2 )
      = ( plus_plus_real @ ( divide_divide_real @ A2 @ C2 ) @ ( divide_divide_real @ B2 @ C2 ) ) ) ).

% add_divide_distrib
thf(fact_1613_add__divide__distrib,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( divide_divide_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C2 )
      = ( plus_plus_rat @ ( divide_divide_rat @ A2 @ C2 ) @ ( divide_divide_rat @ B2 @ C2 ) ) ) ).

% add_divide_distrib
thf(fact_1614_divide__divide__eq__left_H,axiom,
    ! [A2: complex,B2: complex,C2: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A2 @ B2 ) @ C2 )
      = ( divide1717551699836669952omplex @ A2 @ ( times_times_complex @ C2 @ B2 ) ) ) ).

% divide_divide_eq_left'
thf(fact_1615_divide__divide__eq__left_H,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ A2 @ B2 ) @ C2 )
      = ( divide_divide_real @ A2 @ ( times_times_real @ C2 @ B2 ) ) ) ).

% divide_divide_eq_left'
thf(fact_1616_divide__divide__eq__left_H,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A2 @ B2 ) @ C2 )
      = ( divide_divide_rat @ A2 @ ( times_times_rat @ C2 @ B2 ) ) ) ).

% divide_divide_eq_left'
thf(fact_1617_divide__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z: complex,W: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W ) @ ( times_times_complex @ Y @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_1618_divide__divide__times__eq,axiom,
    ! [X: real,Y: real,Z: real,W: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
      = ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_1619_divide__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
      = ( divide_divide_rat @ ( times_times_rat @ X @ W ) @ ( times_times_rat @ Y @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_1620_times__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z: complex,W: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ Y @ W ) ) ) ).

% times_divide_times_eq
thf(fact_1621_times__divide__times__eq,axiom,
    ! [X: real,Y: real,Z: real,W: real] :
      ( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
      = ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ W ) ) ) ).

% times_divide_times_eq
thf(fact_1622_times__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
      = ( divide_divide_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ W ) ) ) ).

% times_divide_times_eq
thf(fact_1623_diff__divide__distrib,axiom,
    ! [A2: complex,B2: complex,C2: complex] :
      ( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A2 @ B2 ) @ C2 )
      = ( minus_minus_complex @ ( divide1717551699836669952omplex @ A2 @ C2 ) @ ( divide1717551699836669952omplex @ B2 @ C2 ) ) ) ).

% diff_divide_distrib
thf(fact_1624_diff__divide__distrib,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( divide_divide_real @ ( minus_minus_real @ A2 @ B2 ) @ C2 )
      = ( minus_minus_real @ ( divide_divide_real @ A2 @ C2 ) @ ( divide_divide_real @ B2 @ C2 ) ) ) ).

% diff_divide_distrib
thf(fact_1625_diff__divide__distrib,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( divide_divide_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C2 )
      = ( minus_minus_rat @ ( divide_divide_rat @ A2 @ C2 ) @ ( divide_divide_rat @ B2 @ C2 ) ) ) ).

% diff_divide_distrib
thf(fact_1626_realpow__pos__nth__unique,axiom,
    ! [N3: nat,A2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ A2 )
       => ? [X4: real] :
            ( ( ord_less_real @ zero_zero_real @ X4 )
            & ( ( power_power_real @ X4 @ N3 )
              = A2 )
            & ! [Y4: real] :
                ( ( ( ord_less_real @ zero_zero_real @ Y4 )
                  & ( ( power_power_real @ Y4 @ N3 )
                    = A2 ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% realpow_pos_nth_unique
thf(fact_1627_realpow__pos__nth,axiom,
    ! [N3: nat,A2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ A2 )
       => ? [R2: real] :
            ( ( ord_less_real @ zero_zero_real @ R2 )
            & ( ( power_power_real @ R2 @ N3 )
              = A2 ) ) ) ) ).

% realpow_pos_nth
thf(fact_1628_singleton__inject,axiom,
    ! [A2: vEBT_VEBT,B2: vEBT_VEBT] :
      ( ( ( insert_VEBT_VEBT @ A2 @ bot_bo8194388402131092736T_VEBT )
        = ( insert_VEBT_VEBT @ B2 @ bot_bo8194388402131092736T_VEBT ) )
     => ( A2 = B2 ) ) ).

% singleton_inject
thf(fact_1629_singleton__inject,axiom,
    ! [A2: real,B2: real] :
      ( ( ( insert_real @ A2 @ bot_bot_set_real )
        = ( insert_real @ B2 @ bot_bot_set_real ) )
     => ( A2 = B2 ) ) ).

% singleton_inject
thf(fact_1630_singleton__inject,axiom,
    ! [A2: $o,B2: $o] :
      ( ( ( insert_o @ A2 @ bot_bot_set_o )
        = ( insert_o @ B2 @ bot_bot_set_o ) )
     => ( A2 = B2 ) ) ).

% singleton_inject
thf(fact_1631_singleton__inject,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( insert_nat @ A2 @ bot_bot_set_nat )
        = ( insert_nat @ B2 @ bot_bot_set_nat ) )
     => ( A2 = B2 ) ) ).

% singleton_inject
thf(fact_1632_singleton__inject,axiom,
    ! [A2: int,B2: int] :
      ( ( ( insert_int @ A2 @ bot_bot_set_int )
        = ( insert_int @ B2 @ bot_bot_set_int ) )
     => ( A2 = B2 ) ) ).

% singleton_inject
thf(fact_1633_insert__not__empty,axiom,
    ! [A2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ( insert_VEBT_VEBT @ A2 @ A )
     != bot_bo8194388402131092736T_VEBT ) ).

% insert_not_empty
thf(fact_1634_insert__not__empty,axiom,
    ! [A2: real,A: set_real] :
      ( ( insert_real @ A2 @ A )
     != bot_bot_set_real ) ).

% insert_not_empty
thf(fact_1635_insert__not__empty,axiom,
    ! [A2: $o,A: set_o] :
      ( ( insert_o @ A2 @ A )
     != bot_bot_set_o ) ).

% insert_not_empty
thf(fact_1636_insert__not__empty,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( insert_nat @ A2 @ A )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_1637_insert__not__empty,axiom,
    ! [A2: int,A: set_int] :
      ( ( insert_int @ A2 @ A )
     != bot_bot_set_int ) ).

% insert_not_empty
thf(fact_1638_doubleton__eq__iff,axiom,
    ! [A2: vEBT_VEBT,B2: vEBT_VEBT,C2: vEBT_VEBT,D2: vEBT_VEBT] :
      ( ( ( insert_VEBT_VEBT @ A2 @ ( insert_VEBT_VEBT @ B2 @ bot_bo8194388402131092736T_VEBT ) )
        = ( insert_VEBT_VEBT @ C2 @ ( insert_VEBT_VEBT @ D2 @ bot_bo8194388402131092736T_VEBT ) ) )
      = ( ( ( A2 = C2 )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_1639_doubleton__eq__iff,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( ( insert_real @ A2 @ ( insert_real @ B2 @ bot_bot_set_real ) )
        = ( insert_real @ C2 @ ( insert_real @ D2 @ bot_bot_set_real ) ) )
      = ( ( ( A2 = C2 )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_1640_doubleton__eq__iff,axiom,
    ! [A2: $o,B2: $o,C2: $o,D2: $o] :
      ( ( ( insert_o @ A2 @ ( insert_o @ B2 @ bot_bot_set_o ) )
        = ( insert_o @ C2 @ ( insert_o @ D2 @ bot_bot_set_o ) ) )
      = ( ( ( A2 = C2 )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_1641_doubleton__eq__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ( insert_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) )
        = ( insert_nat @ C2 @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
      = ( ( ( A2 = C2 )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_1642_doubleton__eq__iff,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ( insert_int @ A2 @ ( insert_int @ B2 @ bot_bot_set_int ) )
        = ( insert_int @ C2 @ ( insert_int @ D2 @ bot_bot_set_int ) ) )
      = ( ( ( A2 = C2 )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_1643_singleton__iff,axiom,
    ! [B2: vEBT_VEBT,A2: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ B2 @ ( insert_VEBT_VEBT @ A2 @ bot_bo8194388402131092736T_VEBT ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_1644_singleton__iff,axiom,
    ! [B2: complex,A2: complex] :
      ( ( member_complex @ B2 @ ( insert_complex @ A2 @ bot_bot_set_complex ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_1645_singleton__iff,axiom,
    ! [B2: real,A2: real] :
      ( ( member_real @ B2 @ ( insert_real @ A2 @ bot_bot_set_real ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_1646_singleton__iff,axiom,
    ! [B2: $o,A2: $o] :
      ( ( member_o @ B2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_1647_singleton__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_1648_singleton__iff,axiom,
    ! [B2: int,A2: int] :
      ( ( member_int @ B2 @ ( insert_int @ A2 @ bot_bot_set_int ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_1649_singletonD,axiom,
    ! [B2: vEBT_VEBT,A2: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ B2 @ ( insert_VEBT_VEBT @ A2 @ bot_bo8194388402131092736T_VEBT ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_1650_singletonD,axiom,
    ! [B2: complex,A2: complex] :
      ( ( member_complex @ B2 @ ( insert_complex @ A2 @ bot_bot_set_complex ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_1651_singletonD,axiom,
    ! [B2: real,A2: real] :
      ( ( member_real @ B2 @ ( insert_real @ A2 @ bot_bot_set_real ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_1652_singletonD,axiom,
    ! [B2: $o,A2: $o] :
      ( ( member_o @ B2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_1653_singletonD,axiom,
    ! [B2: nat,A2: nat] :
      ( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_1654_singletonD,axiom,
    ! [B2: int,A2: int] :
      ( ( member_int @ B2 @ ( insert_int @ A2 @ bot_bot_set_int ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_1655_subset__singleton__iff,axiom,
    ! [X2: set_VEBT_VEBT,A2: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ X2 @ ( insert_VEBT_VEBT @ A2 @ bot_bo8194388402131092736T_VEBT ) )
      = ( ( X2 = bot_bo8194388402131092736T_VEBT )
        | ( X2
          = ( insert_VEBT_VEBT @ A2 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% subset_singleton_iff
thf(fact_1656_subset__singleton__iff,axiom,
    ! [X2: set_real,A2: real] :
      ( ( ord_less_eq_set_real @ X2 @ ( insert_real @ A2 @ bot_bot_set_real ) )
      = ( ( X2 = bot_bot_set_real )
        | ( X2
          = ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ).

% subset_singleton_iff
thf(fact_1657_subset__singleton__iff,axiom,
    ! [X2: set_o,A2: $o] :
      ( ( ord_less_eq_set_o @ X2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
      = ( ( X2 = bot_bot_set_o )
        | ( X2
          = ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).

% subset_singleton_iff
thf(fact_1658_subset__singleton__iff,axiom,
    ! [X2: set_int,A2: int] :
      ( ( ord_less_eq_set_int @ X2 @ ( insert_int @ A2 @ bot_bot_set_int ) )
      = ( ( X2 = bot_bot_set_int )
        | ( X2
          = ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ).

% subset_singleton_iff
thf(fact_1659_subset__singleton__iff,axiom,
    ! [X2: set_nat,A2: nat] :
      ( ( ord_less_eq_set_nat @ X2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
      = ( ( X2 = bot_bot_set_nat )
        | ( X2
          = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_1660_subset__singletonD,axiom,
    ! [A: set_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) )
     => ( ( A = bot_bo8194388402131092736T_VEBT )
        | ( A
          = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% subset_singletonD
thf(fact_1661_subset__singletonD,axiom,
    ! [A: set_real,X: real] :
      ( ( ord_less_eq_set_real @ A @ ( insert_real @ X @ bot_bot_set_real ) )
     => ( ( A = bot_bot_set_real )
        | ( A
          = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).

% subset_singletonD
thf(fact_1662_subset__singletonD,axiom,
    ! [A: set_o,X: $o] :
      ( ( ord_less_eq_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
     => ( ( A = bot_bot_set_o )
        | ( A
          = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).

% subset_singletonD
thf(fact_1663_subset__singletonD,axiom,
    ! [A: set_int,X: int] :
      ( ( ord_less_eq_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
     => ( ( A = bot_bot_set_int )
        | ( A
          = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).

% subset_singletonD
thf(fact_1664_subset__singletonD,axiom,
    ! [A: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
     => ( ( A = bot_bot_set_nat )
        | ( A
          = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% subset_singletonD
thf(fact_1665_insert__Diff__if,axiom,
    ! [X: $o,B: set_o,A: set_o] :
      ( ( ( member_o @ X @ B )
       => ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ B )
          = ( minus_minus_set_o @ A @ B ) ) )
      & ( ~ ( member_o @ X @ B )
       => ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ B )
          = ( insert_o @ X @ ( minus_minus_set_o @ A @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1666_insert__Diff__if,axiom,
    ! [X: real,B: set_real,A: set_real] :
      ( ( ( member_real @ X @ B )
       => ( ( minus_minus_set_real @ ( insert_real @ X @ A ) @ B )
          = ( minus_minus_set_real @ A @ B ) ) )
      & ( ~ ( member_real @ X @ B )
       => ( ( minus_minus_set_real @ ( insert_real @ X @ A ) @ B )
          = ( insert_real @ X @ ( minus_minus_set_real @ A @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1667_insert__Diff__if,axiom,
    ! [X: vEBT_VEBT,B: set_VEBT_VEBT,A: set_VEBT_VEBT] :
      ( ( ( member_VEBT_VEBT @ X @ B )
       => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X @ A ) @ B )
          = ( minus_5127226145743854075T_VEBT @ A @ B ) ) )
      & ( ~ ( member_VEBT_VEBT @ X @ B )
       => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X @ A ) @ B )
          = ( insert_VEBT_VEBT @ X @ ( minus_5127226145743854075T_VEBT @ A @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1668_insert__Diff__if,axiom,
    ! [X: int,B: set_int,A: set_int] :
      ( ( ( member_int @ X @ B )
       => ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ B )
          = ( minus_minus_set_int @ A @ B ) ) )
      & ( ~ ( member_int @ X @ B )
       => ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ B )
          = ( insert_int @ X @ ( minus_minus_set_int @ A @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1669_insert__Diff__if,axiom,
    ! [X: complex,B: set_complex,A: set_complex] :
      ( ( ( member_complex @ X @ B )
       => ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A ) @ B )
          = ( minus_811609699411566653omplex @ A @ B ) ) )
      & ( ~ ( member_complex @ X @ B )
       => ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A ) @ B )
          = ( insert_complex @ X @ ( minus_811609699411566653omplex @ A @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1670_insert__Diff__if,axiom,
    ! [X: nat,B: set_nat,A: set_nat] :
      ( ( ( member_nat @ X @ B )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B )
          = ( minus_minus_set_nat @ A @ B ) ) )
      & ( ~ ( member_nat @ X @ B )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B )
          = ( insert_nat @ X @ ( minus_minus_set_nat @ A @ B ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1671_subset__Diff__insert,axiom,
    ! [A: set_o,B: set_o,X: $o,C: set_o] :
      ( ( ord_less_eq_set_o @ A @ ( minus_minus_set_o @ B @ ( insert_o @ X @ C ) ) )
      = ( ( ord_less_eq_set_o @ A @ ( minus_minus_set_o @ B @ C ) )
        & ~ ( member_o @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_1672_subset__Diff__insert,axiom,
    ! [A: set_real,B: set_real,X: real,C: set_real] :
      ( ( ord_less_eq_set_real @ A @ ( minus_minus_set_real @ B @ ( insert_real @ X @ C ) ) )
      = ( ( ord_less_eq_set_real @ A @ ( minus_minus_set_real @ B @ C ) )
        & ~ ( member_real @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_1673_subset__Diff__insert,axiom,
    ! [A: set_VEBT_VEBT,B: set_VEBT_VEBT,X: vEBT_VEBT,C: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ B @ ( insert_VEBT_VEBT @ X @ C ) ) )
      = ( ( ord_le4337996190870823476T_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ B @ C ) )
        & ~ ( member_VEBT_VEBT @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_1674_subset__Diff__insert,axiom,
    ! [A: set_int,B: set_int,X: int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ ( minus_minus_set_int @ B @ ( insert_int @ X @ C ) ) )
      = ( ( ord_less_eq_set_int @ A @ ( minus_minus_set_int @ B @ C ) )
        & ~ ( member_int @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_1675_subset__Diff__insert,axiom,
    ! [A: set_complex,B: set_complex,X: complex,C: set_complex] :
      ( ( ord_le211207098394363844omplex @ A @ ( minus_811609699411566653omplex @ B @ ( insert_complex @ X @ C ) ) )
      = ( ( ord_le211207098394363844omplex @ A @ ( minus_811609699411566653omplex @ B @ C ) )
        & ~ ( member_complex @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_1676_subset__Diff__insert,axiom,
    ! [A: set_nat,B: set_nat,X: nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ ( insert_nat @ X @ C ) ) )
      = ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ C ) )
        & ~ ( member_nat @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_1677_ent__star__mono,axiom,
    ! [P: assn,P2: assn,Q: assn,Q2: assn] :
      ( ( entails @ P @ P2 )
     => ( ( entails @ Q @ Q2 )
       => ( entails @ ( times_times_assn @ P @ Q ) @ ( times_times_assn @ P2 @ Q2 ) ) ) ) ).

% ent_star_mono
thf(fact_1678_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
       => ( ord_less_eq_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1679_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1680_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1681_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1682_zero__le__mult__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
        | ( ( ord_less_eq_real @ A2 @ zero_zero_real )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).

% zero_le_mult_iff
thf(fact_1683_zero__le__mult__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
        | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) ) ) ).

% zero_le_mult_iff
thf(fact_1684_zero__le__mult__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
          & ( ord_less_eq_int @ zero_zero_int @ B2 ) )
        | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
          & ( ord_less_eq_int @ B2 @ zero_zero_int ) ) ) ) ).

% zero_le_mult_iff
thf(fact_1685_mult__nonneg__nonpos2,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ B2 @ A2 ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_1686_mult__nonneg__nonpos2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ B2 @ A2 ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_1687_mult__nonneg__nonpos2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A2 ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_1688_mult__nonneg__nonpos2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ B2 @ A2 ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_1689_mult__nonpos__nonneg,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ A2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_eq_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).

% mult_nonpos_nonneg
thf(fact_1690_mult__nonpos__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_1691_mult__nonpos__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_1692_mult__nonpos__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% mult_nonpos_nonneg
thf(fact_1693_mult__nonneg__nonpos,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos
thf(fact_1694_mult__nonneg__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_1695_mult__nonneg__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_1696_mult__nonneg__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos
thf(fact_1697_mult__nonneg__nonneg,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_1698_mult__nonneg__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_1699_mult__nonneg__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_1700_mult__nonneg__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_1701_split__mult__neg__le,axiom,
    ! [A2: real,B2: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A2 @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real ) ) ).

% split_mult_neg_le
thf(fact_1702_split__mult__neg__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ).

% split_mult_neg_le
thf(fact_1703_split__mult__neg__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
          & ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
        | ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
          & ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ).

% split_mult_neg_le
thf(fact_1704_split__mult__neg__le,axiom,
    ! [A2: int,B2: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
          & ( ord_less_eq_int @ B2 @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B2 ) ) )
     => ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ).

% split_mult_neg_le
thf(fact_1705_mult__le__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A2 @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).

% mult_le_0_iff
thf(fact_1706_mult__le__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ) ) ).

% mult_le_0_iff
thf(fact_1707_mult__le__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
          & ( ord_less_eq_int @ B2 @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ) ) ).

% mult_le_0_iff
thf(fact_1708_mult__right__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
       => ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ) ).

% mult_right_mono
thf(fact_1709_mult__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) ) ) ) ).

% mult_right_mono
thf(fact_1710_mult__right__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) ) ) ) ).

% mult_right_mono
thf(fact_1711_mult__right__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ) ).

% mult_right_mono
thf(fact_1712_mult__right__mono__neg,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( ord_less_eq_real @ B2 @ A2 )
     => ( ( ord_less_eq_real @ C2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ) ).

% mult_right_mono_neg
thf(fact_1713_mult__right__mono__neg,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) ) ) ) ).

% mult_right_mono_neg
thf(fact_1714_mult__right__mono__neg,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_less_eq_int @ C2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ) ).

% mult_right_mono_neg
thf(fact_1715_mult__left__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
       => ( ord_less_eq_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_1716_mult__left__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_1717_mult__left__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_1718_mult__left__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) ) ) ) ).

% mult_left_mono
thf(fact_1719_mult__nonpos__nonpos,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ A2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_1720_mult__nonpos__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_1721_mult__nonpos__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_1722_mult__left__mono__neg,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( ord_less_eq_real @ B2 @ A2 )
     => ( ( ord_less_eq_real @ C2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_1723_mult__left__mono__neg,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_1724_mult__left__mono__neg,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_less_eq_int @ C2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_1725_split__mult__pos__le,axiom,
    ! [A2: real,B2: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
        | ( ( ord_less_eq_real @ A2 @ zero_zero_real )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) ) ) ).

% split_mult_pos_le
thf(fact_1726_split__mult__pos__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
        | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ).

% split_mult_pos_le
thf(fact_1727_split__mult__pos__le,axiom,
    ! [A2: int,B2: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
          & ( ord_less_eq_int @ zero_zero_int @ B2 ) )
        | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
          & ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ).

% split_mult_pos_le
thf(fact_1728_zero__le__square,axiom,
    ! [A2: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ A2 ) ) ).

% zero_le_square
thf(fact_1729_zero__le__square,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ A2 ) ) ).

% zero_le_square
thf(fact_1730_zero__le__square,axiom,
    ! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ A2 ) ) ).

% zero_le_square
thf(fact_1731_mult__mono_H,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ C2 @ D2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
           => ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_1732_mult__mono_H,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C2 @ D2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_1733_mult__mono_H,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C2 @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_1734_mult__mono_H,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ C2 @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_eq_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_1735_mult__mono,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ C2 @ D2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
           => ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_1736_mult__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C2 @ D2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_1737_mult__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C2 @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_1738_mult__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ C2 @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_eq_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_1739_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_1740_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_1741_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_1742_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ C2 )
       => ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1743_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1744_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1745_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1746_mult__less__cancel__right__disj,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
          & ( ord_less_real @ A2 @ B2 ) )
        | ( ( ord_less_real @ C2 @ zero_zero_real )
          & ( ord_less_real @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_1747_mult__less__cancel__right__disj,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
          & ( ord_less_rat @ A2 @ B2 ) )
        | ( ( ord_less_rat @ C2 @ zero_zero_rat )
          & ( ord_less_rat @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_1748_mult__less__cancel__right__disj,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
          & ( ord_less_int @ A2 @ B2 ) )
        | ( ( ord_less_int @ C2 @ zero_zero_int )
          & ( ord_less_int @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_1749_mult__strict__right__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ C2 )
       => ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ) ).

% mult_strict_right_mono
thf(fact_1750_mult__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) ) ) ) ).

% mult_strict_right_mono
thf(fact_1751_mult__strict__right__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) ) ) ) ).

% mult_strict_right_mono
thf(fact_1752_mult__strict__right__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ) ).

% mult_strict_right_mono
thf(fact_1753_mult__strict__right__mono__neg,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( ord_less_real @ B2 @ A2 )
     => ( ( ord_less_real @ C2 @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_1754_mult__strict__right__mono__neg,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ( ord_less_rat @ C2 @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_1755_mult__strict__right__mono__neg,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( ord_less_int @ C2 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_1756_mult__less__cancel__left__disj,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
          & ( ord_less_real @ A2 @ B2 ) )
        | ( ( ord_less_real @ C2 @ zero_zero_real )
          & ( ord_less_real @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_1757_mult__less__cancel__left__disj,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
          & ( ord_less_rat @ A2 @ B2 ) )
        | ( ( ord_less_rat @ C2 @ zero_zero_rat )
          & ( ord_less_rat @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_1758_mult__less__cancel__left__disj,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
          & ( ord_less_int @ A2 @ B2 ) )
        | ( ( ord_less_int @ C2 @ zero_zero_int )
          & ( ord_less_int @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_1759_mult__strict__left__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ C2 )
       => ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_1760_mult__strict__left__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_1761_mult__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_1762_mult__strict__left__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_1763_mult__strict__left__mono__neg,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( ord_less_real @ B2 @ A2 )
     => ( ( ord_less_real @ C2 @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_1764_mult__strict__left__mono__neg,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ( ord_less_rat @ C2 @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_1765_mult__strict__left__mono__neg,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( ord_less_int @ C2 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_1766_mult__less__cancel__left__pos,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C2 )
     => ( ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
        = ( ord_less_real @ A2 @ B2 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_1767_mult__less__cancel__left__pos,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
        = ( ord_less_rat @ A2 @ B2 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_1768_mult__less__cancel__left__pos,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ C2 )
     => ( ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
        = ( ord_less_int @ A2 @ B2 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_1769_mult__less__cancel__left__neg,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ord_less_real @ C2 @ zero_zero_real )
     => ( ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
        = ( ord_less_real @ B2 @ A2 ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_1770_mult__less__cancel__left__neg,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
        = ( ord_less_rat @ B2 @ A2 ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_1771_mult__less__cancel__left__neg,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_int @ C2 @ zero_zero_int )
     => ( ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
        = ( ord_less_int @ B2 @ A2 ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_1772_zero__less__mult__pos2,axiom,
    ! [B2: real,A2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B2 @ A2 ) )
     => ( ( ord_less_real @ zero_zero_real @ A2 )
       => ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).

% zero_less_mult_pos2
thf(fact_1773_zero__less__mult__pos2,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B2 @ A2 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A2 )
       => ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ).

% zero_less_mult_pos2
thf(fact_1774_zero__less__mult__pos2,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B2 @ A2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A2 )
       => ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).

% zero_less_mult_pos2
thf(fact_1775_zero__less__mult__pos2,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B2 @ A2 ) )
     => ( ( ord_less_int @ zero_zero_int @ A2 )
       => ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).

% zero_less_mult_pos2
thf(fact_1776_zero__less__mult__pos,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) )
     => ( ( ord_less_real @ zero_zero_real @ A2 )
       => ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).

% zero_less_mult_pos
thf(fact_1777_zero__less__mult__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A2 )
       => ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ).

% zero_less_mult_pos
thf(fact_1778_zero__less__mult__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A2 )
       => ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).

% zero_less_mult_pos
thf(fact_1779_zero__less__mult__pos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) )
     => ( ( ord_less_int @ zero_zero_int @ A2 )
       => ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).

% zero_less_mult_pos
thf(fact_1780_zero__less__mult__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A2 )
          & ( ord_less_real @ zero_zero_real @ B2 ) )
        | ( ( ord_less_real @ A2 @ zero_zero_real )
          & ( ord_less_real @ B2 @ zero_zero_real ) ) ) ) ).

% zero_less_mult_iff
thf(fact_1781_zero__less__mult__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
          & ( ord_less_rat @ zero_zero_rat @ B2 ) )
        | ( ( ord_less_rat @ A2 @ zero_zero_rat )
          & ( ord_less_rat @ B2 @ zero_zero_rat ) ) ) ) ).

% zero_less_mult_iff
thf(fact_1782_zero__less__mult__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ A2 )
          & ( ord_less_int @ zero_zero_int @ B2 ) )
        | ( ( ord_less_int @ A2 @ zero_zero_int )
          & ( ord_less_int @ B2 @ zero_zero_int ) ) ) ) ).

% zero_less_mult_iff
thf(fact_1783_mult__pos__neg2,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ B2 @ A2 ) @ zero_zero_real ) ) ) ).

% mult_pos_neg2
thf(fact_1784_mult__pos__neg2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ B2 @ A2 ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg2
thf(fact_1785_mult__pos__neg2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ B2 @ A2 ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg2
thf(fact_1786_mult__pos__neg2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ B2 @ A2 ) @ zero_zero_int ) ) ) ).

% mult_pos_neg2
thf(fact_1787_mult__pos__pos,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) ) ) ) ).

% mult_pos_pos
thf(fact_1788_mult__pos__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ).

% mult_pos_pos
thf(fact_1789_mult__pos__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) ) ) ) ).

% mult_pos_pos
thf(fact_1790_mult__pos__pos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).

% mult_pos_pos
thf(fact_1791_mult__pos__neg,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).

% mult_pos_neg
thf(fact_1792_mult__pos__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg
thf(fact_1793_mult__pos__neg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg
thf(fact_1794_mult__pos__neg,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% mult_pos_neg
thf(fact_1795_mult__neg__pos,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).

% mult_neg_pos
thf(fact_1796_mult__neg__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% mult_neg_pos
thf(fact_1797_mult__neg__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% mult_neg_pos
thf(fact_1798_mult__neg__pos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% mult_neg_pos
thf(fact_1799_mult__less__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ A2 @ B2 ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A2 )
          & ( ord_less_real @ B2 @ zero_zero_real ) )
        | ( ( ord_less_real @ A2 @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B2 ) ) ) ) ).

% mult_less_0_iff
thf(fact_1800_mult__less__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A2 @ B2 ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
          & ( ord_less_rat @ B2 @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A2 @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ) ).

% mult_less_0_iff
thf(fact_1801_mult__less__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int )
      = ( ( ( ord_less_int @ zero_zero_int @ A2 )
          & ( ord_less_int @ B2 @ zero_zero_int ) )
        | ( ( ord_less_int @ A2 @ zero_zero_int )
          & ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).

% mult_less_0_iff
thf(fact_1802_not__square__less__zero,axiom,
    ! [A2: real] :
      ~ ( ord_less_real @ ( times_times_real @ A2 @ A2 ) @ zero_zero_real ) ).

% not_square_less_zero
thf(fact_1803_not__square__less__zero,axiom,
    ! [A2: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A2 @ A2 ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_1804_not__square__less__zero,axiom,
    ! [A2: int] :
      ~ ( ord_less_int @ ( times_times_int @ A2 @ A2 ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_1805_mult__neg__neg,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ zero_zero_real )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) ) ) ) ).

% mult_neg_neg
thf(fact_1806_mult__neg__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ) ).

% mult_neg_neg
thf(fact_1807_mult__neg__neg,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).

% mult_neg_neg
thf(fact_1808_divide__right__mono__neg,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ C2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C2 ) @ ( divide_divide_real @ A2 @ C2 ) ) ) ) ).

% divide_right_mono_neg
thf(fact_1809_divide__right__mono__neg,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C2 ) @ ( divide_divide_rat @ A2 @ C2 ) ) ) ) ).

% divide_right_mono_neg
thf(fact_1810_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_1811_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_1812_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_1813_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_1814_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_1815_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_1816_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_1817_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_1818_zero__le__divide__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A2 @ B2 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) )
        | ( ( ord_less_eq_real @ A2 @ zero_zero_real )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).

% zero_le_divide_iff
thf(fact_1819_zero__le__divide__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A2 @ B2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) )
        | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) ) ) ).

% zero_le_divide_iff
thf(fact_1820_divide__right__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
       => ( ord_less_eq_real @ ( divide_divide_real @ A2 @ C2 ) @ ( divide_divide_real @ B2 @ C2 ) ) ) ) ).

% divide_right_mono
thf(fact_1821_divide__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ A2 @ C2 ) @ ( divide_divide_rat @ B2 @ C2 ) ) ) ) ).

% divide_right_mono
thf(fact_1822_divide__le__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A2 @ B2 ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
          & ( ord_less_eq_real @ B2 @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A2 @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).

% divide_le_0_iff
thf(fact_1823_divide__le__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A2 @ B2 ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ B2 @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ) ) ).

% divide_le_0_iff
thf(fact_1824_divide__strict__right__mono__neg,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( ord_less_real @ B2 @ A2 )
     => ( ( ord_less_real @ C2 @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ A2 @ C2 ) @ ( divide_divide_real @ B2 @ C2 ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_1825_divide__strict__right__mono__neg,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ( ord_less_rat @ C2 @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ A2 @ C2 ) @ ( divide_divide_rat @ B2 @ C2 ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_1826_divide__strict__right__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ C2 )
       => ( ord_less_real @ ( divide_divide_real @ A2 @ C2 ) @ ( divide_divide_real @ B2 @ C2 ) ) ) ) ).

% divide_strict_right_mono
thf(fact_1827_divide__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ ( divide_divide_rat @ A2 @ C2 ) @ ( divide_divide_rat @ B2 @ C2 ) ) ) ) ).

% divide_strict_right_mono
thf(fact_1828_zero__less__divide__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A2 @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A2 )
          & ( ord_less_real @ zero_zero_real @ B2 ) )
        | ( ( ord_less_real @ A2 @ zero_zero_real )
          & ( ord_less_real @ B2 @ zero_zero_real ) ) ) ) ).

% zero_less_divide_iff
thf(fact_1829_zero__less__divide__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A2 @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
          & ( ord_less_rat @ zero_zero_rat @ B2 ) )
        | ( ( ord_less_rat @ A2 @ zero_zero_rat )
          & ( ord_less_rat @ B2 @ zero_zero_rat ) ) ) ) ).

% zero_less_divide_iff
thf(fact_1830_divide__less__cancel,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A2 @ C2 ) @ ( divide_divide_real @ B2 @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ A2 @ B2 ) )
        & ( ( ord_less_real @ C2 @ zero_zero_real )
         => ( ord_less_real @ B2 @ A2 ) )
        & ( C2 != zero_zero_real ) ) ) ).

% divide_less_cancel
thf(fact_1831_divide__less__cancel,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A2 @ C2 ) @ ( divide_divide_rat @ B2 @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ A2 @ B2 ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ A2 ) )
        & ( C2 != zero_zero_rat ) ) ) ).

% divide_less_cancel
thf(fact_1832_divide__less__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A2 @ B2 ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A2 )
          & ( ord_less_real @ B2 @ zero_zero_real ) )
        | ( ( ord_less_real @ A2 @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B2 ) ) ) ) ).

% divide_less_0_iff
thf(fact_1833_divide__less__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A2 @ B2 ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
          & ( ord_less_rat @ B2 @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A2 @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B2 ) ) ) ) ).

% divide_less_0_iff
thf(fact_1834_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_1835_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_1836_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_1837_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_1838_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_1839_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_1840_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_1841_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_1842_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,N3: real,J2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N3 )
     => ( ( ord_less_eq_real @ N3 @ ( plus_plus_real @ J2 @ K ) )
       => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N3 )
         => ( ( ord_less_eq_real @ N3 @ ( plus_plus_real @ J2 @ K ) )
           => ( ord_less_eq_real @ ( minus_minus_real @ N3 @ K ) @ J2 ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1843_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,N3: rat,J2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N3 )
     => ( ( ord_less_eq_rat @ N3 @ ( plus_plus_rat @ J2 @ K ) )
       => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N3 )
         => ( ( ord_less_eq_rat @ N3 @ ( plus_plus_rat @ J2 @ K ) )
           => ( ord_less_eq_rat @ ( minus_minus_rat @ N3 @ K ) @ J2 ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1844_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N3: nat,J2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N3 )
     => ( ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ J2 @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N3 )
         => ( ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ J2 @ K ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N3 @ K ) @ J2 ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1845_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N3: int,J2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N3 )
     => ( ( ord_less_eq_int @ N3 @ ( plus_plus_int @ J2 @ K ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N3 )
         => ( ( ord_less_eq_int @ N3 @ ( plus_plus_int @ J2 @ K ) )
           => ( ord_less_eq_int @ ( minus_minus_int @ N3 @ K ) @ J2 ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1846_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,N3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N3 )
     => ( ord_less_eq_real @ I @ ( minus_minus_real @ N3 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_1847_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,N3: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N3 )
     => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N3 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_1848_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N3 )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N3 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_1849_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N3 )
     => ( ord_less_eq_int @ I @ ( minus_minus_int @ N3 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_1850_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: real,B2: real] :
      ( ~ ( ord_less_real @ A2 @ B2 )
     => ( ( plus_plus_real @ B2 @ ( minus_minus_real @ A2 @ B2 ) )
        = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_1851_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: rat,B2: rat] :
      ( ~ ( ord_less_rat @ A2 @ B2 )
     => ( ( plus_plus_rat @ B2 @ ( minus_minus_rat @ A2 @ B2 ) )
        = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_1852_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: nat,B2: nat] :
      ( ~ ( ord_less_nat @ A2 @ B2 )
     => ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A2 @ B2 ) )
        = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_1853_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: int,B2: int] :
      ( ~ ( ord_less_int @ A2 @ B2 )
     => ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A2 @ B2 ) )
        = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_1854_nonzero__eq__divide__eq,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( A2
          = ( divide1717551699836669952omplex @ B2 @ C2 ) )
        = ( ( times_times_complex @ A2 @ C2 )
          = B2 ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_1855_nonzero__eq__divide__eq,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( A2
          = ( divide_divide_real @ B2 @ C2 ) )
        = ( ( times_times_real @ A2 @ C2 )
          = B2 ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_1856_nonzero__eq__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( A2
          = ( divide_divide_rat @ B2 @ C2 ) )
        = ( ( times_times_rat @ A2 @ C2 )
          = B2 ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_1857_nonzero__divide__eq__eq,axiom,
    ! [C2: complex,B2: complex,A2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( ( divide1717551699836669952omplex @ B2 @ C2 )
          = A2 )
        = ( B2
          = ( times_times_complex @ A2 @ C2 ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_1858_nonzero__divide__eq__eq,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( ( divide_divide_real @ B2 @ C2 )
          = A2 )
        = ( B2
          = ( times_times_real @ A2 @ C2 ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_1859_nonzero__divide__eq__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( ( divide_divide_rat @ B2 @ C2 )
          = A2 )
        = ( B2
          = ( times_times_rat @ A2 @ C2 ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_1860_eq__divide__imp,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( ( times_times_complex @ A2 @ C2 )
          = B2 )
       => ( A2
          = ( divide1717551699836669952omplex @ B2 @ C2 ) ) ) ) ).

% eq_divide_imp
thf(fact_1861_eq__divide__imp,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( ( times_times_real @ A2 @ C2 )
          = B2 )
       => ( A2
          = ( divide_divide_real @ B2 @ C2 ) ) ) ) ).

% eq_divide_imp
thf(fact_1862_eq__divide__imp,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( ( times_times_rat @ A2 @ C2 )
          = B2 )
       => ( A2
          = ( divide_divide_rat @ B2 @ C2 ) ) ) ) ).

% eq_divide_imp
thf(fact_1863_divide__eq__imp,axiom,
    ! [C2: complex,B2: complex,A2: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( B2
          = ( times_times_complex @ A2 @ C2 ) )
       => ( ( divide1717551699836669952omplex @ B2 @ C2 )
          = A2 ) ) ) ).

% divide_eq_imp
thf(fact_1864_divide__eq__imp,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( ( C2 != zero_zero_real )
     => ( ( B2
          = ( times_times_real @ A2 @ C2 ) )
       => ( ( divide_divide_real @ B2 @ C2 )
          = A2 ) ) ) ).

% divide_eq_imp
thf(fact_1865_divide__eq__imp,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( B2
          = ( times_times_rat @ A2 @ C2 ) )
       => ( ( divide_divide_rat @ B2 @ C2 )
          = A2 ) ) ) ).

% divide_eq_imp
thf(fact_1866_eq__divide__eq,axiom,
    ! [A2: complex,B2: complex,C2: complex] :
      ( ( A2
        = ( divide1717551699836669952omplex @ B2 @ C2 ) )
      = ( ( ( C2 != zero_zero_complex )
         => ( ( times_times_complex @ A2 @ C2 )
            = B2 ) )
        & ( ( C2 = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% eq_divide_eq
thf(fact_1867_eq__divide__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( A2
        = ( divide_divide_real @ B2 @ C2 ) )
      = ( ( ( C2 != zero_zero_real )
         => ( ( times_times_real @ A2 @ C2 )
            = B2 ) )
        & ( ( C2 = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% eq_divide_eq
thf(fact_1868_eq__divide__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( A2
        = ( divide_divide_rat @ B2 @ C2 ) )
      = ( ( ( C2 != zero_zero_rat )
         => ( ( times_times_rat @ A2 @ C2 )
            = B2 ) )
        & ( ( C2 = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
thf(fact_1869_divide__eq__eq,axiom,
    ! [B2: complex,C2: complex,A2: complex] :
      ( ( ( divide1717551699836669952omplex @ B2 @ C2 )
        = A2 )
      = ( ( ( C2 != zero_zero_complex )
         => ( B2
            = ( times_times_complex @ A2 @ C2 ) ) )
        & ( ( C2 = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% divide_eq_eq
thf(fact_1870_divide__eq__eq,axiom,
    ! [B2: real,C2: real,A2: real] :
      ( ( ( divide_divide_real @ B2 @ C2 )
        = A2 )
      = ( ( ( C2 != zero_zero_real )
         => ( B2
            = ( times_times_real @ A2 @ C2 ) ) )
        & ( ( C2 = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% divide_eq_eq
thf(fact_1871_divide__eq__eq,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( ( divide_divide_rat @ B2 @ C2 )
        = A2 )
      = ( ( ( C2 != zero_zero_rat )
         => ( B2
            = ( times_times_rat @ A2 @ C2 ) ) )
        & ( ( C2 = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
thf(fact_1872_frac__eq__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( ( divide1717551699836669952omplex @ X @ Y )
            = ( divide1717551699836669952omplex @ W @ Z ) )
          = ( ( times_times_complex @ X @ Z )
            = ( times_times_complex @ W @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_1873_frac__eq__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ( divide_divide_real @ X @ Y )
            = ( divide_divide_real @ W @ Z ) )
          = ( ( times_times_real @ X @ Z )
            = ( times_times_real @ W @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_1874_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ( divide_divide_rat @ X @ Y )
            = ( divide_divide_rat @ W @ Z ) )
          = ( ( times_times_rat @ X @ Z )
            = ( times_times_rat @ W @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_1875_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_1876_L2__set__mult__ineq__lemma,axiom,
    ! [A2: real,C2: real,B2: real,D2: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A2 @ C2 ) ) @ ( times_times_real @ B2 @ D2 ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% L2_set_mult_ineq_lemma
thf(fact_1877_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_1878_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_1879_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_1880_eq__add__iff2,axiom,
    ! [A2: real,E: real,C2: real,B2: real,D2: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A2 @ E ) @ C2 )
        = ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D2 ) )
      = ( C2
        = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_1881_eq__add__iff2,axiom,
    ! [A2: rat,E: rat,C2: rat,B2: rat,D2: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A2 @ E ) @ C2 )
        = ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D2 ) )
      = ( C2
        = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_1882_eq__add__iff2,axiom,
    ! [A2: int,E: int,C2: int,B2: int,D2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C2 )
        = ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D2 ) )
      = ( C2
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_1883_eq__add__iff1,axiom,
    ! [A2: real,E: real,C2: real,B2: real,D2: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A2 @ E ) @ C2 )
        = ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D2 ) )
      = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A2 @ B2 ) @ E ) @ C2 )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_1884_eq__add__iff1,axiom,
    ! [A2: rat,E: rat,C2: rat,B2: rat,D2: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A2 @ E ) @ C2 )
        = ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D2 ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B2 ) @ E ) @ C2 )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_1885_eq__add__iff1,axiom,
    ! [A2: int,E: int,C2: int,B2: int,D2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C2 )
        = ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D2 ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ E ) @ C2 )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_1886_div__mult__le,axiom,
    ! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ B2 ) @ A2 ) ).

% div_mult_le
thf(fact_1887_Diff__insert__absorb,axiom,
    ! [X: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X @ A )
     => ( ( minus_5127226145743854075T_VEBT @ ( insert_VEBT_VEBT @ X @ A ) @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_1888_Diff__insert__absorb,axiom,
    ! [X: complex,A: set_complex] :
      ( ~ ( member_complex @ X @ A )
     => ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A ) @ ( insert_complex @ X @ bot_bot_set_complex ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_1889_Diff__insert__absorb,axiom,
    ! [X: real,A: set_real] :
      ( ~ ( member_real @ X @ A )
     => ( ( minus_minus_set_real @ ( insert_real @ X @ A ) @ ( insert_real @ X @ bot_bot_set_real ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_1890_Diff__insert__absorb,axiom,
    ! [X: $o,A: set_o] :
      ( ~ ( member_o @ X @ A )
     => ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ ( insert_o @ X @ bot_bot_set_o ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_1891_Diff__insert__absorb,axiom,
    ! [X: int,A: set_int] :
      ( ~ ( member_int @ X @ A )
     => ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ ( insert_int @ X @ bot_bot_set_int ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_1892_Diff__insert__absorb,axiom,
    ! [X: nat,A: set_nat] :
      ( ~ ( member_nat @ X @ A )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_1893_Diff__insert2,axiom,
    ! [A: set_VEBT_VEBT,A2: vEBT_VEBT,B: set_VEBT_VEBT] :
      ( ( minus_5127226145743854075T_VEBT @ A @ ( insert_VEBT_VEBT @ A2 @ B ) )
      = ( minus_5127226145743854075T_VEBT @ ( minus_5127226145743854075T_VEBT @ A @ ( insert_VEBT_VEBT @ A2 @ bot_bo8194388402131092736T_VEBT ) ) @ B ) ) ).

% Diff_insert2
thf(fact_1894_Diff__insert2,axiom,
    ! [A: set_real,A2: real,B: set_real] :
      ( ( minus_minus_set_real @ A @ ( insert_real @ A2 @ B ) )
      = ( minus_minus_set_real @ ( minus_minus_set_real @ A @ ( insert_real @ A2 @ bot_bot_set_real ) ) @ B ) ) ).

% Diff_insert2
thf(fact_1895_Diff__insert2,axiom,
    ! [A: set_o,A2: $o,B: set_o] :
      ( ( minus_minus_set_o @ A @ ( insert_o @ A2 @ B ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ B ) ) ).

% Diff_insert2
thf(fact_1896_Diff__insert2,axiom,
    ! [A: set_int,A2: int,B: set_int] :
      ( ( minus_minus_set_int @ A @ ( insert_int @ A2 @ B ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ B ) ) ).

% Diff_insert2
thf(fact_1897_Diff__insert2,axiom,
    ! [A: set_nat,A2: nat,B: set_nat] :
      ( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ B ) ) ).

% Diff_insert2
thf(fact_1898_insert__Diff,axiom,
    ! [A2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ A2 @ A )
     => ( ( insert_VEBT_VEBT @ A2 @ ( minus_5127226145743854075T_VEBT @ A @ ( insert_VEBT_VEBT @ A2 @ bot_bo8194388402131092736T_VEBT ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_1899_insert__Diff,axiom,
    ! [A2: complex,A: set_complex] :
      ( ( member_complex @ A2 @ A )
     => ( ( insert_complex @ A2 @ ( minus_811609699411566653omplex @ A @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_1900_insert__Diff,axiom,
    ! [A2: real,A: set_real] :
      ( ( member_real @ A2 @ A )
     => ( ( insert_real @ A2 @ ( minus_minus_set_real @ A @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_1901_insert__Diff,axiom,
    ! [A2: $o,A: set_o] :
      ( ( member_o @ A2 @ A )
     => ( ( insert_o @ A2 @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_1902_insert__Diff,axiom,
    ! [A2: int,A: set_int] :
      ( ( member_int @ A2 @ A )
     => ( ( insert_int @ A2 @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_1903_insert__Diff,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ A )
     => ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_1904_Diff__insert,axiom,
    ! [A: set_VEBT_VEBT,A2: vEBT_VEBT,B: set_VEBT_VEBT] :
      ( ( minus_5127226145743854075T_VEBT @ A @ ( insert_VEBT_VEBT @ A2 @ B ) )
      = ( minus_5127226145743854075T_VEBT @ ( minus_5127226145743854075T_VEBT @ A @ B ) @ ( insert_VEBT_VEBT @ A2 @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% Diff_insert
thf(fact_1905_Diff__insert,axiom,
    ! [A: set_real,A2: real,B: set_real] :
      ( ( minus_minus_set_real @ A @ ( insert_real @ A2 @ B ) )
      = ( minus_minus_set_real @ ( minus_minus_set_real @ A @ B ) @ ( insert_real @ A2 @ bot_bot_set_real ) ) ) ).

% Diff_insert
thf(fact_1906_Diff__insert,axiom,
    ! [A: set_o,A2: $o,B: set_o] :
      ( ( minus_minus_set_o @ A @ ( insert_o @ A2 @ B ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A @ B ) @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ).

% Diff_insert
thf(fact_1907_Diff__insert,axiom,
    ! [A: set_int,A2: int,B: set_int] :
      ( ( minus_minus_set_int @ A @ ( insert_int @ A2 @ B ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A @ B ) @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ).

% Diff_insert
thf(fact_1908_Diff__insert,axiom,
    ! [A: set_nat,A2: nat,B: set_nat] :
      ( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).

% Diff_insert
thf(fact_1909_psubset__insert__iff,axiom,
    ! [A: set_VEBT_VEBT,X: vEBT_VEBT,B: set_VEBT_VEBT] :
      ( ( ord_le3480810397992357184T_VEBT @ A @ ( insert_VEBT_VEBT @ X @ B ) )
      = ( ( ( member_VEBT_VEBT @ X @ B )
         => ( ord_le3480810397992357184T_VEBT @ A @ B ) )
        & ( ~ ( member_VEBT_VEBT @ X @ B )
         => ( ( ( member_VEBT_VEBT @ X @ A )
             => ( ord_le3480810397992357184T_VEBT @ ( minus_5127226145743854075T_VEBT @ A @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) @ B ) )
            & ( ~ ( member_VEBT_VEBT @ X @ A )
             => ( ord_le4337996190870823476T_VEBT @ A @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1910_psubset__insert__iff,axiom,
    ! [A: set_complex,X: complex,B: set_complex] :
      ( ( ord_less_set_complex @ A @ ( insert_complex @ X @ B ) )
      = ( ( ( member_complex @ X @ B )
         => ( ord_less_set_complex @ A @ B ) )
        & ( ~ ( member_complex @ X @ B )
         => ( ( ( member_complex @ X @ A )
             => ( ord_less_set_complex @ ( minus_811609699411566653omplex @ A @ ( insert_complex @ X @ bot_bot_set_complex ) ) @ B ) )
            & ( ~ ( member_complex @ X @ A )
             => ( ord_le211207098394363844omplex @ A @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1911_psubset__insert__iff,axiom,
    ! [A: set_real,X: real,B: set_real] :
      ( ( ord_less_set_real @ A @ ( insert_real @ X @ B ) )
      = ( ( ( member_real @ X @ B )
         => ( ord_less_set_real @ A @ B ) )
        & ( ~ ( member_real @ X @ B )
         => ( ( ( member_real @ X @ A )
             => ( ord_less_set_real @ ( minus_minus_set_real @ A @ ( insert_real @ X @ bot_bot_set_real ) ) @ B ) )
            & ( ~ ( member_real @ X @ A )
             => ( ord_less_eq_set_real @ A @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1912_psubset__insert__iff,axiom,
    ! [A: set_o,X: $o,B: set_o] :
      ( ( ord_less_set_o @ A @ ( insert_o @ X @ B ) )
      = ( ( ( member_o @ X @ B )
         => ( ord_less_set_o @ A @ B ) )
        & ( ~ ( member_o @ X @ B )
         => ( ( ( member_o @ X @ A )
             => ( ord_less_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ B ) )
            & ( ~ ( member_o @ X @ A )
             => ( ord_less_eq_set_o @ A @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1913_psubset__insert__iff,axiom,
    ! [A: set_int,X: int,B: set_int] :
      ( ( ord_less_set_int @ A @ ( insert_int @ X @ B ) )
      = ( ( ( member_int @ X @ B )
         => ( ord_less_set_int @ A @ B ) )
        & ( ~ ( member_int @ X @ B )
         => ( ( ( member_int @ X @ A )
             => ( ord_less_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ B ) )
            & ( ~ ( member_int @ X @ A )
             => ( ord_less_eq_set_int @ A @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1914_psubset__insert__iff,axiom,
    ! [A: set_nat,X: nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ ( insert_nat @ X @ B ) )
      = ( ( ( member_nat @ X @ B )
         => ( ord_less_set_nat @ A @ B ) )
        & ( ~ ( member_nat @ X @ B )
         => ( ( ( member_nat @ X @ A )
             => ( ord_less_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) )
            & ( ~ ( member_nat @ X @ A )
             => ( ord_less_eq_set_nat @ A @ B ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1915_Diff__single__insert,axiom,
    ! [A: set_VEBT_VEBT,X: vEBT_VEBT,B: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( minus_5127226145743854075T_VEBT @ A @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) @ B )
     => ( ord_le4337996190870823476T_VEBT @ A @ ( insert_VEBT_VEBT @ X @ B ) ) ) ).

% Diff_single_insert
thf(fact_1916_Diff__single__insert,axiom,
    ! [A: set_real,X: real,B: set_real] :
      ( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A @ ( insert_real @ X @ bot_bot_set_real ) ) @ B )
     => ( ord_less_eq_set_real @ A @ ( insert_real @ X @ B ) ) ) ).

% Diff_single_insert
thf(fact_1917_Diff__single__insert,axiom,
    ! [A: set_o,X: $o,B: set_o] :
      ( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ B )
     => ( ord_less_eq_set_o @ A @ ( insert_o @ X @ B ) ) ) ).

% Diff_single_insert
thf(fact_1918_Diff__single__insert,axiom,
    ! [A: set_int,X: int,B: set_int] :
      ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ B )
     => ( ord_less_eq_set_int @ A @ ( insert_int @ X @ B ) ) ) ).

% Diff_single_insert
thf(fact_1919_Diff__single__insert,axiom,
    ! [A: set_nat,X: nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B )
     => ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B ) ) ) ).

% Diff_single_insert
thf(fact_1920_subset__insert__iff,axiom,
    ! [A: set_VEBT_VEBT,X: vEBT_VEBT,B: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A @ ( insert_VEBT_VEBT @ X @ B ) )
      = ( ( ( member_VEBT_VEBT @ X @ A )
         => ( ord_le4337996190870823476T_VEBT @ ( minus_5127226145743854075T_VEBT @ A @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) @ B ) )
        & ( ~ ( member_VEBT_VEBT @ X @ A )
         => ( ord_le4337996190870823476T_VEBT @ A @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_1921_subset__insert__iff,axiom,
    ! [A: set_complex,X: complex,B: set_complex] :
      ( ( ord_le211207098394363844omplex @ A @ ( insert_complex @ X @ B ) )
      = ( ( ( member_complex @ X @ A )
         => ( ord_le211207098394363844omplex @ ( minus_811609699411566653omplex @ A @ ( insert_complex @ X @ bot_bot_set_complex ) ) @ B ) )
        & ( ~ ( member_complex @ X @ A )
         => ( ord_le211207098394363844omplex @ A @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_1922_subset__insert__iff,axiom,
    ! [A: set_real,X: real,B: set_real] :
      ( ( ord_less_eq_set_real @ A @ ( insert_real @ X @ B ) )
      = ( ( ( member_real @ X @ A )
         => ( ord_less_eq_set_real @ ( minus_minus_set_real @ A @ ( insert_real @ X @ bot_bot_set_real ) ) @ B ) )
        & ( ~ ( member_real @ X @ A )
         => ( ord_less_eq_set_real @ A @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_1923_subset__insert__iff,axiom,
    ! [A: set_o,X: $o,B: set_o] :
      ( ( ord_less_eq_set_o @ A @ ( insert_o @ X @ B ) )
      = ( ( ( member_o @ X @ A )
         => ( ord_less_eq_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ B ) )
        & ( ~ ( member_o @ X @ A )
         => ( ord_less_eq_set_o @ A @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_1924_subset__insert__iff,axiom,
    ! [A: set_int,X: int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ ( insert_int @ X @ B ) )
      = ( ( ( member_int @ X @ A )
         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ B ) )
        & ( ~ ( member_int @ X @ A )
         => ( ord_less_eq_set_int @ A @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_1925_subset__insert__iff,axiom,
    ! [A: set_nat,X: nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B ) )
      = ( ( ( member_nat @ X @ A )
         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) )
        & ( ~ ( member_nat @ X @ A )
         => ( ord_less_eq_set_nat @ A @ B ) ) ) ) ).

% subset_insert_iff
thf(fact_1926_field__le__epsilon,axiom,
    ! [X: real,Y: real] :
      ( ! [E2: real] :
          ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E2 ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_epsilon
thf(fact_1927_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y @ E2 ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_epsilon
thf(fact_1928_mult__less__le__imp__less,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ C2 @ D2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
         => ( ( ord_less_real @ zero_zero_real @ C2 )
           => ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_1929_mult__less__le__imp__less,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C2 @ D2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
         => ( ( ord_less_rat @ zero_zero_rat @ C2 )
           => ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_1930_mult__less__le__imp__less,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C2 @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
         => ( ( ord_less_nat @ zero_zero_nat @ C2 )
           => ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_1931_mult__less__le__imp__less,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ C2 @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
         => ( ( ord_less_int @ zero_zero_int @ C2 )
           => ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_1932_mult__le__less__imp__less,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_real @ C2 @ D2 )
       => ( ( ord_less_real @ zero_zero_real @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
           => ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_1933_mult__le__less__imp__less,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ C2 @ D2 )
       => ( ( ord_less_rat @ zero_zero_rat @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_1934_mult__le__less__imp__less,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ C2 @ D2 )
       => ( ( ord_less_nat @ zero_zero_nat @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_1935_mult__le__less__imp__less,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_int @ C2 @ D2 )
       => ( ( ord_less_int @ zero_zero_int @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_1936_mult__right__le__imp__le,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
     => ( ( ord_less_real @ zero_zero_real @ C2 )
       => ( ord_less_eq_real @ A2 @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_1937_mult__right__le__imp__le,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_1938_mult__right__le__imp__le,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_1939_mult__right__le__imp__le,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ A2 @ B2 ) ) ) ).

% mult_right_le_imp_le
thf(fact_1940_mult__left__le__imp__le,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
     => ( ( ord_less_real @ zero_zero_real @ C2 )
       => ( ord_less_eq_real @ A2 @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_1941_mult__left__le__imp__le,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_1942_mult__left__le__imp__le,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_1943_mult__left__le__imp__le,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ A2 @ B2 ) ) ) ).

% mult_left_le_imp_le
thf(fact_1944_mult__le__cancel__left__pos,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C2 )
     => ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
        = ( ord_less_eq_real @ A2 @ B2 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_1945_mult__le__cancel__left__pos,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
        = ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_1946_mult__le__cancel__left__pos,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ C2 )
     => ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
        = ( ord_less_eq_int @ A2 @ B2 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_1947_mult__le__cancel__left__neg,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ord_less_real @ C2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
        = ( ord_less_eq_real @ B2 @ A2 ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_1948_mult__le__cancel__left__neg,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
        = ( ord_less_eq_rat @ B2 @ A2 ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_1949_mult__le__cancel__left__neg,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_int @ C2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
        = ( ord_less_eq_int @ B2 @ A2 ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_1950_mult__less__cancel__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ A2 @ B2 ) )
        & ( ( ord_less_eq_real @ C2 @ zero_zero_real )
         => ( ord_less_real @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_right
thf(fact_1951_mult__less__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ A2 @ B2 ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_right
thf(fact_1952_mult__less__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ A2 @ B2 ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_right
thf(fact_1953_mult__strict__mono_H,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_real @ C2 @ D2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
           => ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_1954_mult__strict__mono_H,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ C2 @ D2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_1955_mult__strict__mono_H,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ C2 @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_1956_mult__strict__mono_H,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ C2 @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_1957_mult__right__less__imp__less,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
       => ( ord_less_real @ A2 @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_1958_mult__right__less__imp__less,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ A2 @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_1959_mult__right__less__imp__less,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ A2 @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_1960_mult__right__less__imp__less,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ A2 @ B2 ) ) ) ).

% mult_right_less_imp_less
thf(fact_1961_mult__less__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ A2 @ B2 ) )
        & ( ( ord_less_eq_real @ C2 @ zero_zero_real )
         => ( ord_less_real @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left
thf(fact_1962_mult__less__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ A2 @ B2 ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left
thf(fact_1963_mult__less__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ A2 @ B2 ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ B2 @ A2 ) ) ) ) ).

% mult_less_cancel_left
thf(fact_1964_mult__strict__mono,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_real @ C2 @ D2 )
       => ( ( ord_less_real @ zero_zero_real @ B2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
           => ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_1965_mult__strict__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ C2 @ D2 )
       => ( ( ord_less_rat @ zero_zero_rat @ B2 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_1966_mult__strict__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ C2 @ D2 )
       => ( ( ord_less_nat @ zero_zero_nat @ B2 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_1967_mult__strict__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ C2 @ D2 )
       => ( ( ord_less_int @ zero_zero_int @ B2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_1968_mult__left__less__imp__less,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
       => ( ord_less_real @ A2 @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_1969_mult__left__less__imp__less,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ A2 @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_1970_mult__left__less__imp__less,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ A2 @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_1971_mult__left__less__imp__less,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ A2 @ B2 ) ) ) ).

% mult_left_less_imp_less
thf(fact_1972_mult__le__cancel__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ A2 @ B2 ) )
        & ( ( ord_less_real @ C2 @ zero_zero_real )
         => ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_right
thf(fact_1973_mult__le__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ A2 @ B2 ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_right
thf(fact_1974_mult__le__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ A2 @ B2 ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_right
thf(fact_1975_mult__le__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ A2 @ B2 ) )
        & ( ( ord_less_real @ C2 @ zero_zero_real )
         => ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_left
thf(fact_1976_mult__le__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ A2 @ B2 ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_left
thf(fact_1977_mult__le__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ A2 @ B2 ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).

% mult_le_cancel_left
thf(fact_1978_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_1979_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_1980_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_1981_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_1982_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_1983_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_1984_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_1985_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_1986_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_1987_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_1988_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_1989_divide__le__cancel,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A2 @ C2 ) @ ( divide_divide_real @ B2 @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ A2 @ B2 ) )
        & ( ( ord_less_real @ C2 @ zero_zero_real )
         => ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).

% divide_le_cancel
thf(fact_1990_divide__le__cancel,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A2 @ C2 ) @ ( divide_divide_rat @ B2 @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ A2 @ B2 ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).

% divide_le_cancel
thf(fact_1991_frac__less2,axiom,
    ! [X: real,Y: real,W: real,Z: 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 @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).

% frac_less2
thf(fact_1992_frac__less2,axiom,
    ! [X: rat,Y: rat,W: rat,Z: 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 @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).

% frac_less2
thf(fact_1993_frac__less,axiom,
    ! [X: real,Y: real,W: real,Z: 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 @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).

% frac_less
thf(fact_1994_frac__less,axiom,
    ! [X: rat,Y: rat,W: rat,Z: 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 @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).

% frac_less
thf(fact_1995_frac__le,axiom,
    ! [Y: real,X: real,W: real,Z: 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 @ Z )
           => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).

% frac_le
thf(fact_1996_frac__le,axiom,
    ! [Y: rat,X: rat,W: rat,Z: 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 @ Z )
           => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).

% frac_le
thf(fact_1997_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_1998_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_1999_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_2000_divide__strict__left__mono__neg,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_real @ C2 @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) )
         => ( ord_less_real @ ( divide_divide_real @ C2 @ A2 ) @ ( divide_divide_real @ C2 @ B2 ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_2001_divide__strict__left__mono__neg,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ C2 @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C2 @ A2 ) @ ( divide_divide_rat @ C2 @ B2 ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_2002_divide__strict__left__mono,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( ord_less_real @ B2 @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ C2 )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) )
         => ( ord_less_real @ ( divide_divide_real @ C2 @ A2 ) @ ( divide_divide_real @ C2 @ B2 ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_2003_divide__strict__left__mono,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C2 @ A2 ) @ ( divide_divide_rat @ C2 @ B2 ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_2004_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ ( times_times_real @ Z @ Y ) @ X )
       => ( ord_less_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_2005_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y ) @ X )
       => ( ord_less_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_2006_mult__imp__div__pos__less,axiom,
    ! [Y: real,X: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ X @ ( times_times_real @ Z @ Y ) )
       => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_2007_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ X @ ( times_times_rat @ Z @ Y ) )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_2008_pos__less__divide__eq,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C2 )
     => ( ( ord_less_real @ A2 @ ( divide_divide_real @ B2 @ C2 ) )
        = ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ B2 ) ) ) ).

% pos_less_divide_eq
thf(fact_2009_pos__less__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B2 @ C2 ) )
        = ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ B2 ) ) ) ).

% pos_less_divide_eq
thf(fact_2010_pos__divide__less__eq,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( ( ord_less_real @ zero_zero_real @ C2 )
     => ( ( ord_less_real @ ( divide_divide_real @ B2 @ C2 ) @ A2 )
        = ( ord_less_real @ B2 @ ( times_times_real @ A2 @ C2 ) ) ) ) ).

% pos_divide_less_eq
thf(fact_2011_pos__divide__less__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C2 ) @ A2 )
        = ( ord_less_rat @ B2 @ ( times_times_rat @ A2 @ C2 ) ) ) ) ).

% pos_divide_less_eq
thf(fact_2012_neg__less__divide__eq,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ord_less_real @ C2 @ zero_zero_real )
     => ( ( ord_less_real @ A2 @ ( divide_divide_real @ B2 @ C2 ) )
        = ( ord_less_real @ B2 @ ( times_times_real @ A2 @ C2 ) ) ) ) ).

% neg_less_divide_eq
thf(fact_2013_neg__less__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B2 @ C2 ) )
        = ( ord_less_rat @ B2 @ ( times_times_rat @ A2 @ C2 ) ) ) ) ).

% neg_less_divide_eq
thf(fact_2014_neg__divide__less__eq,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( ( ord_less_real @ C2 @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B2 @ C2 ) @ A2 )
        = ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ B2 ) ) ) ).

% neg_divide_less_eq
thf(fact_2015_neg__divide__less__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C2 ) @ A2 )
        = ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ B2 ) ) ) ).

% neg_divide_less_eq
thf(fact_2016_less__divide__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_real @ A2 @ ( divide_divide_real @ B2 @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ B2 @ ( times_times_real @ A2 @ C2 ) ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ A2 @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_2017_less__divide__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B2 @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ B2 @ ( times_times_rat @ A2 @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ A2 @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_2018_divide__less__eq,axiom,
    ! [B2: real,C2: real,A2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ C2 ) @ A2 )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ B2 @ ( times_times_real @ A2 @ C2 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ B2 ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A2 ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_2019_divide__less__eq,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C2 ) @ A2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ B2 @ ( times_times_rat @ A2 @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A2 ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_2020_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A2: real,E: real,C2: real,B2: real,D2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D2 ) )
      = ( ord_less_eq_real @ C2 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_2021_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A2: rat,E: rat,C2: rat,B2: rat,D2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D2 ) )
      = ( ord_less_eq_rat @ C2 @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_2022_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A2: int,E: int,C2: int,B2: int,D2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D2 ) )
      = ( ord_less_eq_int @ C2 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_2023_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A2: real,E: real,C2: real,B2: real,D2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D2 ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A2 @ B2 ) @ E ) @ C2 ) @ D2 ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_2024_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A2: rat,E: rat,C2: rat,B2: rat,D2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D2 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B2 ) @ E ) @ C2 ) @ D2 ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_2025_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A2: int,E: int,C2: int,B2: int,D2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D2 ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ E ) @ C2 ) @ D2 ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_2026_divide__add__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2027_divide__add__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X @ Z ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2028_divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2029_add__divide__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2030_add__divide__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z ) )
        = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2031_add__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2032_add__num__frac,axiom,
    ! [Y: complex,Z: complex,X: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X @ Y ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_2033_add__num__frac,axiom,
    ! [Y: real,Z: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X @ Y ) )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_2034_add__num__frac,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X @ Y ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_2035_add__frac__num,axiom,
    ! [Y: complex,X: complex,Z: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ Z )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_2036_add__frac__num,axiom,
    ! [Y: real,X: real,Z: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_2037_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_2038_add__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2039_add__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2040_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2041_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: complex,A2: complex,B2: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ A2 @ ( divide1717551699836669952omplex @ B2 @ Z ) )
          = A2 ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ A2 @ ( divide1717551699836669952omplex @ B2 @ Z ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A2 @ Z ) @ B2 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2042_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: real,A2: real,B2: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ A2 @ ( divide_divide_real @ B2 @ Z ) )
          = A2 ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ A2 @ ( divide_divide_real @ B2 @ Z ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A2 @ Z ) @ B2 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2043_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: rat,A2: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ A2 @ ( divide_divide_rat @ B2 @ Z ) )
          = A2 ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ A2 @ ( divide_divide_rat @ B2 @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ Z ) @ B2 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2044_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: complex,A2: complex,B2: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A2 @ Z ) @ B2 )
          = B2 ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A2 @ Z ) @ B2 )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A2 @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2045_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: real,A2: real,B2: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A2 @ Z ) @ B2 )
          = B2 ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A2 @ Z ) @ B2 )
          = ( divide_divide_real @ ( plus_plus_real @ A2 @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2046_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: rat,A2: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A2 @ Z ) @ B2 )
          = B2 ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A2 @ Z ) @ B2 )
          = ( divide_divide_rat @ ( plus_plus_rat @ A2 @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2047_less__add__iff2,axiom,
    ! [A2: real,E: real,C2: real,B2: real,D2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D2 ) )
      = ( ord_less_real @ C2 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_2048_less__add__iff2,axiom,
    ! [A2: rat,E: rat,C2: rat,B2: rat,D2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D2 ) )
      = ( ord_less_rat @ C2 @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_2049_less__add__iff2,axiom,
    ! [A2: int,E: int,C2: int,B2: int,D2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D2 ) )
      = ( ord_less_int @ C2 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A2 ) @ E ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_2050_less__add__iff1,axiom,
    ! [A2: real,E: real,C2: real,B2: real,D2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B2 @ E ) @ D2 ) )
      = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A2 @ B2 ) @ E ) @ C2 ) @ D2 ) ) ).

% less_add_iff1
thf(fact_2051_less__add__iff1,axiom,
    ! [A2: rat,E: rat,C2: rat,B2: rat,D2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B2 @ E ) @ D2 ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B2 ) @ E ) @ C2 ) @ D2 ) ) ).

% less_add_iff1
thf(fact_2052_less__add__iff1,axiom,
    ! [A2: int,E: int,C2: int,B2: int,D2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D2 ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ E ) @ C2 ) @ D2 ) ) ).

% less_add_iff1
thf(fact_2053_divide__diff__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_2054_divide__diff__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ X @ Z ) @ Y )
        = ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_2055_divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_2056_diff__divide__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_2057_diff__divide__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_2058_diff__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_2059_diff__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_2060_diff__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_2061_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_2062_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: complex,A2: complex,B2: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ A2 @ ( divide1717551699836669952omplex @ B2 @ Z ) )
          = A2 ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ A2 @ ( divide1717551699836669952omplex @ B2 @ Z ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A2 @ Z ) @ B2 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_2063_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: real,A2: real,B2: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ A2 @ ( divide_divide_real @ B2 @ Z ) )
          = A2 ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ A2 @ ( divide_divide_real @ B2 @ Z ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A2 @ Z ) @ B2 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_2064_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: rat,A2: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ A2 @ ( divide_divide_rat @ B2 @ Z ) )
          = A2 ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ A2 @ ( divide_divide_rat @ B2 @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A2 @ Z ) @ B2 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_2065_td__gal__lt,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C2 )
     => ( ( ord_less_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) )
        = ( ord_less_nat @ ( divide_divide_nat @ A2 @ C2 ) @ B2 ) ) ) ).

% td_gal_lt
thf(fact_2066_divide__left__mono__neg,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ C2 @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C2 @ A2 ) @ ( divide_divide_real @ C2 @ B2 ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_2067_divide__left__mono__neg,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C2 @ A2 ) @ ( divide_divide_rat @ C2 @ B2 ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_2068_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y ) @ X )
       => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_2069_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y ) @ X )
       => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_2070_mult__imp__div__pos__le,axiom,
    ! [Y: real,X: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ X @ ( times_times_real @ Z @ Y ) )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_2071_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z @ Y ) )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_2072_pos__le__divide__eq,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C2 )
     => ( ( ord_less_eq_real @ A2 @ ( divide_divide_real @ B2 @ C2 ) )
        = ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ B2 ) ) ) ).

% pos_le_divide_eq
thf(fact_2073_pos__le__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B2 @ C2 ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ B2 ) ) ) ).

% pos_le_divide_eq
thf(fact_2074_pos__divide__le__eq,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( ( ord_less_real @ zero_zero_real @ C2 )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C2 ) @ A2 )
        = ( ord_less_eq_real @ B2 @ ( times_times_real @ A2 @ C2 ) ) ) ) ).

% pos_divide_le_eq
thf(fact_2075_pos__divide__le__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C2 ) @ A2 )
        = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A2 @ C2 ) ) ) ) ).

% pos_divide_le_eq
thf(fact_2076_neg__le__divide__eq,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ord_less_real @ C2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ A2 @ ( divide_divide_real @ B2 @ C2 ) )
        = ( ord_less_eq_real @ B2 @ ( times_times_real @ A2 @ C2 ) ) ) ) ).

% neg_le_divide_eq
thf(fact_2077_neg__le__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B2 @ C2 ) )
        = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A2 @ C2 ) ) ) ) ).

% neg_le_divide_eq
thf(fact_2078_neg__divide__le__eq,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( ( ord_less_real @ C2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C2 ) @ A2 )
        = ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ B2 ) ) ) ).

% neg_divide_le_eq
thf(fact_2079_neg__divide__le__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C2 ) @ A2 )
        = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ B2 ) ) ) ).

% neg_divide_le_eq
thf(fact_2080_divide__left__mono,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( ord_less_eq_real @ B2 @ A2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B2 ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C2 @ A2 ) @ ( divide_divide_real @ C2 @ B2 ) ) ) ) ) ).

% divide_left_mono
thf(fact_2081_divide__left__mono,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B2 ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C2 @ A2 ) @ ( divide_divide_rat @ C2 @ B2 ) ) ) ) ) ).

% divide_left_mono
thf(fact_2082_le__divide__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_eq_real @ A2 @ ( divide_divide_real @ B2 @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ B2 @ ( times_times_real @ A2 @ C2 ) ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_2083_le__divide__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B2 @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A2 @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_2084_divide__le__eq,axiom,
    ! [B2: real,C2: real,A2: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C2 ) @ A2 )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ B2 @ ( times_times_real @ A2 @ C2 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ B2 ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_2085_divide__le__eq,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C2 ) @ A2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A2 @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_2086_frac__le__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
          = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_le_eq
thf(fact_2087_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_le_eq
thf(fact_2088_frac__less__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
          = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_less_eq
thf(fact_2089_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_less_eq
thf(fact_2090_pos2,axiom,
    ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).

% pos2
thf(fact_2091_td__gal,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C2 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ B2 @ C2 ) @ A2 )
        = ( ord_less_eq_nat @ B2 @ ( divide_divide_nat @ A2 @ C2 ) ) ) ) ).

% td_gal
thf(fact_2092_power__sub,axiom,
    ! [N3: nat,M: nat,A2: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( ord_less_nat @ zero_zero_nat @ A2 )
       => ( ( power_power_nat @ A2 @ ( minus_minus_nat @ M @ N3 ) )
          = ( divide_divide_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N3 ) ) ) ) ) ).

% power_sub
thf(fact_2093_scaling__mono,axiom,
    ! [U: real,V: real,R3: real,S2: real] :
      ( ( ord_less_eq_real @ U @ V )
     => ( ( ord_less_eq_real @ zero_zero_real @ R3 )
       => ( ( ord_less_eq_real @ R3 @ S2 )
         => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R3 @ ( minus_minus_real @ V @ U ) ) @ S2 ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_2094_scaling__mono,axiom,
    ! [U: rat,V: rat,R3: rat,S2: rat] :
      ( ( ord_less_eq_rat @ U @ V )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ R3 )
       => ( ( ord_less_eq_rat @ R3 @ S2 )
         => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R3 @ ( minus_minus_rat @ V @ U ) ) @ S2 ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_2095_power__minus__is__div,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A2 @ B2 ) )
        = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).

% power_minus_is_div
thf(fact_2096_two__pow__div__gt__le,axiom,
    ! [V: nat,N3: nat,M: nat] :
      ( ( ord_less_nat @ V @ ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) )
     => ( ord_less_eq_nat @ M @ N3 ) ) ).

% two_pow_div_gt_le
thf(fact_2097_less__two__pow__divI,axiom,
    ! [X: nat,N3: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) ) )
     => ( ( ord_less_eq_nat @ M @ N3 )
       => ( ord_less_nat @ X @ ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).

% less_two_pow_divI
thf(fact_2098_less__two__pow__divD,axiom,
    ! [X: nat,N3: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) )
     => ( ( ord_less_eq_nat @ M @ N3 )
        & ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ).

% less_two_pow_divD
thf(fact_2099_less__eq__option__Some,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_le2843612097646854710et_nat @ ( some_set_nat @ X ) @ ( some_set_nat @ Y ) )
      = ( ord_less_eq_set_nat @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_2100_less__eq__option__Some,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_le2406147912482264968on_rat @ ( some_rat @ X ) @ ( some_rat @ Y ) )
      = ( ord_less_eq_rat @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_2101_less__eq__option__Some,axiom,
    ! [X: num,Y: num] :
      ( ( ord_le6622620407824499402on_num @ ( some_num @ X ) @ ( some_num @ Y ) )
      = ( ord_less_eq_num @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_2102_less__eq__option__Some,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_le5914376470875661696on_nat @ ( some_nat @ X ) @ ( some_nat @ Y ) )
      = ( ord_less_eq_nat @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_2103_less__eq__option__Some,axiom,
    ! [X: int,Y: int] :
      ( ( ord_le1736525451366464988on_int @ ( some_int @ X ) @ ( some_int @ Y ) )
      = ( ord_less_eq_int @ X @ Y ) ) ).

% less_eq_option_Some
thf(fact_2104_nat__less__power__trans,axiom,
    ! [N3: nat,M: nat,K: nat] :
      ( ( ord_less_nat @ N3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ K ) ) )
     => ( ( ord_less_eq_nat @ K @ M )
       => ( ord_less_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ N3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% nat_less_power_trans
thf(fact_2105_diff__add__zero,axiom,
    ! [A2: nat,B2: nat] :
      ( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_2106_diff__gt__0__iff__gt,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B2 ) )
      = ( ord_less_real @ B2 @ A2 ) ) ).

% diff_gt_0_iff_gt
thf(fact_2107_diff__gt__0__iff__gt,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B2 ) )
      = ( ord_less_rat @ B2 @ A2 ) ) ).

% diff_gt_0_iff_gt
thf(fact_2108_diff__gt__0__iff__gt,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B2 ) )
      = ( ord_less_int @ B2 @ A2 ) ) ).

% diff_gt_0_iff_gt
thf(fact_2109_diff__ge__0__iff__ge,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B2 ) )
      = ( ord_less_eq_real @ B2 @ A2 ) ) ).

% diff_ge_0_iff_ge
thf(fact_2110_diff__ge__0__iff__ge,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B2 ) )
      = ( ord_less_eq_rat @ B2 @ A2 ) ) ).

% diff_ge_0_iff_ge
thf(fact_2111_diff__ge__0__iff__ge,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B2 ) )
      = ( ord_less_eq_int @ B2 @ A2 ) ) ).

% diff_ge_0_iff_ge
thf(fact_2112_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ A2 ) )
      = ( ord_less_real @ zero_zero_real @ A2 ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_2113_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ A2 ) )
      = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_2114_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
      = ( ord_less_int @ zero_zero_int @ A2 ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_2115_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A2 @ A2 ) @ zero_zero_real )
      = ( ord_less_real @ A2 @ zero_zero_real ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_2116_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ A2 ) @ zero_zero_rat )
      = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_2117_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
      = ( ord_less_int @ A2 @ zero_zero_int ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_2118_less__add__same__cancel2,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ ( plus_plus_real @ B2 @ A2 ) )
      = ( ord_less_real @ zero_zero_real @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_2119_less__add__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ ( plus_plus_rat @ B2 @ A2 ) )
      = ( ord_less_rat @ zero_zero_rat @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_2120_less__add__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
      = ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_2121_less__add__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
      = ( ord_less_int @ zero_zero_int @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_2122_less__add__same__cancel1,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ ( plus_plus_real @ A2 @ B2 ) )
      = ( ord_less_real @ zero_zero_real @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_2123_less__add__same__cancel1,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ ( plus_plus_rat @ A2 @ B2 ) )
      = ( ord_less_rat @ zero_zero_rat @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_2124_less__add__same__cancel1,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
      = ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_2125_less__add__same__cancel1,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
      = ( ord_less_int @ zero_zero_int @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_2126_add__less__same__cancel2,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A2 @ B2 ) @ B2 )
      = ( ord_less_real @ A2 @ zero_zero_real ) ) ).

% add_less_same_cancel2
thf(fact_2127_add__less__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ B2 ) @ B2 )
      = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).

% add_less_same_cancel2
thf(fact_2128_add__less__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
      = ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_2129_add__less__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
      = ( ord_less_int @ A2 @ zero_zero_int ) ) ).

% add_less_same_cancel2
thf(fact_2130_add__left__cancel,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ( plus_plus_real @ A2 @ B2 )
        = ( plus_plus_real @ A2 @ C2 ) )
      = ( B2 = C2 ) ) ).

% add_left_cancel
thf(fact_2131_add__left__cancel,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ( plus_plus_rat @ A2 @ B2 )
        = ( plus_plus_rat @ A2 @ C2 ) )
      = ( B2 = C2 ) ) ).

% add_left_cancel
thf(fact_2132_add__left__cancel,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ( plus_plus_nat @ A2 @ B2 )
        = ( plus_plus_nat @ A2 @ C2 ) )
      = ( B2 = C2 ) ) ).

% add_left_cancel
thf(fact_2133_add__left__cancel,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ( plus_plus_int @ A2 @ B2 )
        = ( plus_plus_int @ A2 @ C2 ) )
      = ( B2 = C2 ) ) ).

% add_left_cancel
thf(fact_2134_add__right__cancel,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( ( plus_plus_real @ B2 @ A2 )
        = ( plus_plus_real @ C2 @ A2 ) )
      = ( B2 = C2 ) ) ).

% add_right_cancel
thf(fact_2135_add__right__cancel,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ( plus_plus_rat @ B2 @ A2 )
        = ( plus_plus_rat @ C2 @ A2 ) )
      = ( B2 = C2 ) ) ).

% add_right_cancel
thf(fact_2136_add__right__cancel,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( ( plus_plus_nat @ B2 @ A2 )
        = ( plus_plus_nat @ C2 @ A2 ) )
      = ( B2 = C2 ) ) ).

% add_right_cancel
thf(fact_2137_add__right__cancel,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ( plus_plus_int @ B2 @ A2 )
        = ( plus_plus_int @ C2 @ A2 ) )
      = ( B2 = C2 ) ) ).

% add_right_cancel
thf(fact_2138_real__divide__square__eq,axiom,
    ! [R3: real,A2: real] :
      ( ( divide_divide_real @ ( times_times_real @ R3 @ A2 ) @ ( times_times_real @ R3 @ R3 ) )
      = ( divide_divide_real @ A2 @ R3 ) ) ).

% real_divide_square_eq
thf(fact_2139_le__zero__eq,axiom,
    ! [N3: nat] :
      ( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
      = ( N3 = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_2140_not__gr__zero,axiom,
    ! [N3: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
      = ( N3 = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_2141_add_Oright__neutral,axiom,
    ! [A2: complex] :
      ( ( plus_plus_complex @ A2 @ zero_zero_complex )
      = A2 ) ).

% add.right_neutral
thf(fact_2142_add_Oright__neutral,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ A2 @ zero_zero_real )
      = A2 ) ).

% add.right_neutral
thf(fact_2143_add_Oright__neutral,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ A2 @ zero_zero_rat )
      = A2 ) ).

% add.right_neutral
thf(fact_2144_add_Oright__neutral,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% add.right_neutral
thf(fact_2145_add_Oright__neutral,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% add.right_neutral
thf(fact_2146_double__zero__sym,axiom,
    ! [A2: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A2 @ A2 ) )
      = ( A2 = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_2147_double__zero__sym,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A2 @ A2 ) )
      = ( A2 = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_2148_double__zero__sym,axiom,
    ! [A2: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A2 @ A2 ) )
      = ( A2 = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_2149_add__cancel__left__left,axiom,
    ! [B2: complex,A2: complex] :
      ( ( ( plus_plus_complex @ B2 @ A2 )
        = A2 )
      = ( B2 = zero_zero_complex ) ) ).

% add_cancel_left_left
thf(fact_2150_add__cancel__left__left,axiom,
    ! [B2: real,A2: real] :
      ( ( ( plus_plus_real @ B2 @ A2 )
        = A2 )
      = ( B2 = zero_zero_real ) ) ).

% add_cancel_left_left
thf(fact_2151_add__cancel__left__left,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ( plus_plus_rat @ B2 @ A2 )
        = A2 )
      = ( B2 = zero_zero_rat ) ) ).

% add_cancel_left_left
thf(fact_2152_add__cancel__left__left,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ( plus_plus_nat @ B2 @ A2 )
        = A2 )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_2153_add__cancel__left__left,axiom,
    ! [B2: int,A2: int] :
      ( ( ( plus_plus_int @ B2 @ A2 )
        = A2 )
      = ( B2 = zero_zero_int ) ) ).

% add_cancel_left_left
thf(fact_2154_add__cancel__left__right,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ( plus_plus_complex @ A2 @ B2 )
        = A2 )
      = ( B2 = zero_zero_complex ) ) ).

% add_cancel_left_right
thf(fact_2155_add__cancel__left__right,axiom,
    ! [A2: real,B2: real] :
      ( ( ( plus_plus_real @ A2 @ B2 )
        = A2 )
      = ( B2 = zero_zero_real ) ) ).

% add_cancel_left_right
thf(fact_2156_add__cancel__left__right,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( plus_plus_rat @ A2 @ B2 )
        = A2 )
      = ( B2 = zero_zero_rat ) ) ).

% add_cancel_left_right
thf(fact_2157_add__cancel__left__right,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( plus_plus_nat @ A2 @ B2 )
        = A2 )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_2158_add__cancel__left__right,axiom,
    ! [A2: int,B2: int] :
      ( ( ( plus_plus_int @ A2 @ B2 )
        = A2 )
      = ( B2 = zero_zero_int ) ) ).

% add_cancel_left_right
thf(fact_2159_add__cancel__right__left,axiom,
    ! [A2: complex,B2: complex] :
      ( ( A2
        = ( plus_plus_complex @ B2 @ A2 ) )
      = ( B2 = zero_zero_complex ) ) ).

% add_cancel_right_left
thf(fact_2160_add__cancel__right__left,axiom,
    ! [A2: real,B2: real] :
      ( ( A2
        = ( plus_plus_real @ B2 @ A2 ) )
      = ( B2 = zero_zero_real ) ) ).

% add_cancel_right_left
thf(fact_2161_add__cancel__right__left,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2
        = ( plus_plus_rat @ B2 @ A2 ) )
      = ( B2 = zero_zero_rat ) ) ).

% add_cancel_right_left
thf(fact_2162_add__cancel__right__left,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2
        = ( plus_plus_nat @ B2 @ A2 ) )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_2163_add__cancel__right__left,axiom,
    ! [A2: int,B2: int] :
      ( ( A2
        = ( plus_plus_int @ B2 @ A2 ) )
      = ( B2 = zero_zero_int ) ) ).

% add_cancel_right_left
thf(fact_2164_add__cancel__right__right,axiom,
    ! [A2: complex,B2: complex] :
      ( ( A2
        = ( plus_plus_complex @ A2 @ B2 ) )
      = ( B2 = zero_zero_complex ) ) ).

% add_cancel_right_right
thf(fact_2165_add__cancel__right__right,axiom,
    ! [A2: real,B2: real] :
      ( ( A2
        = ( plus_plus_real @ A2 @ B2 ) )
      = ( B2 = zero_zero_real ) ) ).

% add_cancel_right_right
thf(fact_2166_add__cancel__right__right,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2
        = ( plus_plus_rat @ A2 @ B2 ) )
      = ( B2 = zero_zero_rat ) ) ).

% add_cancel_right_right
thf(fact_2167_add__cancel__right__right,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2
        = ( plus_plus_nat @ A2 @ B2 ) )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_2168_add__cancel__right__right,axiom,
    ! [A2: int,B2: int] :
      ( ( A2
        = ( plus_plus_int @ A2 @ B2 ) )
      = ( B2 = zero_zero_int ) ) ).

% add_cancel_right_right
thf(fact_2169_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_2170_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_2171_add__0,axiom,
    ! [A2: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A2 )
      = A2 ) ).

% add_0
thf(fact_2172_add__0,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ zero_zero_real @ A2 )
      = A2 ) ).

% add_0
thf(fact_2173_add__0,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A2 )
      = A2 ) ).

% add_0
thf(fact_2174_add__0,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A2 )
      = A2 ) ).

% add_0
thf(fact_2175_add__0,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ zero_zero_int @ A2 )
      = A2 ) ).

% add_0
thf(fact_2176_add__le__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A2 ) @ ( plus_plus_real @ C2 @ B2 ) )
      = ( ord_less_eq_real @ A2 @ B2 ) ) ).

% add_le_cancel_left
thf(fact_2177_add__le__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C2 @ A2 ) @ ( plus_plus_rat @ C2 @ B2 ) )
      = ( ord_less_eq_rat @ A2 @ B2 ) ) ).

% add_le_cancel_left
thf(fact_2178_add__le__cancel__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B2 ) )
      = ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% add_le_cancel_left
thf(fact_2179_add__le__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A2 ) @ ( plus_plus_int @ C2 @ B2 ) )
      = ( ord_less_eq_int @ A2 @ B2 ) ) ).

% add_le_cancel_left
thf(fact_2180_add__le__cancel__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B2 @ C2 ) )
      = ( ord_less_eq_real @ A2 @ B2 ) ) ).

% add_le_cancel_right
thf(fact_2181_add__le__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C2 ) @ ( plus_plus_rat @ B2 @ C2 ) )
      = ( ord_less_eq_rat @ A2 @ B2 ) ) ).

% add_le_cancel_right
thf(fact_2182_add__le__cancel__right,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
      = ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% add_le_cancel_right
thf(fact_2183_add__le__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ C2 ) )
      = ( ord_less_eq_int @ A2 @ B2 ) ) ).

% add_le_cancel_right
thf(fact_2184_add__less__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C2 @ A2 ) @ ( plus_plus_real @ C2 @ B2 ) )
      = ( ord_less_real @ A2 @ B2 ) ) ).

% add_less_cancel_left
thf(fact_2185_add__less__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C2 @ A2 ) @ ( plus_plus_rat @ C2 @ B2 ) )
      = ( ord_less_rat @ A2 @ B2 ) ) ).

% add_less_cancel_left
thf(fact_2186_add__less__cancel__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B2 ) )
      = ( ord_less_nat @ A2 @ B2 ) ) ).

% add_less_cancel_left
thf(fact_2187_add__less__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C2 @ A2 ) @ ( plus_plus_int @ C2 @ B2 ) )
      = ( ord_less_int @ A2 @ B2 ) ) ).

% add_less_cancel_left
thf(fact_2188_add__less__cancel__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B2 @ C2 ) )
      = ( ord_less_real @ A2 @ B2 ) ) ).

% add_less_cancel_right
thf(fact_2189_add__less__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ C2 ) @ ( plus_plus_rat @ B2 @ C2 ) )
      = ( ord_less_rat @ A2 @ B2 ) ) ).

% add_less_cancel_right
thf(fact_2190_add__less__cancel__right,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
      = ( ord_less_nat @ A2 @ B2 ) ) ).

% add_less_cancel_right
thf(fact_2191_add__less__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ C2 ) )
      = ( ord_less_int @ A2 @ B2 ) ) ).

% add_less_cancel_right
thf(fact_2192_diff__self,axiom,
    ! [A2: complex] :
      ( ( minus_minus_complex @ A2 @ A2 )
      = zero_zero_complex ) ).

% diff_self
thf(fact_2193_diff__self,axiom,
    ! [A2: real] :
      ( ( minus_minus_real @ A2 @ A2 )
      = zero_zero_real ) ).

% diff_self
thf(fact_2194_diff__self,axiom,
    ! [A2: rat] :
      ( ( minus_minus_rat @ A2 @ A2 )
      = zero_zero_rat ) ).

% diff_self
thf(fact_2195_diff__self,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ A2 @ A2 )
      = zero_zero_int ) ).

% diff_self
thf(fact_2196_diff__0__right,axiom,
    ! [A2: complex] :
      ( ( minus_minus_complex @ A2 @ zero_zero_complex )
      = A2 ) ).

% diff_0_right
thf(fact_2197_diff__0__right,axiom,
    ! [A2: real] :
      ( ( minus_minus_real @ A2 @ zero_zero_real )
      = A2 ) ).

% diff_0_right
thf(fact_2198_diff__0__right,axiom,
    ! [A2: rat] :
      ( ( minus_minus_rat @ A2 @ zero_zero_rat )
      = A2 ) ).

% diff_0_right
thf(fact_2199_diff__0__right,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% diff_0_right
thf(fact_2200_zero__diff,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_2201_diff__zero,axiom,
    ! [A2: complex] :
      ( ( minus_minus_complex @ A2 @ zero_zero_complex )
      = A2 ) ).

% diff_zero
thf(fact_2202_diff__zero,axiom,
    ! [A2: real] :
      ( ( minus_minus_real @ A2 @ zero_zero_real )
      = A2 ) ).

% diff_zero
thf(fact_2203_diff__zero,axiom,
    ! [A2: rat] :
      ( ( minus_minus_rat @ A2 @ zero_zero_rat )
      = A2 ) ).

% diff_zero
thf(fact_2204_diff__zero,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% diff_zero
thf(fact_2205_diff__zero,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% diff_zero
thf(fact_2206_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: complex] :
      ( ( minus_minus_complex @ A2 @ A2 )
      = zero_zero_complex ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_2207_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: real] :
      ( ( minus_minus_real @ A2 @ A2 )
      = zero_zero_real ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_2208_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: rat] :
      ( ( minus_minus_rat @ A2 @ A2 )
      = zero_zero_rat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_2209_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ A2 @ A2 )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_2210_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ A2 @ A2 )
      = zero_zero_int ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_2211_add__diff__cancel,axiom,
    ! [A2: real,B2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel
thf(fact_2212_add__diff__cancel,axiom,
    ! [A2: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel
thf(fact_2213_add__diff__cancel,axiom,
    ! [A2: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel
thf(fact_2214_diff__add__cancel,axiom,
    ! [A2: real,B2: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% diff_add_cancel
thf(fact_2215_diff__add__cancel,axiom,
    ! [A2: rat,B2: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% diff_add_cancel
thf(fact_2216_diff__add__cancel,axiom,
    ! [A2: int,B2: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% diff_add_cancel
thf(fact_2217_add__diff__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ C2 @ A2 ) @ ( plus_plus_real @ C2 @ B2 ) )
      = ( minus_minus_real @ A2 @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_2218_add__diff__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ C2 @ A2 ) @ ( plus_plus_rat @ C2 @ B2 ) )
      = ( minus_minus_rat @ A2 @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_2219_add__diff__cancel__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B2 ) )
      = ( minus_minus_nat @ A2 @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_2220_add__diff__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ C2 @ A2 ) @ ( plus_plus_int @ C2 @ B2 ) )
      = ( minus_minus_int @ A2 @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_2221_add__diff__cancel__left_H,axiom,
    ! [A2: real,B2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A2 @ B2 ) @ A2 )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_2222_add__diff__cancel__left_H,axiom,
    ! [A2: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ A2 )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_2223_add__diff__cancel__left_H,axiom,
    ! [A2: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ A2 )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_2224_add__diff__cancel__left_H,axiom,
    ! [A2: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ A2 )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_2225_add__diff__cancel__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B2 @ C2 ) )
      = ( minus_minus_real @ A2 @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_2226_add__diff__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C2 ) @ ( plus_plus_rat @ B2 @ C2 ) )
      = ( minus_minus_rat @ A2 @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_2227_add__diff__cancel__right,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
      = ( minus_minus_nat @ A2 @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_2228_add__diff__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ C2 ) )
      = ( minus_minus_int @ A2 @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_2229_add__diff__cancel__right_H,axiom,
    ! [A2: real,B2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_2230_add__diff__cancel__right_H,axiom,
    ! [A2: rat,B2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_2231_add__diff__cancel__right_H,axiom,
    ! [A2: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_2232_add__diff__cancel__right_H,axiom,
    ! [A2: int,B2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_2233_less__option__Some,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_option_real @ ( some_real @ X ) @ ( some_real @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% less_option_Some
thf(fact_2234_less__option__Some,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_option_rat @ ( some_rat @ X ) @ ( some_rat @ Y ) )
      = ( ord_less_rat @ X @ Y ) ) ).

% less_option_Some
thf(fact_2235_less__option__Some,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_option_num @ ( some_num @ X ) @ ( some_num @ Y ) )
      = ( ord_less_num @ X @ Y ) ) ).

% less_option_Some
thf(fact_2236_less__option__Some,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_option_nat @ ( some_nat @ X ) @ ( some_nat @ Y ) )
      = ( ord_less_nat @ X @ Y ) ) ).

% less_option_Some
thf(fact_2237_less__option__Some,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_option_int @ ( some_int @ X ) @ ( some_int @ Y ) )
      = ( ord_less_int @ X @ Y ) ) ).

% less_option_Some
thf(fact_2238_add__le__same__cancel1,axiom,
    ! [B2: real,A2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ B2 @ A2 ) @ B2 )
      = ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).

% add_le_same_cancel1
thf(fact_2239_add__le__same__cancel1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ B2 @ A2 ) @ B2 )
      = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).

% add_le_same_cancel1
thf(fact_2240_add__le__same__cancel1,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
      = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_2241_add__le__same__cancel1,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
      = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).

% add_le_same_cancel1
thf(fact_2242_add__le__same__cancel2,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B2 ) @ B2 )
      = ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).

% add_le_same_cancel2
thf(fact_2243_add__le__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B2 ) @ B2 )
      = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).

% add_le_same_cancel2
thf(fact_2244_add__le__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
      = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_2245_add__le__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
      = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).

% add_le_same_cancel2
thf(fact_2246_le__add__same__cancel1,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ A2 @ ( plus_plus_real @ A2 @ B2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_2247_le__add__same__cancel1,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ A2 @ B2 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_2248_le__add__same__cancel1,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_2249_le__add__same__cancel1,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).

% le_add_same_cancel1
thf(fact_2250_le__add__same__cancel2,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ A2 @ ( plus_plus_real @ B2 @ A2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_2251_le__add__same__cancel2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ B2 @ A2 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_2252_le__add__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_2253_le__add__same__cancel2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).

% le_add_same_cancel2
thf(fact_2254_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ A2 ) @ zero_zero_real )
      = ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_2255_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ A2 ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_2256_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
      = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_2257_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A2 @ A2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_2258_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ A2 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_2259_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_2260_add__less__same__cancel1,axiom,
    ! [B2: real,A2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ B2 @ A2 ) @ B2 )
      = ( ord_less_real @ A2 @ zero_zero_real ) ) ).

% add_less_same_cancel1
thf(fact_2261_add__less__same__cancel1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ B2 @ A2 ) @ B2 )
      = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).

% add_less_same_cancel1
thf(fact_2262_add__less__same__cancel1,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
      = ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_2263_add__less__same__cancel1,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
      = ( ord_less_int @ A2 @ zero_zero_int ) ) ).

% add_less_same_cancel1
thf(fact_2264_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_2265_zero__reorient,axiom,
    ! [X: complex] :
      ( ( zero_zero_complex = X )
      = ( X = zero_zero_complex ) ) ).

% zero_reorient
thf(fact_2266_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
      = ( X = zero_zero_real ) ) ).

% zero_reorient
thf(fact_2267_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
      = ( X = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_2268_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_2269_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
      = ( X = zero_zero_int ) ) ).

% zero_reorient
thf(fact_2270_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J2: real,K: real,L2: real] :
      ( ( ( I = J2 )
        & ( K = L2 ) )
     => ( ( plus_plus_real @ I @ K )
        = ( plus_plus_real @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_2271_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J2: rat,K: rat,L2: rat] :
      ( ( ( I = J2 )
        & ( K = L2 ) )
     => ( ( plus_plus_rat @ I @ K )
        = ( plus_plus_rat @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_2272_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J2: nat,K: nat,L2: nat] :
      ( ( ( I = J2 )
        & ( K = L2 ) )
     => ( ( plus_plus_nat @ I @ K )
        = ( plus_plus_nat @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_2273_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J2: int,K: int,L2: int] :
      ( ( ( I = J2 )
        & ( K = L2 ) )
     => ( ( plus_plus_int @ I @ K )
        = ( plus_plus_int @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_2274_group__cancel_Oadd1,axiom,
    ! [A: real,K: real,A2: real,B2: real] :
      ( ( A
        = ( plus_plus_real @ K @ A2 ) )
     => ( ( plus_plus_real @ A @ B2 )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A2 @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_2275_group__cancel_Oadd1,axiom,
    ! [A: rat,K: rat,A2: rat,B2: rat] :
      ( ( A
        = ( plus_plus_rat @ K @ A2 ) )
     => ( ( plus_plus_rat @ A @ B2 )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A2 @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_2276_group__cancel_Oadd1,axiom,
    ! [A: nat,K: nat,A2: nat,B2: nat] :
      ( ( A
        = ( plus_plus_nat @ K @ A2 ) )
     => ( ( plus_plus_nat @ A @ B2 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_2277_group__cancel_Oadd1,axiom,
    ! [A: int,K: int,A2: int,B2: int] :
      ( ( A
        = ( plus_plus_int @ K @ A2 ) )
     => ( ( plus_plus_int @ A @ B2 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_2278_group__cancel_Oadd2,axiom,
    ! [B: real,K: real,B2: real,A2: real] :
      ( ( B
        = ( plus_plus_real @ K @ B2 ) )
     => ( ( plus_plus_real @ A2 @ B )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A2 @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_2279_group__cancel_Oadd2,axiom,
    ! [B: rat,K: rat,B2: rat,A2: rat] :
      ( ( B
        = ( plus_plus_rat @ K @ B2 ) )
     => ( ( plus_plus_rat @ A2 @ B )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A2 @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_2280_group__cancel_Oadd2,axiom,
    ! [B: nat,K: nat,B2: nat,A2: nat] :
      ( ( B
        = ( plus_plus_nat @ K @ B2 ) )
     => ( ( plus_plus_nat @ A2 @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_2281_group__cancel_Oadd2,axiom,
    ! [B: int,K: int,B2: int,A2: int] :
      ( ( B
        = ( plus_plus_int @ K @ B2 ) )
     => ( ( plus_plus_int @ A2 @ B )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_2282_add_Oassoc,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A2 @ B2 ) @ C2 )
      = ( plus_plus_real @ A2 @ ( plus_plus_real @ B2 @ C2 ) ) ) ).

% add.assoc
thf(fact_2283_add_Oassoc,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C2 )
      = ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B2 @ C2 ) ) ) ).

% add.assoc
thf(fact_2284_add_Oassoc,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 )
      = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).

% add.assoc
thf(fact_2285_add_Oassoc,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 )
      = ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C2 ) ) ) ).

% add.assoc
thf(fact_2286_add_Oleft__cancel,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ( plus_plus_real @ A2 @ B2 )
        = ( plus_plus_real @ A2 @ C2 ) )
      = ( B2 = C2 ) ) ).

% add.left_cancel
thf(fact_2287_add_Oleft__cancel,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ( plus_plus_rat @ A2 @ B2 )
        = ( plus_plus_rat @ A2 @ C2 ) )
      = ( B2 = C2 ) ) ).

% add.left_cancel
thf(fact_2288_add_Oleft__cancel,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ( plus_plus_int @ A2 @ B2 )
        = ( plus_plus_int @ A2 @ C2 ) )
      = ( B2 = C2 ) ) ).

% add.left_cancel
thf(fact_2289_add_Oright__cancel,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( ( plus_plus_real @ B2 @ A2 )
        = ( plus_plus_real @ C2 @ A2 ) )
      = ( B2 = C2 ) ) ).

% add.right_cancel
thf(fact_2290_add_Oright__cancel,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ( plus_plus_rat @ B2 @ A2 )
        = ( plus_plus_rat @ C2 @ A2 ) )
      = ( B2 = C2 ) ) ).

% add.right_cancel
thf(fact_2291_add_Oright__cancel,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ( plus_plus_int @ B2 @ A2 )
        = ( plus_plus_int @ C2 @ A2 ) )
      = ( B2 = C2 ) ) ).

% add.right_cancel
thf(fact_2292_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A7: real,B7: real] : ( plus_plus_real @ B7 @ A7 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_2293_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A7: rat,B7: rat] : ( plus_plus_rat @ B7 @ A7 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_2294_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A7: nat,B7: nat] : ( plus_plus_nat @ B7 @ A7 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_2295_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A7: int,B7: int] : ( plus_plus_int @ B7 @ A7 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_2296_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( plus_plus_real @ B2 @ ( plus_plus_real @ A2 @ C2 ) )
      = ( plus_plus_real @ A2 @ ( plus_plus_real @ B2 @ C2 ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_2297_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( plus_plus_rat @ B2 @ ( plus_plus_rat @ A2 @ C2 ) )
      = ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B2 @ C2 ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_2298_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A2 @ C2 ) )
      = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_2299_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( plus_plus_int @ B2 @ ( plus_plus_int @ A2 @ C2 ) )
      = ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C2 ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_2300_add__left__imp__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ( plus_plus_real @ A2 @ B2 )
        = ( plus_plus_real @ A2 @ C2 ) )
     => ( B2 = C2 ) ) ).

% add_left_imp_eq
thf(fact_2301_add__left__imp__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ( plus_plus_rat @ A2 @ B2 )
        = ( plus_plus_rat @ A2 @ C2 ) )
     => ( B2 = C2 ) ) ).

% add_left_imp_eq
thf(fact_2302_add__left__imp__eq,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ( plus_plus_nat @ A2 @ B2 )
        = ( plus_plus_nat @ A2 @ C2 ) )
     => ( B2 = C2 ) ) ).

% add_left_imp_eq
thf(fact_2303_add__left__imp__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ( plus_plus_int @ A2 @ B2 )
        = ( plus_plus_int @ A2 @ C2 ) )
     => ( B2 = C2 ) ) ).

% add_left_imp_eq
thf(fact_2304_add__right__imp__eq,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( ( plus_plus_real @ B2 @ A2 )
        = ( plus_plus_real @ C2 @ A2 ) )
     => ( B2 = C2 ) ) ).

% add_right_imp_eq
thf(fact_2305_add__right__imp__eq,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ( plus_plus_rat @ B2 @ A2 )
        = ( plus_plus_rat @ C2 @ A2 ) )
     => ( B2 = C2 ) ) ).

% add_right_imp_eq
thf(fact_2306_add__right__imp__eq,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( ( plus_plus_nat @ B2 @ A2 )
        = ( plus_plus_nat @ C2 @ A2 ) )
     => ( B2 = C2 ) ) ).

% add_right_imp_eq
thf(fact_2307_add__right__imp__eq,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ( plus_plus_int @ B2 @ A2 )
        = ( plus_plus_int @ C2 @ A2 ) )
     => ( B2 = C2 ) ) ).

% add_right_imp_eq
thf(fact_2308_mult_Oassoc,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( times_times_real @ ( times_times_real @ A2 @ B2 ) @ C2 )
      = ( times_times_real @ A2 @ ( times_times_real @ B2 @ C2 ) ) ) ).

% mult.assoc
thf(fact_2309_mult_Oassoc,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A2 @ B2 ) @ C2 )
      = ( times_times_rat @ A2 @ ( times_times_rat @ B2 @ C2 ) ) ) ).

% mult.assoc
thf(fact_2310_mult_Oassoc,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 )
      = ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) ) ) ).

% mult.assoc
thf(fact_2311_mult_Oassoc,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( times_times_int @ ( times_times_int @ A2 @ B2 ) @ C2 )
      = ( times_times_int @ A2 @ ( times_times_int @ B2 @ C2 ) ) ) ).

% mult.assoc
thf(fact_2312_mult_Oassoc,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A2 @ B2 ) @ C2 )
      = ( times_times_assn @ A2 @ ( times_times_assn @ B2 @ C2 ) ) ) ).

% mult.assoc
thf(fact_2313_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A7: real,B7: real] : ( times_times_real @ B7 @ A7 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_2314_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_rat
    = ( ^ [A7: rat,B7: rat] : ( times_times_rat @ B7 @ A7 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_2315_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A7: nat,B7: nat] : ( times_times_nat @ B7 @ A7 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_2316_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A7: int,B7: int] : ( times_times_int @ B7 @ A7 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_2317_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_assn
    = ( ^ [A7: assn,B7: assn] : ( times_times_assn @ B7 @ A7 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_2318_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( times_times_real @ B2 @ ( times_times_real @ A2 @ C2 ) )
      = ( times_times_real @ A2 @ ( times_times_real @ B2 @ C2 ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_2319_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( times_times_rat @ B2 @ ( times_times_rat @ A2 @ C2 ) )
      = ( times_times_rat @ A2 @ ( times_times_rat @ B2 @ C2 ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_2320_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( times_times_nat @ B2 @ ( times_times_nat @ A2 @ C2 ) )
      = ( times_times_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_2321_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( times_times_int @ B2 @ ( times_times_int @ A2 @ C2 ) )
      = ( times_times_int @ A2 @ ( times_times_int @ B2 @ C2 ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_2322_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B2: assn,A2: assn,C2: assn] :
      ( ( times_times_assn @ B2 @ ( times_times_assn @ A2 @ C2 ) )
      = ( times_times_assn @ A2 @ ( times_times_assn @ B2 @ C2 ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_2323_diff__eq__diff__eq,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( ( minus_minus_real @ A2 @ B2 )
        = ( minus_minus_real @ C2 @ D2 ) )
     => ( ( A2 = B2 )
        = ( C2 = D2 ) ) ) ).

% diff_eq_diff_eq
thf(fact_2324_diff__eq__diff__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ( minus_minus_rat @ A2 @ B2 )
        = ( minus_minus_rat @ C2 @ D2 ) )
     => ( ( A2 = B2 )
        = ( C2 = D2 ) ) ) ).

% diff_eq_diff_eq
thf(fact_2325_diff__eq__diff__eq,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ( minus_minus_int @ A2 @ B2 )
        = ( minus_minus_int @ C2 @ D2 ) )
     => ( ( A2 = B2 )
        = ( C2 = D2 ) ) ) ).

% diff_eq_diff_eq
thf(fact_2326_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A2 @ C2 ) @ B2 )
      = ( minus_minus_real @ ( minus_minus_real @ A2 @ B2 ) @ C2 ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_2327_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A2 @ C2 ) @ B2 )
      = ( minus_minus_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C2 ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_2328_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C2 ) @ B2 )
      = ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C2 ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_2329_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A2 @ C2 ) @ B2 )
      = ( minus_minus_int @ ( minus_minus_int @ A2 @ B2 ) @ C2 ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_2330_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_2331_gr__zeroI,axiom,
    ! [N3: nat] :
      ( ( N3 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).

% gr_zeroI
thf(fact_2332_not__less__zero,axiom,
    ! [N3: nat] :
      ~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).

% not_less_zero
thf(fact_2333_gr__implies__not__zero,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( N3 != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_2334_zero__less__iff__neq__zero,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
      = ( N3 != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_2335_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_2336_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ zero_zero_real @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_2337_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_2338_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_2339_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ zero_zero_int @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_2340_add_Ocomm__neutral,axiom,
    ! [A2: complex] :
      ( ( plus_plus_complex @ A2 @ zero_zero_complex )
      = A2 ) ).

% add.comm_neutral
thf(fact_2341_add_Ocomm__neutral,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ A2 @ zero_zero_real )
      = A2 ) ).

% add.comm_neutral
thf(fact_2342_add_Ocomm__neutral,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ A2 @ zero_zero_rat )
      = A2 ) ).

% add.comm_neutral
thf(fact_2343_add_Ocomm__neutral,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% add.comm_neutral
thf(fact_2344_add_Ocomm__neutral,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% add.comm_neutral
thf(fact_2345_add_Ogroup__left__neutral,axiom,
    ! [A2: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A2 )
      = A2 ) ).

% add.group_left_neutral
thf(fact_2346_add_Ogroup__left__neutral,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ zero_zero_real @ A2 )
      = A2 ) ).

% add.group_left_neutral
thf(fact_2347_add_Ogroup__left__neutral,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A2 )
      = A2 ) ).

% add.group_left_neutral
thf(fact_2348_add_Ogroup__left__neutral,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ zero_zero_int @ A2 )
      = A2 ) ).

% add.group_left_neutral
thf(fact_2349_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J2: real,K: real,L2: real] :
      ( ( ( ord_less_eq_real @ I @ J2 )
        & ( K = L2 ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2350_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J2: rat,K: rat,L2: rat] :
      ( ( ( ord_less_eq_rat @ I @ J2 )
        & ( K = L2 ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2351_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J2: nat,K: nat,L2: nat] :
      ( ( ( ord_less_eq_nat @ I @ J2 )
        & ( K = L2 ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2352_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J2: int,K: int,L2: int] :
      ( ( ( ord_less_eq_int @ I @ J2 )
        & ( K = L2 ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_2353_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J2: real,K: real,L2: real] :
      ( ( ( I = J2 )
        & ( ord_less_eq_real @ K @ L2 ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2354_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J2: rat,K: rat,L2: rat] :
      ( ( ( I = J2 )
        & ( ord_less_eq_rat @ K @ L2 ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2355_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J2: nat,K: nat,L2: nat] :
      ( ( ( I = J2 )
        & ( ord_less_eq_nat @ K @ L2 ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2356_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J2: int,K: int,L2: int] :
      ( ( ( I = J2 )
        & ( ord_less_eq_int @ K @ L2 ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_2357_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J2: real,K: real,L2: real] :
      ( ( ( ord_less_eq_real @ I @ J2 )
        & ( ord_less_eq_real @ K @ L2 ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2358_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J2: rat,K: rat,L2: rat] :
      ( ( ( ord_less_eq_rat @ I @ J2 )
        & ( ord_less_eq_rat @ K @ L2 ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2359_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J2: nat,K: nat,L2: nat] :
      ( ( ( ord_less_eq_nat @ I @ J2 )
        & ( ord_less_eq_nat @ K @ L2 ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2360_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J2: int,K: int,L2: int] :
      ( ( ( ord_less_eq_int @ I @ J2 )
        & ( ord_less_eq_int @ K @ L2 ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_2361_add__mono,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ C2 @ D2 )
       => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B2 @ D2 ) ) ) ) ).

% add_mono
thf(fact_2362_add__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C2 @ D2 )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C2 ) @ ( plus_plus_rat @ B2 @ D2 ) ) ) ) ).

% add_mono
thf(fact_2363_add__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C2 @ D2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).

% add_mono
thf(fact_2364_add__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ C2 @ D2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ D2 ) ) ) ) ).

% add_mono
thf(fact_2365_add__left__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A2 ) @ ( plus_plus_real @ C2 @ B2 ) ) ) ).

% add_left_mono
thf(fact_2366_add__left__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ C2 @ A2 ) @ ( plus_plus_rat @ C2 @ B2 ) ) ) ).

% add_left_mono
thf(fact_2367_add__left__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B2 ) ) ) ).

% add_left_mono
thf(fact_2368_add__left__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A2 ) @ ( plus_plus_int @ C2 @ B2 ) ) ) ).

% add_left_mono
thf(fact_2369_less__eqE,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ~ ! [C4: nat] :
            ( B2
           != ( plus_plus_nat @ A2 @ C4 ) ) ) ).

% less_eqE
thf(fact_2370_add__right__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B2 @ C2 ) ) ) ).

% add_right_mono
thf(fact_2371_add__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C2 ) @ ( plus_plus_rat @ B2 @ C2 ) ) ) ).

% add_right_mono
thf(fact_2372_add__right__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).

% add_right_mono
thf(fact_2373_add__right__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ C2 ) ) ) ).

% add_right_mono
thf(fact_2374_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A7: nat,B7: nat] :
        ? [C5: nat] :
          ( B7
          = ( plus_plus_nat @ A7 @ C5 ) ) ) ) ).

% le_iff_add
thf(fact_2375_add__le__imp__le__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A2 ) @ ( plus_plus_real @ C2 @ B2 ) )
     => ( ord_less_eq_real @ A2 @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_2376_add__le__imp__le__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C2 @ A2 ) @ ( plus_plus_rat @ C2 @ B2 ) )
     => ( ord_less_eq_rat @ A2 @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_2377_add__le__imp__le__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B2 ) )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_2378_add__le__imp__le__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A2 ) @ ( plus_plus_int @ C2 @ B2 ) )
     => ( ord_less_eq_int @ A2 @ B2 ) ) ).

% add_le_imp_le_left
thf(fact_2379_add__le__imp__le__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B2 @ C2 ) )
     => ( ord_less_eq_real @ A2 @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_2380_add__le__imp__le__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C2 ) @ ( plus_plus_rat @ B2 @ C2 ) )
     => ( ord_less_eq_rat @ A2 @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_2381_add__le__imp__le__right,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_2382_add__le__imp__le__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ C2 ) )
     => ( ord_less_eq_int @ A2 @ B2 ) ) ).

% add_le_imp_le_right
thf(fact_2383_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J2: real,K: real,L2: real] :
      ( ( ( ord_less_real @ I @ J2 )
        & ( ord_less_real @ K @ L2 ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_2384_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J2: rat,K: rat,L2: rat] :
      ( ( ( ord_less_rat @ I @ J2 )
        & ( ord_less_rat @ K @ L2 ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_2385_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J2: nat,K: nat,L2: nat] :
      ( ( ( ord_less_nat @ I @ J2 )
        & ( ord_less_nat @ K @ L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_2386_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J2: int,K: int,L2: int] :
      ( ( ( ord_less_int @ I @ J2 )
        & ( ord_less_int @ K @ L2 ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_2387_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J2: real,K: real,L2: real] :
      ( ( ( I = J2 )
        & ( ord_less_real @ K @ L2 ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_2388_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J2: rat,K: rat,L2: rat] :
      ( ( ( I = J2 )
        & ( ord_less_rat @ K @ L2 ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_2389_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J2: nat,K: nat,L2: nat] :
      ( ( ( I = J2 )
        & ( ord_less_nat @ K @ L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_2390_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J2: int,K: int,L2: int] :
      ( ( ( I = J2 )
        & ( ord_less_int @ K @ L2 ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_2391_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J2: real,K: real,L2: real] :
      ( ( ( ord_less_real @ I @ J2 )
        & ( K = L2 ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_2392_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J2: rat,K: rat,L2: rat] :
      ( ( ( ord_less_rat @ I @ J2 )
        & ( K = L2 ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_2393_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J2: nat,K: nat,L2: nat] :
      ( ( ( ord_less_nat @ I @ J2 )
        & ( K = L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_2394_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J2: int,K: int,L2: int] :
      ( ( ( ord_less_int @ I @ J2 )
        & ( K = L2 ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_2395_add__strict__mono,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_real @ C2 @ D2 )
       => ( ord_less_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B2 @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_2396_add__strict__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ C2 @ D2 )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C2 ) @ ( plus_plus_rat @ B2 @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_2397_add__strict__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ C2 @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_2398_add__strict__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ C2 @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_2399_add__strict__left__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ord_less_real @ ( plus_plus_real @ C2 @ A2 ) @ ( plus_plus_real @ C2 @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_2400_add__strict__left__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ord_less_rat @ ( plus_plus_rat @ C2 @ A2 ) @ ( plus_plus_rat @ C2 @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_2401_add__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_2402_add__strict__left__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ord_less_int @ ( plus_plus_int @ C2 @ A2 ) @ ( plus_plus_int @ C2 @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_2403_add__strict__right__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ord_less_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B2 @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_2404_add__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C2 ) @ ( plus_plus_rat @ B2 @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_2405_add__strict__right__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_2406_add__strict__right__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ord_less_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_2407_add__less__imp__less__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C2 @ A2 ) @ ( plus_plus_real @ C2 @ B2 ) )
     => ( ord_less_real @ A2 @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_2408_add__less__imp__less__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C2 @ A2 ) @ ( plus_plus_rat @ C2 @ B2 ) )
     => ( ord_less_rat @ A2 @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_2409_add__less__imp__less__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A2 ) @ ( plus_plus_nat @ C2 @ B2 ) )
     => ( ord_less_nat @ A2 @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_2410_add__less__imp__less__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C2 @ A2 ) @ ( plus_plus_int @ C2 @ B2 ) )
     => ( ord_less_int @ A2 @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_2411_add__less__imp__less__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B2 @ C2 ) )
     => ( ord_less_real @ A2 @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_2412_add__less__imp__less__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ C2 ) @ ( plus_plus_rat @ B2 @ C2 ) )
     => ( ord_less_rat @ A2 @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_2413_add__less__imp__less__right,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ C2 ) )
     => ( ord_less_nat @ A2 @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_2414_add__less__imp__less__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ C2 ) )
     => ( ord_less_int @ A2 @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_2415_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: complex,Z3: complex] : Y5 = Z3 )
    = ( ^ [A7: complex,B7: complex] :
          ( ( minus_minus_complex @ A7 @ B7 )
          = zero_zero_complex ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2416_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: real,Z3: real] : Y5 = Z3 )
    = ( ^ [A7: real,B7: real] :
          ( ( minus_minus_real @ A7 @ B7 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2417_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: rat,Z3: rat] : Y5 = Z3 )
    = ( ^ [A7: rat,B7: rat] :
          ( ( minus_minus_rat @ A7 @ B7 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2418_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: int,Z3: int] : Y5 = Z3 )
    = ( ^ [A7: int,B7: int] :
          ( ( minus_minus_int @ A7 @ B7 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_2419_diff__mono,axiom,
    ! [A2: real,B2: real,D2: real,C2: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ D2 @ C2 )
       => ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C2 ) @ ( minus_minus_real @ B2 @ D2 ) ) ) ) ).

% diff_mono
thf(fact_2420_diff__mono,axiom,
    ! [A2: rat,B2: rat,D2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ D2 @ C2 )
       => ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ C2 ) @ ( minus_minus_rat @ B2 @ D2 ) ) ) ) ).

% diff_mono
thf(fact_2421_diff__mono,axiom,
    ! [A2: int,B2: int,D2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ D2 @ C2 )
       => ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C2 ) @ ( minus_minus_int @ B2 @ D2 ) ) ) ) ).

% diff_mono
thf(fact_2422_diff__left__mono,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( ord_less_eq_real @ B2 @ A2 )
     => ( ord_less_eq_real @ ( minus_minus_real @ C2 @ A2 ) @ ( minus_minus_real @ C2 @ B2 ) ) ) ).

% diff_left_mono
thf(fact_2423_diff__left__mono,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ C2 @ A2 ) @ ( minus_minus_rat @ C2 @ B2 ) ) ) ).

% diff_left_mono
thf(fact_2424_diff__left__mono,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ord_less_eq_int @ ( minus_minus_int @ C2 @ A2 ) @ ( minus_minus_int @ C2 @ B2 ) ) ) ).

% diff_left_mono
thf(fact_2425_diff__right__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C2 ) @ ( minus_minus_real @ B2 @ C2 ) ) ) ).

% diff_right_mono
thf(fact_2426_diff__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ C2 ) @ ( minus_minus_rat @ B2 @ C2 ) ) ) ).

% diff_right_mono
thf(fact_2427_diff__right__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C2 ) @ ( minus_minus_int @ B2 @ C2 ) ) ) ).

% diff_right_mono
thf(fact_2428_diff__eq__diff__less__eq,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( ( minus_minus_real @ A2 @ B2 )
        = ( minus_minus_real @ C2 @ D2 ) )
     => ( ( ord_less_eq_real @ A2 @ B2 )
        = ( ord_less_eq_real @ C2 @ D2 ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_2429_diff__eq__diff__less__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ( minus_minus_rat @ A2 @ B2 )
        = ( minus_minus_rat @ C2 @ D2 ) )
     => ( ( ord_less_eq_rat @ A2 @ B2 )
        = ( ord_less_eq_rat @ C2 @ D2 ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_2430_diff__eq__diff__less__eq,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ( minus_minus_int @ A2 @ B2 )
        = ( minus_minus_int @ C2 @ D2 ) )
     => ( ( ord_less_eq_int @ A2 @ B2 )
        = ( ord_less_eq_int @ C2 @ D2 ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_2431_diff__strict__mono,axiom,
    ! [A2: real,B2: real,D2: real,C2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_real @ D2 @ C2 )
       => ( ord_less_real @ ( minus_minus_real @ A2 @ C2 ) @ ( minus_minus_real @ B2 @ D2 ) ) ) ) ).

% diff_strict_mono
thf(fact_2432_diff__strict__mono,axiom,
    ! [A2: rat,B2: rat,D2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ D2 @ C2 )
       => ( ord_less_rat @ ( minus_minus_rat @ A2 @ C2 ) @ ( minus_minus_rat @ B2 @ D2 ) ) ) ) ).

% diff_strict_mono
thf(fact_2433_diff__strict__mono,axiom,
    ! [A2: int,B2: int,D2: int,C2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ D2 @ C2 )
       => ( ord_less_int @ ( minus_minus_int @ A2 @ C2 ) @ ( minus_minus_int @ B2 @ D2 ) ) ) ) ).

% diff_strict_mono
thf(fact_2434_diff__eq__diff__less,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( ( minus_minus_real @ A2 @ B2 )
        = ( minus_minus_real @ C2 @ D2 ) )
     => ( ( ord_less_real @ A2 @ B2 )
        = ( ord_less_real @ C2 @ D2 ) ) ) ).

% diff_eq_diff_less
thf(fact_2435_diff__eq__diff__less,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ( minus_minus_rat @ A2 @ B2 )
        = ( minus_minus_rat @ C2 @ D2 ) )
     => ( ( ord_less_rat @ A2 @ B2 )
        = ( ord_less_rat @ C2 @ D2 ) ) ) ).

% diff_eq_diff_less
thf(fact_2436_diff__eq__diff__less,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ( minus_minus_int @ A2 @ B2 )
        = ( minus_minus_int @ C2 @ D2 ) )
     => ( ( ord_less_int @ A2 @ B2 )
        = ( ord_less_int @ C2 @ D2 ) ) ) ).

% diff_eq_diff_less
thf(fact_2437_diff__strict__left__mono,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( ord_less_real @ B2 @ A2 )
     => ( ord_less_real @ ( minus_minus_real @ C2 @ A2 ) @ ( minus_minus_real @ C2 @ B2 ) ) ) ).

% diff_strict_left_mono
thf(fact_2438_diff__strict__left__mono,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ord_less_rat @ ( minus_minus_rat @ C2 @ A2 ) @ ( minus_minus_rat @ C2 @ B2 ) ) ) ).

% diff_strict_left_mono
thf(fact_2439_diff__strict__left__mono,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ord_less_int @ ( minus_minus_int @ C2 @ A2 ) @ ( minus_minus_int @ C2 @ B2 ) ) ) ).

% diff_strict_left_mono
thf(fact_2440_diff__strict__right__mono,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ord_less_real @ ( minus_minus_real @ A2 @ C2 ) @ ( minus_minus_real @ B2 @ C2 ) ) ) ).

% diff_strict_right_mono
thf(fact_2441_diff__strict__right__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ord_less_rat @ ( minus_minus_rat @ A2 @ C2 ) @ ( minus_minus_rat @ B2 @ C2 ) ) ) ).

% diff_strict_right_mono
thf(fact_2442_diff__strict__right__mono,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ord_less_int @ ( minus_minus_int @ A2 @ C2 ) @ ( minus_minus_int @ B2 @ C2 ) ) ) ).

% diff_strict_right_mono
thf(fact_2443_group__cancel_Osub1,axiom,
    ! [A: real,K: real,A2: real,B2: real] :
      ( ( A
        = ( plus_plus_real @ K @ A2 ) )
     => ( ( minus_minus_real @ A @ B2 )
        = ( plus_plus_real @ K @ ( minus_minus_real @ A2 @ B2 ) ) ) ) ).

% group_cancel.sub1
thf(fact_2444_group__cancel_Osub1,axiom,
    ! [A: rat,K: rat,A2: rat,B2: rat] :
      ( ( A
        = ( plus_plus_rat @ K @ A2 ) )
     => ( ( minus_minus_rat @ A @ B2 )
        = ( plus_plus_rat @ K @ ( minus_minus_rat @ A2 @ B2 ) ) ) ) ).

% group_cancel.sub1
thf(fact_2445_group__cancel_Osub1,axiom,
    ! [A: int,K: int,A2: int,B2: int] :
      ( ( A
        = ( plus_plus_int @ K @ A2 ) )
     => ( ( minus_minus_int @ A @ B2 )
        = ( plus_plus_int @ K @ ( minus_minus_int @ A2 @ B2 ) ) ) ) ).

% group_cancel.sub1
thf(fact_2446_diff__eq__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ( minus_minus_real @ A2 @ B2 )
        = C2 )
      = ( A2
        = ( plus_plus_real @ C2 @ B2 ) ) ) ).

% diff_eq_eq
thf(fact_2447_diff__eq__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ( minus_minus_rat @ A2 @ B2 )
        = C2 )
      = ( A2
        = ( plus_plus_rat @ C2 @ B2 ) ) ) ).

% diff_eq_eq
thf(fact_2448_diff__eq__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ( minus_minus_int @ A2 @ B2 )
        = C2 )
      = ( A2
        = ( plus_plus_int @ C2 @ B2 ) ) ) ).

% diff_eq_eq
thf(fact_2449_eq__diff__eq,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( A2
        = ( minus_minus_real @ C2 @ B2 ) )
      = ( ( plus_plus_real @ A2 @ B2 )
        = C2 ) ) ).

% eq_diff_eq
thf(fact_2450_eq__diff__eq,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( A2
        = ( minus_minus_rat @ C2 @ B2 ) )
      = ( ( plus_plus_rat @ A2 @ B2 )
        = C2 ) ) ).

% eq_diff_eq
thf(fact_2451_eq__diff__eq,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( A2
        = ( minus_minus_int @ C2 @ B2 ) )
      = ( ( plus_plus_int @ A2 @ B2 )
        = C2 ) ) ).

% eq_diff_eq
thf(fact_2452_add__diff__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( plus_plus_real @ A2 @ ( minus_minus_real @ B2 @ C2 ) )
      = ( minus_minus_real @ ( plus_plus_real @ A2 @ B2 ) @ C2 ) ) ).

% add_diff_eq
thf(fact_2453_add__diff__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( plus_plus_rat @ A2 @ ( minus_minus_rat @ B2 @ C2 ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C2 ) ) ).

% add_diff_eq
thf(fact_2454_add__diff__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( plus_plus_int @ A2 @ ( minus_minus_int @ B2 @ C2 ) )
      = ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 ) ) ).

% add_diff_eq
thf(fact_2455_diff__diff__eq2,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( minus_minus_real @ A2 @ ( minus_minus_real @ B2 @ C2 ) )
      = ( minus_minus_real @ ( plus_plus_real @ A2 @ C2 ) @ B2 ) ) ).

% diff_diff_eq2
thf(fact_2456_diff__diff__eq2,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( minus_minus_rat @ A2 @ ( minus_minus_rat @ B2 @ C2 ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C2 ) @ B2 ) ) ).

% diff_diff_eq2
thf(fact_2457_diff__diff__eq2,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( minus_minus_int @ A2 @ ( minus_minus_int @ B2 @ C2 ) )
      = ( minus_minus_int @ ( plus_plus_int @ A2 @ C2 ) @ B2 ) ) ).

% diff_diff_eq2
thf(fact_2458_diff__add__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A2 @ B2 ) @ C2 )
      = ( minus_minus_real @ ( plus_plus_real @ A2 @ C2 ) @ B2 ) ) ).

% diff_add_eq
thf(fact_2459_diff__add__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C2 )
      = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C2 ) @ B2 ) ) ).

% diff_add_eq
thf(fact_2460_diff__add__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ C2 )
      = ( minus_minus_int @ ( plus_plus_int @ A2 @ C2 ) @ B2 ) ) ).

% diff_add_eq
thf(fact_2461_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( minus_minus_real @ A2 @ ( plus_plus_real @ B2 @ C2 ) )
      = ( minus_minus_real @ ( minus_minus_real @ A2 @ C2 ) @ B2 ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_2462_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( minus_minus_rat @ A2 @ ( plus_plus_rat @ B2 @ C2 ) )
      = ( minus_minus_rat @ ( minus_minus_rat @ A2 @ C2 ) @ B2 ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_2463_diff__add__eq__diff__diff__swap,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( minus_minus_int @ A2 @ ( plus_plus_int @ B2 @ C2 ) )
      = ( minus_minus_int @ ( minus_minus_int @ A2 @ C2 ) @ B2 ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_2464_add__implies__diff,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( ( ( plus_plus_real @ C2 @ B2 )
        = A2 )
     => ( C2
        = ( minus_minus_real @ A2 @ B2 ) ) ) ).

% add_implies_diff
thf(fact_2465_add__implies__diff,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( ( plus_plus_rat @ C2 @ B2 )
        = A2 )
     => ( C2
        = ( minus_minus_rat @ A2 @ B2 ) ) ) ).

% add_implies_diff
thf(fact_2466_add__implies__diff,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( ( plus_plus_nat @ C2 @ B2 )
        = A2 )
     => ( C2
        = ( minus_minus_nat @ A2 @ B2 ) ) ) ).

% add_implies_diff
thf(fact_2467_add__implies__diff,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( ( plus_plus_int @ C2 @ B2 )
        = A2 )
     => ( C2
        = ( minus_minus_int @ A2 @ B2 ) ) ) ).

% add_implies_diff
thf(fact_2468_diff__diff__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A2 @ B2 ) @ C2 )
      = ( minus_minus_real @ A2 @ ( plus_plus_real @ B2 @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_2469_diff__diff__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C2 )
      = ( minus_minus_rat @ A2 @ ( plus_plus_rat @ B2 @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_2470_diff__diff__eq,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C2 )
      = ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B2 @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_2471_diff__diff__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A2 @ B2 ) @ C2 )
      = ( minus_minus_int @ A2 @ ( plus_plus_int @ B2 @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_2472_diff__diff__less,axiom,
    ! [I: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ M @ ( minus_minus_nat @ M @ N3 ) ) )
      = ( ( ord_less_nat @ I @ M )
        & ( ord_less_nat @ I @ N3 ) ) ) ).

% diff_diff_less
thf(fact_2473_add__decreasing,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( ord_less_eq_real @ A2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ C2 @ B2 )
       => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C2 ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_2474_add__decreasing,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ C2 @ B2 )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C2 ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_2475_add__decreasing,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C2 @ B2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_2476_add__decreasing,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ C2 @ B2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C2 ) @ B2 ) ) ) ).

% add_decreasing
thf(fact_2477_add__increasing,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ B2 @ C2 )
       => ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A2 @ C2 ) ) ) ) ).

% add_increasing
thf(fact_2478_add__increasing,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ B2 @ C2 )
       => ( ord_less_eq_rat @ B2 @ ( plus_plus_rat @ A2 @ C2 ) ) ) ) ).

% add_increasing
thf(fact_2479_add__increasing,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).

% add_increasing
thf(fact_2480_add__increasing,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ B2 @ C2 )
       => ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A2 @ C2 ) ) ) ) ).

% add_increasing
thf(fact_2481_add__decreasing2,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ord_less_eq_real @ C2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ A2 @ B2 )
       => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C2 ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_2482_add__decreasing2,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A2 @ B2 )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C2 ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_2483_add__decreasing2,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C2 ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_2484_add__decreasing2,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ C2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ A2 @ B2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C2 ) @ B2 ) ) ) ).

% add_decreasing2
thf(fact_2485_add__increasing2,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ C2 )
     => ( ( ord_less_eq_real @ B2 @ A2 )
       => ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A2 @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_2486_add__increasing2,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ B2 @ A2 )
       => ( ord_less_eq_rat @ B2 @ ( plus_plus_rat @ A2 @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_2487_add__increasing2,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
     => ( ( ord_less_eq_nat @ B2 @ A2 )
       => ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_2488_add__increasing2,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C2 )
     => ( ( ord_less_eq_int @ B2 @ A2 )
       => ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A2 @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_2489_add__nonneg__nonneg,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_2490_add__nonneg__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_2491_add__nonneg__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_2492_add__nonneg__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_2493_add__nonpos__nonpos,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ A2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).

% add_nonpos_nonpos
thf(fact_2494_add__nonpos__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% add_nonpos_nonpos
thf(fact_2495_add__nonpos__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_2496_add__nonpos__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% add_nonpos_nonpos
thf(fact_2497_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_2498_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_2499_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_2500_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_2501_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_2502_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_2503_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_2504_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_2505_add__neg__neg,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ zero_zero_real )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).

% add_neg_neg
thf(fact_2506_add__neg__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% add_neg_neg
thf(fact_2507_add__neg__neg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_2508_add__neg__neg,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% add_neg_neg
thf(fact_2509_add__pos__pos,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_2510_add__pos__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_2511_add__pos__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_2512_add__pos__pos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_2513_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ~ ! [C4: nat] :
            ( ( B2
              = ( plus_plus_nat @ A2 @ C4 ) )
           => ( C4 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_2514_pos__add__strict,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ B2 @ C2 )
       => ( ord_less_real @ B2 @ ( plus_plus_real @ A2 @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_2515_pos__add__strict,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ B2 @ C2 )
       => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A2 @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_2516_pos__add__strict,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_2517_pos__add__strict,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B2 @ C2 )
       => ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_2518_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J2: real,K: real,L2: real] :
      ( ( ( ord_less_eq_real @ I @ J2 )
        & ( ord_less_real @ K @ L2 ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2519_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J2: rat,K: rat,L2: rat] :
      ( ( ( ord_less_eq_rat @ I @ J2 )
        & ( ord_less_rat @ K @ L2 ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2520_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J2: nat,K: nat,L2: nat] :
      ( ( ( ord_less_eq_nat @ I @ J2 )
        & ( ord_less_nat @ K @ L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2521_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J2: int,K: int,L2: int] :
      ( ( ( ord_less_eq_int @ I @ J2 )
        & ( ord_less_int @ K @ L2 ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_2522_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J2: real,K: real,L2: real] :
      ( ( ( ord_less_real @ I @ J2 )
        & ( ord_less_eq_real @ K @ L2 ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2523_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J2: rat,K: rat,L2: rat] :
      ( ( ( ord_less_rat @ I @ J2 )
        & ( ord_less_eq_rat @ K @ L2 ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2524_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J2: nat,K: nat,L2: nat] :
      ( ( ( ord_less_nat @ I @ J2 )
        & ( ord_less_eq_nat @ K @ L2 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2525_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J2: int,K: int,L2: int] :
      ( ( ( ord_less_int @ I @ J2 )
        & ( ord_less_eq_int @ K @ L2 ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J2 @ L2 ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_2526_add__le__less__mono,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_real @ C2 @ D2 )
       => ( ord_less_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B2 @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_2527_add__le__less__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ C2 @ D2 )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C2 ) @ ( plus_plus_rat @ B2 @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_2528_add__le__less__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ C2 @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_2529_add__le__less__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_int @ C2 @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_2530_add__less__le__mono,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ C2 @ D2 )
       => ( ord_less_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B2 @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_2531_add__less__le__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ C2 @ D2 )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C2 ) @ ( plus_plus_rat @ B2 @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_2532_add__less__le__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C2 @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C2 ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_2533_add__less__le__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ C2 @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_2534_le__iff__diff__le__0,axiom,
    ( ord_less_eq_real
    = ( ^ [A7: real,B7: real] : ( ord_less_eq_real @ ( minus_minus_real @ A7 @ B7 ) @ zero_zero_real ) ) ) ).

% le_iff_diff_le_0
thf(fact_2535_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A7: rat,B7: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A7 @ B7 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_2536_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A7: int,B7: int] : ( ord_less_eq_int @ ( minus_minus_int @ A7 @ B7 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_2537_less__iff__diff__less__0,axiom,
    ( ord_less_real
    = ( ^ [A7: real,B7: real] : ( ord_less_real @ ( minus_minus_real @ A7 @ B7 ) @ zero_zero_real ) ) ) ).

% less_iff_diff_less_0
thf(fact_2538_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A7: rat,B7: rat] : ( ord_less_rat @ ( minus_minus_rat @ A7 @ B7 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_2539_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A7: int,B7: int] : ( ord_less_int @ ( minus_minus_int @ A7 @ B7 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_2540_diff__le__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_eq_real @ ( minus_minus_real @ A2 @ B2 ) @ C2 )
      = ( ord_less_eq_real @ A2 @ ( plus_plus_real @ C2 @ B2 ) ) ) ).

% diff_le_eq
thf(fact_2541_diff__le__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C2 )
      = ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ C2 @ B2 ) ) ) ).

% diff_le_eq
thf(fact_2542_diff__le__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B2 ) @ C2 )
      = ( ord_less_eq_int @ A2 @ ( plus_plus_int @ C2 @ B2 ) ) ) ).

% diff_le_eq
thf(fact_2543_le__diff__eq,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( ord_less_eq_real @ A2 @ ( minus_minus_real @ C2 @ B2 ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B2 ) @ C2 ) ) ).

% le_diff_eq
thf(fact_2544_le__diff__eq,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( minus_minus_rat @ C2 @ B2 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C2 ) ) ).

% le_diff_eq
thf(fact_2545_le__diff__eq,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ ( minus_minus_int @ C2 @ B2 ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 ) ) ).

% le_diff_eq
thf(fact_2546_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ A2 )
        = B2 ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_2547_le__add__diff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C2 ) @ A2 ) ) ) ).

% le_add_diff
thf(fact_2548_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B2 @ A2 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A2 ) @ B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_2549_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B2 @ A2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B2 ) @ A2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_2550_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B2 ) @ A2 )
        = ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B2 @ A2 ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_2551_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C2 )
        = ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C2 ) @ A2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_2552_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C2 ) @ A2 )
        = ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_2553_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B2 @ A2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A2 ) @ B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_2554_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B2 @ A2 ) )
        = B2 ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_2555_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( ( ( minus_minus_nat @ B2 @ A2 )
            = C2 )
          = ( B2
            = ( plus_plus_nat @ C2 @ A2 ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_2556_diff__less__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_real @ ( minus_minus_real @ A2 @ B2 ) @ C2 )
      = ( ord_less_real @ A2 @ ( plus_plus_real @ C2 @ B2 ) ) ) ).

% diff_less_eq
thf(fact_2557_diff__less__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ A2 @ B2 ) @ C2 )
      = ( ord_less_rat @ A2 @ ( plus_plus_rat @ C2 @ B2 ) ) ) ).

% diff_less_eq
thf(fact_2558_diff__less__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A2 @ B2 ) @ C2 )
      = ( ord_less_int @ A2 @ ( plus_plus_int @ C2 @ B2 ) ) ) ).

% diff_less_eq
thf(fact_2559_less__diff__eq,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( ord_less_real @ A2 @ ( minus_minus_real @ C2 @ B2 ) )
      = ( ord_less_real @ ( plus_plus_real @ A2 @ B2 ) @ C2 ) ) ).

% less_diff_eq
thf(fact_2560_less__diff__eq,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ ( minus_minus_rat @ C2 @ B2 ) )
      = ( ord_less_rat @ ( plus_plus_rat @ A2 @ B2 ) @ C2 ) ) ).

% less_diff_eq
thf(fact_2561_less__diff__eq,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ord_less_int @ A2 @ ( minus_minus_int @ C2 @ B2 ) )
      = ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 ) ) ).

% less_diff_eq
thf(fact_2562_add__neg__nonpos,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).

% add_neg_nonpos
thf(fact_2563_add__neg__nonpos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% add_neg_nonpos
thf(fact_2564_add__neg__nonpos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_2565_add__neg__nonpos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% add_neg_nonpos
thf(fact_2566_add__nonneg__pos,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_2567_add__nonneg__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_2568_add__nonneg__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_2569_add__nonneg__pos,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).

% add_nonneg_pos
thf(fact_2570_add__nonpos__neg,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ A2 @ zero_zero_real )
     => ( ( ord_less_real @ B2 @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).

% add_nonpos_neg
thf(fact_2571_add__nonpos__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_rat @ B2 @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% add_nonpos_neg
thf(fact_2572_add__nonpos__neg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_2573_add__nonpos__neg,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% add_nonpos_neg
thf(fact_2574_add__pos__nonneg,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_2575_add__pos__nonneg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_2576_add__pos__nonneg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_2577_add__pos__nonneg,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).

% add_pos_nonneg
thf(fact_2578_add__strict__increasing,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ B2 @ C2 )
       => ( ord_less_real @ B2 @ ( plus_plus_real @ A2 @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_2579_add__strict__increasing,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ B2 @ C2 )
       => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A2 @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_2580_add__strict__increasing,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_2581_add__strict__increasing,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ B2 @ C2 )
       => ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_2582_add__strict__increasing2,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ B2 @ C2 )
       => ( ord_less_real @ B2 @ ( plus_plus_real @ A2 @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_2583_add__strict__increasing2,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ B2 @ C2 )
       => ( ord_less_rat @ B2 @ ( plus_plus_rat @ A2 @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_2584_add__strict__increasing2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_2585_add__strict__increasing2,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B2 @ C2 )
       => ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_2586_n__less__equal__power__2,axiom,
    ! [N3: nat] : ( ord_less_nat @ N3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% n_less_equal_power_2
thf(fact_2587_msrevs_I1_J,axiom,
    ! [N3: nat,K: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ K @ N3 ) @ M ) @ N3 )
        = ( plus_plus_nat @ ( divide_divide_nat @ M @ N3 ) @ K ) ) ) ).

% msrevs(1)
thf(fact_2588_nat__mult__power__less__eq,axiom,
    ! [B2: nat,A2: nat,N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ B2 @ N3 ) ) @ ( power_power_nat @ B2 @ M ) )
        = ( ord_less_nat @ A2 @ ( power_power_nat @ B2 @ ( minus_minus_nat @ M @ N3 ) ) ) ) ) ).

% nat_mult_power_less_eq
thf(fact_2589_nat__add__offset__less,axiom,
    ! [Y: nat,N3: nat,X: nat,M: nat,Sz: nat] :
      ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
       => ( ( Sz
            = ( plus_plus_nat @ M @ N3 ) )
         => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) @ Y ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Sz ) ) ) ) ) ).

% nat_add_offset_less
thf(fact_2590_nat__power__less__diff,axiom,
    ! [N3: nat,Q3: nat,M: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ Q3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
     => ( ord_less_nat @ Q3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N3 ) ) ) ) ).

% nat_power_less_diff
thf(fact_2591_nat__le__power__trans,axiom,
    ! [N3: nat,M: nat,K: nat] :
      ( ( ord_less_eq_nat @ N3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ K ) ) )
     => ( ( ord_less_eq_nat @ K @ M )
       => ( ord_less_eq_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ N3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% nat_le_power_trans
thf(fact_2592_real__average__minus__second,axiom,
    ! [B2: real,A2: real] :
      ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B2 @ A2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A2 )
      = ( divide_divide_real @ ( minus_minus_real @ B2 @ A2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% real_average_minus_second
thf(fact_2593_real__average__minus__first,axiom,
    ! [A2: real,B2: real] :
      ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A2 @ B2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A2 )
      = ( divide_divide_real @ ( minus_minus_real @ B2 @ A2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% real_average_minus_first
thf(fact_2594_nat__bit__induct,axiom,
    ! [P: nat > $o,N3: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N: nat] :
            ( ( P @ N )
           => ( ( ord_less_nat @ zero_zero_nat @ N )
             => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
       => ( ! [N: nat] :
              ( ( P @ N )
             => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
         => ( P @ N3 ) ) ) ) ).

% nat_bit_induct
thf(fact_2595_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_2596_div__exp__eq,axiom,
    ! [A2: code_integer,M: nat,N3: nat] :
      ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) )
      = ( divide6298287555418463151nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N3 ) ) ) ) ).

% div_exp_eq
thf(fact_2597_div__exp__eq,axiom,
    ! [A2: nat,M: nat,N3: nat] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( divide_divide_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N3 ) ) ) ) ).

% div_exp_eq
thf(fact_2598_div__exp__eq,axiom,
    ! [A2: int,M: nat,N3: nat] :
      ( ( divide_divide_int @ ( divide_divide_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
      = ( divide_divide_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N3 ) ) ) ) ).

% div_exp_eq
thf(fact_2599_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N3: nat,M: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) )
       != zero_z3403309356797280102nteger ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_2600_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N3: nat,M: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) )
       != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_2601_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N3: nat,M: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) )
       != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_2602_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N3 ) )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
       != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_left
thf(fact_2603_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N3 ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
thf(fact_2604_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N3 ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
thf(fact_2605_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N3 ) )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 )
       != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_right
thf(fact_2606_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N3 ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
thf(fact_2607_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N3 ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
thf(fact_2608_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_2609_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_2610_cnt__non__neg,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_eq_real @ zero_zero_real @ ( vEBT_VEBT_cnt @ T ) ) ).

% cnt_non_neg
thf(fact_2611_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_2612_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_2613_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_2614_bits__div__by__0,axiom,
    ! [A2: nat] :
      ( ( divide_divide_nat @ A2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% bits_div_by_0
thf(fact_2615_bits__div__by__0,axiom,
    ! [A2: int] :
      ( ( divide_divide_int @ A2 @ zero_zero_int )
      = zero_zero_int ) ).

% bits_div_by_0
thf(fact_2616_bits__div__0,axiom,
    ! [A2: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% bits_div_0
thf(fact_2617_bits__div__0,axiom,
    ! [A2: int] :
      ( ( divide_divide_int @ zero_zero_int @ A2 )
      = zero_zero_int ) ).

% bits_div_0
thf(fact_2618__C7_Oprems_C,axiom,
    vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ treeList @ summary ) @ na ).

% "7.prems"
thf(fact_2619_complete__real,axiom,
    ! [S: set_real] :
      ( ? [X5: real] : ( member_real @ X5 @ S )
     => ( ? [Z4: real] :
          ! [X4: real] :
            ( ( member_real @ X4 @ S )
           => ( ord_less_eq_real @ X4 @ Z4 ) )
       => ? [Y3: real] :
            ( ! [X5: real] :
                ( ( member_real @ X5 @ S )
               => ( ord_less_eq_real @ X5 @ Y3 ) )
            & ! [Z4: real] :
                ( ! [X4: real] :
                    ( ( member_real @ X4 @ S )
                   => ( ord_less_eq_real @ X4 @ Z4 ) )
               => ( ord_less_eq_real @ Y3 @ Z4 ) ) ) ) ) ).

% complete_real
thf(fact_2620_field__lbound__gt__zero,axiom,
    ! [D1: real,D22: real] :
      ( ( ord_less_real @ zero_zero_real @ D1 )
     => ( ( ord_less_real @ zero_zero_real @ D22 )
       => ? [E2: real] :
            ( ( ord_less_real @ zero_zero_real @ E2 )
            & ( ord_less_real @ E2 @ D1 )
            & ( ord_less_real @ E2 @ D22 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_2621_field__lbound__gt__zero,axiom,
    ! [D1: rat,D22: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ D1 )
     => ( ( ord_less_rat @ zero_zero_rat @ D22 )
       => ? [E2: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ E2 )
            & ( ord_less_rat @ E2 @ D1 )
            & ( ord_less_rat @ E2 @ D22 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_2622_not__exp__less__eq__0__int,axiom,
    ! [N3: nat] :
      ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_int ) ).

% not_exp_less_eq_0_int
thf(fact_2623_add__diff__add,axiom,
    ! [A2: real,C2: real,B2: real,D2: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B2 @ D2 ) )
      = ( plus_plus_real @ ( minus_minus_real @ A2 @ B2 ) @ ( minus_minus_real @ C2 @ D2 ) ) ) ).

% add_diff_add
thf(fact_2624_add__diff__add,axiom,
    ! [A2: rat,C2: rat,B2: rat,D2: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C2 ) @ ( plus_plus_rat @ B2 @ D2 ) )
      = ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B2 ) @ ( minus_minus_rat @ C2 @ D2 ) ) ) ).

% add_diff_add
thf(fact_2625_add__diff__add,axiom,
    ! [A2: int,C2: int,B2: int,D2: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A2 @ C2 ) @ ( plus_plus_int @ B2 @ D2 ) )
      = ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ ( minus_minus_int @ C2 @ D2 ) ) ) ).

% add_diff_add
thf(fact_2626_less__eq__real__def,axiom,
    ( ord_less_eq_real
    = ( ^ [X3: real,Y2: real] :
          ( ( ord_less_real @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% less_eq_real_def
thf(fact_2627_mult__diff__mult,axiom,
    ! [X: real,Y: real,A2: real,B2: real] :
      ( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A2 @ B2 ) )
      = ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B2 ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A2 ) @ B2 ) ) ) ).

% mult_diff_mult
thf(fact_2628_mult__diff__mult,axiom,
    ! [X: rat,Y: rat,A2: rat,B2: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ Y ) @ ( times_times_rat @ A2 @ B2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ X @ ( minus_minus_rat @ Y @ B2 ) ) @ ( times_times_rat @ ( minus_minus_rat @ X @ A2 ) @ B2 ) ) ) ).

% mult_diff_mult
thf(fact_2629_mult__diff__mult,axiom,
    ! [X: int,Y: int,A2: int,B2: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ Y ) @ ( times_times_int @ A2 @ B2 ) )
      = ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y @ B2 ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A2 ) @ B2 ) ) ) ).

% mult_diff_mult
thf(fact_2630_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_2631_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_2632_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,L2: 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 ) ) ) )
                  = L2 )
               => ( ( 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 ) @ L2 ) )
                   => ( ( 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_2633_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_2634_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L2: 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 ) ) ) )
                = L2 )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
               => ( ~ ( 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_2635_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N3: nat,Va: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N3 )
     => ( ( N3
          = ( suc @ ( suc @ Va ) ) )
       => ( ~ ( 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 @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ).

% nested_mint
thf(fact_2636_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_2637_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_2638_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_2639_count__buildup,axiom,
    ! [N3: nat] : ( ord_less_eq_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% count_buildup
thf(fact_2640_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N3 )
     => ( ( Mi != Ma )
       => ( ( ord_less_nat @ Mi @ Ma )
          & ? [M4: nat] :
              ( ( ( some_nat @ M4 )
                = ( vEBT_vebt_mint @ Summary ) )
              & ( ord_less_nat @ M4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% mintlistlength
thf(fact_2641_valid__0__not,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_0_not
thf(fact_2642_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_tree_deg_neq_0
thf(fact_2643_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N3 )
     => ( Deg = N3 ) ) ).

% deg_deg_n
thf(fact_2644_delete__pres__valid,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ T @ X ) @ N3 ) ) ).

% delete_pres_valid
thf(fact_2645_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ~ ( vEBT_vebt_member @ T @ X ) ) ).

% min_Null_member
thf(fact_2646_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).

% deg_not_0
thf(fact_2647_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N3 ) ) )
     => ? [Info2: option4927543243414619207at_nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
          ( Tree
          = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N3 ) ) @ TreeList3 @ S3 ) ) ) ).

% deg_SUcn_Node
thf(fact_2648_dele__member__cont__corr,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_vebt_member @ ( vEBT_vebt_delete @ T @ X ) @ Y )
        = ( ( X != Y )
          & ( vEBT_vebt_member @ T @ Y ) ) ) ) ).

% dele_member_cont_corr
thf(fact_2649_buildup__gives__valid,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N3 ) @ N3 ) ) ).

% buildup_gives_valid
thf(fact_2650_mint__member,axiom,
    ! [T: vEBT_VEBT,N3: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% mint_member
thf(fact_2651_maxt__member,axiom,
    ! [T: vEBT_VEBT,N3: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% maxt_member
thf(fact_2652_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N3: nat,Mini: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Mini ) )
       => ( ( vEBT_vebt_member @ T @ X )
         => ( ord_less_eq_nat @ Mini @ X ) ) ) ) ).

% mint_corr_help
thf(fact_2653_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N3: nat,Maxi: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( ( vEBT_vebt_member @ T @ X )
         => ( ord_less_eq_nat @ X @ Maxi ) ) ) ) ).

% maxt_corr_help
thf(fact_2654_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N3: nat] :
      ( ( vEBT_vebt_member @ Tree @ X )
     => ( ( vEBT_invar_vebt @ Tree @ N3 )
       => ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% member_bound
thf(fact_2655_misiz,axiom,
    ! [T: vEBT_VEBT,N3: nat,M: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( some_nat @ M )
          = ( vEBT_vebt_mint @ T ) )
       => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% misiz
thf(fact_2656_helpyd,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Y ) )
       => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% helpyd
thf(fact_2657_helpypredd,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Y ) )
       => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% helpypredd
thf(fact_2658_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
         => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X ) @ Y )
           => ( ( vEBT_vebt_member @ T @ Y )
              | ( X = Y ) ) ) ) ) ) ).

% post_member_pre_member
thf(fact_2659_sumprop,axiom,
    vEBT_invar_vebt @ summary @ ( minus_minus_nat @ na @ ( divide_divide_nat @ na @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% sumprop
thf(fact_2660_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N3 )
     => ( ( 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_2661_member__correct,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_vebt_member @ T @ X )
        = ( member_nat @ X @ ( vEBT_set_vebt @ T ) ) ) ) ).

% member_correct
thf(fact_2662_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_2663_div__neg__neg__trivial,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L2 @ K )
       => ( ( divide_divide_int @ K @ L2 )
          = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
thf(fact_2664_div__pos__pos__trivial,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L2 )
       => ( ( divide_divide_int @ K @ L2 )
          = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
thf(fact_2665_zdiv__mult__self,axiom,
    ! [M: int,A2: int,N3: int] :
      ( ( M != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ ( times_times_int @ M @ N3 ) ) @ M )
        = ( plus_plus_int @ ( divide_divide_int @ A2 @ M ) @ N3 ) ) ) ).

% zdiv_mult_self
thf(fact_2666_zdiv__le__dividend,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B2 ) @ A2 ) ) ) ).

% zdiv_le_dividend
thf(fact_2667_zdiv__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C2 )
     => ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ C2 ) )
        = ( divide_divide_int @ ( divide_divide_int @ A2 @ B2 ) @ C2 ) ) ) ).

% zdiv_zmult2_eq
thf(fact_2668_delete__correct_H,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( 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_2669_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
       => ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X ) ) ) ) ).

% maxt_sound
thf(fact_2670_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X ) )
       => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).

% maxt_corr
thf(fact_2671_mint__corr,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X ) )
       => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).

% mint_corr
thf(fact_2672_mint__sound,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
       => ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X ) ) ) ) ).

% mint_sound
thf(fact_2673_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_2674_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_2675_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_2676_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_2677_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( ( vEBT_vebt_mint @ T )
        = none_nat )
     => ( vEBT_VEBT_minNull @ T ) ) ).

% minminNull
thf(fact_2678_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ T )
     => ( ( vEBT_vebt_mint @ T )
        = none_nat ) ) ).

% minNullmin
thf(fact_2679_buildup__gives__empty,axiom,
    ! [N3: nat] :
      ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N3 ) )
      = bot_bot_set_nat ) ).

% buildup_gives_empty
thf(fact_2680_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_set_vebt @ T )
        = ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_set_vebt'_valid
thf(fact_2681_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_mint @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
thf(fact_2682_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
thf(fact_2683_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N3 )
     => ( ( 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_2684_delete__correct,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( 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_2685_max__Suc__Suc,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N3 ) )
      = ( suc @ ( ord_max_nat @ M @ N3 ) ) ) ).

% max_Suc_Suc
thf(fact_2686_max__0R,axiom,
    ! [N3: nat] :
      ( ( ord_max_nat @ N3 @ zero_zero_nat )
      = N3 ) ).

% max_0R
thf(fact_2687_max__0L,axiom,
    ! [N3: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N3 )
      = N3 ) ).

% max_0L
thf(fact_2688_max__nat_Oright__neutral,axiom,
    ! [A2: nat] :
      ( ( ord_max_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% max_nat.right_neutral
thf(fact_2689_max__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( zero_zero_nat
        = ( ord_max_nat @ A2 @ B2 ) )
      = ( ( A2 = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
thf(fact_2690_max__nat_Oleft__neutral,axiom,
    ! [A2: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A2 )
      = A2 ) ).

% max_nat.left_neutral
thf(fact_2691_max__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( ord_max_nat @ A2 @ B2 )
        = zero_zero_nat )
      = ( ( A2 = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
thf(fact_2692_less__eq__option__None__code,axiom,
    ! [X: option_nat] : ( ord_le5914376470875661696on_nat @ none_nat @ X ) ).

% less_eq_option_None_code
thf(fact_2693_less__option__None,axiom,
    ! [X: option_nat] :
      ~ ( ord_less_option_nat @ X @ none_nat ) ).

% less_option_None
thf(fact_2694_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_2695_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_2696_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_2697_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_2698_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_2699_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_2700_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_2701_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_2702_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_2703_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_2704_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_2705_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_2706_less__eq__option__Some__None,axiom,
    ! [X: nat] :
      ~ ( ord_le5914376470875661696on_nat @ ( some_nat @ X ) @ none_nat ) ).

% less_eq_option_Some_None
thf(fact_2707_less__option__None__Some__code,axiom,
    ! [X: nat] : ( ord_less_option_nat @ none_nat @ ( some_nat @ X ) ) ).

% less_option_None_Some_code
thf(fact_2708_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 )
        & ! [Z5: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z5 )
              & ( ord_less_nat @ Z5 @ X ) )
           => ( ord_less_eq_nat @ Z5 @ Y ) ) ) ) ).

% pred_member
thf(fact_2709_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 )
        & ! [Z5: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z5 )
              & ( ord_less_nat @ X @ Z5 ) )
           => ( ord_less_eq_nat @ Y @ Z5 ) ) ) ) ).

% succ_member
thf(fact_2710_succ__corr,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% succ_corr
thf(fact_2711_pred__corr,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat,Px: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Px ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Px ) ) ) ).

% pred_corr
thf(fact_2712_pred__correct,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% pred_correct
thf(fact_2713_succ__correct,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% succ_correct
thf(fact_2714_max__add__distrib__right,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( plus_plus_real @ X @ ( ord_max_real @ Y @ Z ) )
      = ( ord_max_real @ ( plus_plus_real @ X @ Y ) @ ( plus_plus_real @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_2715_max__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y @ Z ) )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_2716_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z ) )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_2717_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z ) )
      = ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_2718_max__add__distrib__left,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( plus_plus_real @ ( ord_max_real @ X @ Y ) @ Z )
      = ( ord_max_real @ ( plus_plus_real @ X @ Z ) @ ( plus_plus_real @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_2719_max__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Z ) @ ( plus_plus_rat @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_2720_max__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_2721_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ord_max_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_2722_max__diff__distrib__left,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( minus_minus_real @ ( ord_max_real @ X @ Y ) @ Z )
      = ( ord_max_real @ ( minus_minus_real @ X @ Z ) @ ( minus_minus_real @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_2723_max__diff__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( minus_minus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
      = ( ord_max_rat @ ( minus_minus_rat @ X @ Z ) @ ( minus_minus_rat @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_2724_max__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( minus_minus_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ord_max_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_2725_nat__add__max__right,axiom,
    ! [M: nat,N3: nat,Q3: nat] :
      ( ( plus_plus_nat @ M @ ( ord_max_nat @ N3 @ Q3 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ N3 ) @ ( plus_plus_nat @ M @ Q3 ) ) ) ).

% nat_add_max_right
thf(fact_2726_nat__add__max__left,axiom,
    ! [M: nat,N3: nat,Q3: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M @ N3 ) @ Q3 )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ Q3 ) @ ( plus_plus_nat @ N3 @ Q3 ) ) ) ).

% nat_add_max_left
thf(fact_2727_nat__mult__max__right,axiom,
    ! [M: nat,N3: nat,Q3: nat] :
      ( ( times_times_nat @ M @ ( ord_max_nat @ N3 @ Q3 ) )
      = ( ord_max_nat @ ( times_times_nat @ M @ N3 ) @ ( times_times_nat @ M @ Q3 ) ) ) ).

% nat_mult_max_right
thf(fact_2728_nat__mult__max__left,axiom,
    ! [M: nat,N3: nat,Q3: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M @ N3 ) @ Q3 )
      = ( ord_max_nat @ ( times_times_nat @ M @ Q3 ) @ ( times_times_nat @ N3 @ Q3 ) ) ) ).

% nat_mult_max_left
thf(fact_2729_less__eq__option__None,axiom,
    ! [X: option_nat] : ( ord_le5914376470875661696on_nat @ none_nat @ X ) ).

% less_eq_option_None
thf(fact_2730_less__eq__option__None__is__None,axiom,
    ! [X: option_nat] :
      ( ( ord_le5914376470875661696on_nat @ X @ none_nat )
     => ( X = none_nat ) ) ).

% less_eq_option_None_is_None
thf(fact_2731_nat__minus__add__max,axiom,
    ! [N3: nat,M: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ N3 @ M ) @ M )
      = ( ord_max_nat @ N3 @ M ) ) ).

% nat_minus_add_max
thf(fact_2732_less__option__None__Some,axiom,
    ! [X: nat] : ( ord_less_option_nat @ none_nat @ ( some_nat @ X ) ) ).

% less_option_None_Some
thf(fact_2733_less__option__None__is__Some,axiom,
    ! [X: option_nat] :
      ( ( ord_less_option_nat @ none_nat @ X )
     => ? [Z2: nat] :
          ( X
          = ( some_nat @ Z2 ) ) ) ).

% less_option_None_is_Some
thf(fact_2734_cnt__bound_H,axiom,
    ! [T: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_real @ ( vEBT_VEBT_cnt @ T ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( minus_minus_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) @ one_one_real ) ) ) ) ).

% cnt_bound'
thf(fact_2735_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_2736_cnt__bound,axiom,
    ! [T: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_real @ ( vEBT_VEBT_cnt @ T ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( minus_minus_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% cnt_bound
thf(fact_2737_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( ( some_nat @ ( the_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_2738_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_2739_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_2740_listI__assn__wrap__insert,axiom,
    ! [P: assn,Uu: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs2: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F: assn,C2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ Uu @ Uua ) @ Xi ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L1528199826722428489_VEBTi @ I3 @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( vEBT_vebt_insert @ Uu @ Uua ) ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F ) @ C2 @ Q )
           => ( hoare_hoare_triple_o @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_wrap_insert
thf(fact_2741_listI__assn__wrap__insert,axiom,
    ! [P: assn,Uu: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs2: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F: assn,C2: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ Uu @ Uua ) @ Xi ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( vEBT_L1528199826722428489_VEBTi @ I3 @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( vEBT_vebt_insert @ Uu @ Uua ) ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F ) @ C2 @ Q )
           => ( hoare_1429296392585015714_VEBTi @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_wrap_insert
thf(fact_2742_listI__assn__wrap__insert,axiom,
    ! [P: assn,Uu: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs2: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F: assn,C2: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ Uu @ Uua ) @ Xi ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( hoare_7629718768684598413on_nat @ ( times_times_assn @ ( vEBT_L1528199826722428489_VEBTi @ I3 @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( vEBT_vebt_insert @ Uu @ Uua ) ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F ) @ C2 @ Q )
           => ( hoare_7629718768684598413on_nat @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_wrap_insert
thf(fact_2743_listI__assn__wrap__insert,axiom,
    ! [P: assn,Uu: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs2: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi,F: assn,C2: heap_Time_Heap_nat,Q: nat > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ Uu @ Uua ) @ Xi ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( hoare_3067605981109127869le_nat @ ( times_times_assn @ ( vEBT_L1528199826722428489_VEBTi @ I3 @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( vEBT_vebt_insert @ Uu @ Uua ) ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F ) @ C2 @ Q )
           => ( hoare_3067605981109127869le_nat @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_wrap_insert
thf(fact_2744_two__powr__height__bound__deg,axiom,
    ! [T: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( vEBT_VEBT_height @ T ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% two_powr_height_bound_deg
thf(fact_2745_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N3 )
     => ( ( 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_2746_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_2747_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ T )
     => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).

% not_min_Null_member
thf(fact_2748_maxbmo,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( ( vEBT_vebt_maxt @ T )
        = ( some_nat @ X ) )
     => ( vEBT_V8194947554948674370ptions @ T @ X ) ) ).

% maxbmo
thf(fact_2749_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T @ X ) @ Y )
        = ( ( X != Y )
          & ( vEBT_V8194947554948674370ptions @ T @ Y ) ) ) ) ).

% dele_bmo_cont_corr
thf(fact_2750_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X )
       => ( vEBT_vebt_member @ T @ X ) ) ) ).

% valid_member_both_member_options
thf(fact_2751_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X )
        = ( vEBT_vebt_member @ T @ X ) ) ) ).

% both_member_options_equiv_member
thf(fact_2752_semiring__norm_I90_J,axiom,
    ! [M: num,N3: num] :
      ( ( ( bit1 @ M )
        = ( bit1 @ N3 ) )
      = ( M = N3 ) ) ).

% semiring_norm(90)
thf(fact_2753_option_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( some_nat @ X22 )
        = ( some_nat @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_2754_option_Oinject,axiom,
    ! [X22: product_prod_nat_nat,Y22: product_prod_nat_nat] :
      ( ( ( some_P7363390416028606310at_nat @ X22 )
        = ( some_P7363390416028606310at_nat @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_2755_mult_Oright__neutral,axiom,
    ! [A2: real] :
      ( ( times_times_real @ A2 @ one_one_real )
      = A2 ) ).

% mult.right_neutral
thf(fact_2756_mult_Oright__neutral,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ A2 @ one_one_rat )
      = A2 ) ).

% mult.right_neutral
thf(fact_2757_mult_Oright__neutral,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ A2 @ one_one_nat )
      = A2 ) ).

% mult.right_neutral
thf(fact_2758_mult_Oright__neutral,axiom,
    ! [A2: int] :
      ( ( times_times_int @ A2 @ one_one_int )
      = A2 ) ).

% mult.right_neutral
thf(fact_2759_mult_Oright__neutral,axiom,
    ! [A2: assn] :
      ( ( times_times_assn @ A2 @ one_one_assn )
      = A2 ) ).

% mult.right_neutral
thf(fact_2760_mult__1,axiom,
    ! [A2: real] :
      ( ( times_times_real @ one_one_real @ A2 )
      = A2 ) ).

% mult_1
thf(fact_2761_mult__1,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ one_one_rat @ A2 )
      = A2 ) ).

% mult_1
thf(fact_2762_mult__1,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ one_one_nat @ A2 )
      = A2 ) ).

% mult_1
thf(fact_2763_mult__1,axiom,
    ! [A2: int] :
      ( ( times_times_int @ one_one_int @ A2 )
      = A2 ) ).

% mult_1
thf(fact_2764_mult__1,axiom,
    ! [A2: assn] :
      ( ( times_times_assn @ one_one_assn @ A2 )
      = A2 ) ).

% mult_1
thf(fact_2765_div__by__1,axiom,
    ! [A2: complex] :
      ( ( divide1717551699836669952omplex @ A2 @ one_one_complex )
      = A2 ) ).

% div_by_1
thf(fact_2766_div__by__1,axiom,
    ! [A2: real] :
      ( ( divide_divide_real @ A2 @ one_one_real )
      = A2 ) ).

% div_by_1
thf(fact_2767_div__by__1,axiom,
    ! [A2: rat] :
      ( ( divide_divide_rat @ A2 @ one_one_rat )
      = A2 ) ).

% div_by_1
thf(fact_2768_div__by__1,axiom,
    ! [A2: nat] :
      ( ( divide_divide_nat @ A2 @ one_one_nat )
      = A2 ) ).

% div_by_1
thf(fact_2769_div__by__1,axiom,
    ! [A2: int] :
      ( ( divide_divide_int @ A2 @ one_one_int )
      = A2 ) ).

% div_by_1
thf(fact_2770_bits__div__by__1,axiom,
    ! [A2: nat] :
      ( ( divide_divide_nat @ A2 @ one_one_nat )
      = A2 ) ).

% bits_div_by_1
thf(fact_2771_bits__div__by__1,axiom,
    ! [A2: int] :
      ( ( divide_divide_int @ A2 @ one_one_int )
      = A2 ) ).

% bits_div_by_1
thf(fact_2772_power__one,axiom,
    ! [N3: nat] :
      ( ( power_power_assn @ one_one_assn @ N3 )
      = one_one_assn ) ).

% power_one
thf(fact_2773_power__one,axiom,
    ! [N3: nat] :
      ( ( power_power_rat @ one_one_rat @ N3 )
      = one_one_rat ) ).

% power_one
thf(fact_2774_power__one,axiom,
    ! [N3: nat] :
      ( ( power_power_nat @ one_one_nat @ N3 )
      = one_one_nat ) ).

% power_one
thf(fact_2775_power__one,axiom,
    ! [N3: nat] :
      ( ( power_power_real @ one_one_real @ N3 )
      = one_one_real ) ).

% power_one
thf(fact_2776_power__one,axiom,
    ! [N3: nat] :
      ( ( power_power_int @ one_one_int @ N3 )
      = one_one_int ) ).

% power_one
thf(fact_2777_power__one,axiom,
    ! [N3: nat] :
      ( ( power_power_complex @ one_one_complex @ N3 )
      = one_one_complex ) ).

% power_one
thf(fact_2778_power__one,axiom,
    ! [N3: nat] :
      ( ( power_8256067586552552935nteger @ one_one_Code_integer @ N3 )
      = one_one_Code_integer ) ).

% power_one
thf(fact_2779_semiring__norm_I89_J,axiom,
    ! [M: num,N3: num] :
      ( ( bit1 @ M )
     != ( bit0 @ N3 ) ) ).

% semiring_norm(89)
thf(fact_2780_semiring__norm_I88_J,axiom,
    ! [M: num,N3: num] :
      ( ( bit0 @ M )
     != ( bit1 @ N3 ) ) ).

% semiring_norm(88)
thf(fact_2781_semiring__norm_I86_J,axiom,
    ! [M: num] :
      ( ( bit1 @ M )
     != one ) ).

% semiring_norm(86)
thf(fact_2782_semiring__norm_I84_J,axiom,
    ! [N3: num] :
      ( one
     != ( bit1 @ N3 ) ) ).

% semiring_norm(84)
thf(fact_2783_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
       => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X ) @ X ) ) ) ).

% valid_insert_both_member_options_add
thf(fact_2784_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
         => ( ( vEBT_V8194947554948674370ptions @ T @ X )
           => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y ) @ X ) ) ) ) ) ).

% valid_insert_both_member_options_pres
thf(fact_2785_not__None__eq,axiom,
    ! [X: option_nat] :
      ( ( X != none_nat )
      = ( ? [Y2: nat] :
            ( X
            = ( some_nat @ Y2 ) ) ) ) ).

% not_None_eq
thf(fact_2786_not__None__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( X != none_P5556105721700978146at_nat )
      = ( ? [Y2: product_prod_nat_nat] :
            ( X
            = ( some_P7363390416028606310at_nat @ Y2 ) ) ) ) ).

% not_None_eq
thf(fact_2787_not__Some__eq,axiom,
    ! [X: option_nat] :
      ( ( ! [Y2: nat] :
            ( X
           != ( some_nat @ Y2 ) ) )
      = ( X = none_nat ) ) ).

% not_Some_eq
thf(fact_2788_not__Some__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( ! [Y2: product_prod_nat_nat] :
            ( X
           != ( some_P7363390416028606310at_nat @ Y2 ) ) )
      = ( X = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
thf(fact_2789_semiring__norm_I80_J,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
      = ( ord_less_num @ M @ N3 ) ) ).

% semiring_norm(80)
thf(fact_2790_semiring__norm_I73_J,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
      = ( ord_less_eq_num @ M @ N3 ) ) ).

% semiring_norm(73)
thf(fact_2791_mult__cancel__left1,axiom,
    ! [C2: complex,B2: complex] :
      ( ( C2
        = ( times_times_complex @ C2 @ B2 ) )
      = ( ( C2 = zero_zero_complex )
        | ( B2 = one_one_complex ) ) ) ).

% mult_cancel_left1
thf(fact_2792_mult__cancel__left1,axiom,
    ! [C2: real,B2: real] :
      ( ( C2
        = ( times_times_real @ C2 @ B2 ) )
      = ( ( C2 = zero_zero_real )
        | ( B2 = one_one_real ) ) ) ).

% mult_cancel_left1
thf(fact_2793_mult__cancel__left1,axiom,
    ! [C2: rat,B2: rat] :
      ( ( C2
        = ( times_times_rat @ C2 @ B2 ) )
      = ( ( C2 = zero_zero_rat )
        | ( B2 = one_one_rat ) ) ) ).

% mult_cancel_left1
thf(fact_2794_mult__cancel__left1,axiom,
    ! [C2: int,B2: int] :
      ( ( C2
        = ( times_times_int @ C2 @ B2 ) )
      = ( ( C2 = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_left1
thf(fact_2795_mult__cancel__left2,axiom,
    ! [C2: complex,A2: complex] :
      ( ( ( times_times_complex @ C2 @ A2 )
        = C2 )
      = ( ( C2 = zero_zero_complex )
        | ( A2 = one_one_complex ) ) ) ).

% mult_cancel_left2
thf(fact_2796_mult__cancel__left2,axiom,
    ! [C2: real,A2: real] :
      ( ( ( times_times_real @ C2 @ A2 )
        = C2 )
      = ( ( C2 = zero_zero_real )
        | ( A2 = one_one_real ) ) ) ).

% mult_cancel_left2
thf(fact_2797_mult__cancel__left2,axiom,
    ! [C2: rat,A2: rat] :
      ( ( ( times_times_rat @ C2 @ A2 )
        = C2 )
      = ( ( C2 = zero_zero_rat )
        | ( A2 = one_one_rat ) ) ) ).

% mult_cancel_left2
thf(fact_2798_mult__cancel__left2,axiom,
    ! [C2: int,A2: int] :
      ( ( ( times_times_int @ C2 @ A2 )
        = C2 )
      = ( ( C2 = zero_zero_int )
        | ( A2 = one_one_int ) ) ) ).

% mult_cancel_left2
thf(fact_2799_mult__cancel__right1,axiom,
    ! [C2: complex,B2: complex] :
      ( ( C2
        = ( times_times_complex @ B2 @ C2 ) )
      = ( ( C2 = zero_zero_complex )
        | ( B2 = one_one_complex ) ) ) ).

% mult_cancel_right1
thf(fact_2800_mult__cancel__right1,axiom,
    ! [C2: real,B2: real] :
      ( ( C2
        = ( times_times_real @ B2 @ C2 ) )
      = ( ( C2 = zero_zero_real )
        | ( B2 = one_one_real ) ) ) ).

% mult_cancel_right1
thf(fact_2801_mult__cancel__right1,axiom,
    ! [C2: rat,B2: rat] :
      ( ( C2
        = ( times_times_rat @ B2 @ C2 ) )
      = ( ( C2 = zero_zero_rat )
        | ( B2 = one_one_rat ) ) ) ).

% mult_cancel_right1
thf(fact_2802_mult__cancel__right1,axiom,
    ! [C2: int,B2: int] :
      ( ( C2
        = ( times_times_int @ B2 @ C2 ) )
      = ( ( C2 = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_right1
thf(fact_2803_mult__cancel__right2,axiom,
    ! [A2: complex,C2: complex] :
      ( ( ( times_times_complex @ A2 @ C2 )
        = C2 )
      = ( ( C2 = zero_zero_complex )
        | ( A2 = one_one_complex ) ) ) ).

% mult_cancel_right2
thf(fact_2804_mult__cancel__right2,axiom,
    ! [A2: real,C2: real] :
      ( ( ( times_times_real @ A2 @ C2 )
        = C2 )
      = ( ( C2 = zero_zero_real )
        | ( A2 = one_one_real ) ) ) ).

% mult_cancel_right2
thf(fact_2805_mult__cancel__right2,axiom,
    ! [A2: rat,C2: rat] :
      ( ( ( times_times_rat @ A2 @ C2 )
        = C2 )
      = ( ( C2 = zero_zero_rat )
        | ( A2 = one_one_rat ) ) ) ).

% mult_cancel_right2
thf(fact_2806_mult__cancel__right2,axiom,
    ! [A2: int,C2: int] :
      ( ( ( times_times_int @ A2 @ C2 )
        = C2 )
      = ( ( C2 = zero_zero_int )
        | ( A2 = one_one_int ) ) ) ).

% mult_cancel_right2
thf(fact_2807_numeral__eq__one__iff,axiom,
    ! [N3: num] :
      ( ( ( numera6690914467698888265omplex @ N3 )
        = one_one_complex )
      = ( N3 = one ) ) ).

% numeral_eq_one_iff
thf(fact_2808_numeral__eq__one__iff,axiom,
    ! [N3: num] :
      ( ( ( numeral_numeral_real @ N3 )
        = one_one_real )
      = ( N3 = one ) ) ).

% numeral_eq_one_iff
thf(fact_2809_numeral__eq__one__iff,axiom,
    ! [N3: num] :
      ( ( ( numeral_numeral_rat @ N3 )
        = one_one_rat )
      = ( N3 = one ) ) ).

% numeral_eq_one_iff
thf(fact_2810_numeral__eq__one__iff,axiom,
    ! [N3: num] :
      ( ( ( numeral_numeral_nat @ N3 )
        = one_one_nat )
      = ( N3 = one ) ) ).

% numeral_eq_one_iff
thf(fact_2811_numeral__eq__one__iff,axiom,
    ! [N3: num] :
      ( ( ( numeral_numeral_int @ N3 )
        = one_one_int )
      = ( N3 = one ) ) ).

% numeral_eq_one_iff
thf(fact_2812_one__eq__numeral__iff,axiom,
    ! [N3: num] :
      ( ( one_one_complex
        = ( numera6690914467698888265omplex @ N3 ) )
      = ( one = N3 ) ) ).

% one_eq_numeral_iff
thf(fact_2813_one__eq__numeral__iff,axiom,
    ! [N3: num] :
      ( ( one_one_real
        = ( numeral_numeral_real @ N3 ) )
      = ( one = N3 ) ) ).

% one_eq_numeral_iff
thf(fact_2814_one__eq__numeral__iff,axiom,
    ! [N3: num] :
      ( ( one_one_rat
        = ( numeral_numeral_rat @ N3 ) )
      = ( one = N3 ) ) ).

% one_eq_numeral_iff
thf(fact_2815_one__eq__numeral__iff,axiom,
    ! [N3: num] :
      ( ( one_one_nat
        = ( numeral_numeral_nat @ N3 ) )
      = ( one = N3 ) ) ).

% one_eq_numeral_iff
thf(fact_2816_one__eq__numeral__iff,axiom,
    ! [N3: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ N3 ) )
      = ( one = N3 ) ) ).

% one_eq_numeral_iff
thf(fact_2817_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
    = zero_zero_complex ) ).

% diff_numeral_special(9)
thf(fact_2818_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_real @ one_one_real @ one_one_real )
    = zero_zero_real ) ).

% diff_numeral_special(9)
thf(fact_2819_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_2820_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_2821_divide__eq__1__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ( divide1717551699836669952omplex @ A2 @ B2 )
        = one_one_complex )
      = ( ( B2 != zero_zero_complex )
        & ( A2 = B2 ) ) ) ).

% divide_eq_1_iff
thf(fact_2822_divide__eq__1__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ( divide_divide_real @ A2 @ B2 )
        = one_one_real )
      = ( ( B2 != zero_zero_real )
        & ( A2 = B2 ) ) ) ).

% divide_eq_1_iff
thf(fact_2823_divide__eq__1__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( divide_divide_rat @ A2 @ B2 )
        = one_one_rat )
      = ( ( B2 != zero_zero_rat )
        & ( A2 = B2 ) ) ) ).

% divide_eq_1_iff
thf(fact_2824_div__self,axiom,
    ! [A2: complex] :
      ( ( A2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A2 @ A2 )
        = one_one_complex ) ) ).

% div_self
thf(fact_2825_div__self,axiom,
    ! [A2: real] :
      ( ( A2 != zero_zero_real )
     => ( ( divide_divide_real @ A2 @ A2 )
        = one_one_real ) ) ).

% div_self
thf(fact_2826_div__self,axiom,
    ! [A2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( divide_divide_rat @ A2 @ A2 )
        = one_one_rat ) ) ).

% div_self
thf(fact_2827_div__self,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( divide_divide_nat @ A2 @ A2 )
        = one_one_nat ) ) ).

% div_self
thf(fact_2828_div__self,axiom,
    ! [A2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( divide_divide_int @ A2 @ A2 )
        = one_one_int ) ) ).

% div_self
thf(fact_2829_one__eq__divide__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( one_one_complex
        = ( divide1717551699836669952omplex @ A2 @ B2 ) )
      = ( ( B2 != zero_zero_complex )
        & ( A2 = B2 ) ) ) ).

% one_eq_divide_iff
thf(fact_2830_one__eq__divide__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( one_one_real
        = ( divide_divide_real @ A2 @ B2 ) )
      = ( ( B2 != zero_zero_real )
        & ( A2 = B2 ) ) ) ).

% one_eq_divide_iff
thf(fact_2831_one__eq__divide__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ A2 @ B2 ) )
      = ( ( B2 != zero_zero_rat )
        & ( A2 = B2 ) ) ) ).

% one_eq_divide_iff
thf(fact_2832_divide__self,axiom,
    ! [A2: complex] :
      ( ( A2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A2 @ A2 )
        = one_one_complex ) ) ).

% divide_self
thf(fact_2833_divide__self,axiom,
    ! [A2: real] :
      ( ( A2 != zero_zero_real )
     => ( ( divide_divide_real @ A2 @ A2 )
        = one_one_real ) ) ).

% divide_self
thf(fact_2834_divide__self,axiom,
    ! [A2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( divide_divide_rat @ A2 @ A2 )
        = one_one_rat ) ) ).

% divide_self
thf(fact_2835_divide__self__if,axiom,
    ! [A2: complex] :
      ( ( ( A2 = zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ A2 @ A2 )
          = zero_zero_complex ) )
      & ( ( A2 != zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ A2 @ A2 )
          = one_one_complex ) ) ) ).

% divide_self_if
thf(fact_2836_divide__self__if,axiom,
    ! [A2: real] :
      ( ( ( A2 = zero_zero_real )
       => ( ( divide_divide_real @ A2 @ A2 )
          = zero_zero_real ) )
      & ( ( A2 != zero_zero_real )
       => ( ( divide_divide_real @ A2 @ A2 )
          = one_one_real ) ) ) ).

% divide_self_if
thf(fact_2837_divide__self__if,axiom,
    ! [A2: rat] :
      ( ( ( A2 = zero_zero_rat )
       => ( ( divide_divide_rat @ A2 @ A2 )
          = zero_zero_rat ) )
      & ( ( A2 != zero_zero_rat )
       => ( ( divide_divide_rat @ A2 @ A2 )
          = one_one_rat ) ) ) ).

% divide_self_if
thf(fact_2838_divide__eq__eq__1,axiom,
    ! [B2: real,A2: real] :
      ( ( ( divide_divide_real @ B2 @ A2 )
        = one_one_real )
      = ( ( A2 != zero_zero_real )
        & ( A2 = B2 ) ) ) ).

% divide_eq_eq_1
thf(fact_2839_divide__eq__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ( divide_divide_rat @ B2 @ A2 )
        = one_one_rat )
      = ( ( A2 != zero_zero_rat )
        & ( A2 = B2 ) ) ) ).

% divide_eq_eq_1
thf(fact_2840_eq__divide__eq__1,axiom,
    ! [B2: real,A2: real] :
      ( ( one_one_real
        = ( divide_divide_real @ B2 @ A2 ) )
      = ( ( A2 != zero_zero_real )
        & ( A2 = B2 ) ) ) ).

% eq_divide_eq_1
thf(fact_2841_eq__divide__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ B2 @ A2 ) )
      = ( ( A2 != zero_zero_rat )
        & ( A2 = B2 ) ) ) ).

% eq_divide_eq_1
thf(fact_2842_one__divide__eq__0__iff,axiom,
    ! [A2: real] :
      ( ( ( divide_divide_real @ one_one_real @ A2 )
        = zero_zero_real )
      = ( A2 = zero_zero_real ) ) ).

% one_divide_eq_0_iff
thf(fact_2843_one__divide__eq__0__iff,axiom,
    ! [A2: rat] :
      ( ( ( divide_divide_rat @ one_one_rat @ A2 )
        = zero_zero_rat )
      = ( A2 = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
thf(fact_2844_zero__eq__1__divide__iff,axiom,
    ! [A2: real] :
      ( ( zero_zero_real
        = ( divide_divide_real @ one_one_real @ A2 ) )
      = ( A2 = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
thf(fact_2845_zero__eq__1__divide__iff,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat
        = ( divide_divide_rat @ one_one_rat @ A2 ) )
      = ( A2 = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
thf(fact_2846_power__inject__exp,axiom,
    ! [A2: code_integer,M: nat,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
     => ( ( ( power_8256067586552552935nteger @ A2 @ M )
          = ( power_8256067586552552935nteger @ A2 @ N3 ) )
        = ( M = N3 ) ) ) ).

% power_inject_exp
thf(fact_2847_power__inject__exp,axiom,
    ! [A2: real,M: nat,N3: nat] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ( power_power_real @ A2 @ M )
          = ( power_power_real @ A2 @ N3 ) )
        = ( M = N3 ) ) ) ).

% power_inject_exp
thf(fact_2848_power__inject__exp,axiom,
    ! [A2: rat,M: nat,N3: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ( ( power_power_rat @ A2 @ M )
          = ( power_power_rat @ A2 @ N3 ) )
        = ( M = N3 ) ) ) ).

% power_inject_exp
thf(fact_2849_power__inject__exp,axiom,
    ! [A2: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ( ( power_power_nat @ A2 @ M )
          = ( power_power_nat @ A2 @ N3 ) )
        = ( M = N3 ) ) ) ).

% power_inject_exp
thf(fact_2850_power__inject__exp,axiom,
    ! [A2: int,M: nat,N3: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ( ( power_power_int @ A2 @ M )
          = ( power_power_int @ A2 @ N3 ) )
        = ( M = N3 ) ) ) ).

% power_inject_exp
thf(fact_2851_max__0__1_I1_J,axiom,
    ( ( ord_max_real @ zero_zero_real @ one_one_real )
    = one_one_real ) ).

% max_0_1(1)
thf(fact_2852_max__0__1_I1_J,axiom,
    ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
    = one_one_rat ) ).

% max_0_1(1)
thf(fact_2853_max__0__1_I1_J,axiom,
    ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
    = one_one_nat ) ).

% max_0_1(1)
thf(fact_2854_max__0__1_I1_J,axiom,
    ( ( ord_max_int @ zero_zero_int @ one_one_int )
    = one_one_int ) ).

% max_0_1(1)
thf(fact_2855_max__0__1_I2_J,axiom,
    ( ( ord_max_real @ one_one_real @ zero_zero_real )
    = one_one_real ) ).

% max_0_1(2)
thf(fact_2856_max__0__1_I2_J,axiom,
    ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
    = one_one_rat ) ).

% max_0_1(2)
thf(fact_2857_max__0__1_I2_J,axiom,
    ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
    = one_one_nat ) ).

% max_0_1(2)
thf(fact_2858_max__0__1_I2_J,axiom,
    ( ( ord_max_int @ one_one_int @ zero_zero_int )
    = one_one_int ) ).

% max_0_1(2)
thf(fact_2859_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_2860_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_2861_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_2862_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_2863_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_2864_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_2865_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_2866_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_2867_not__Some__eq2,axiom,
    ! [V: option2651255830984564193nteger] :
      ( ( ! [X3: code_integer,Y2: code_integer] :
            ( V
           != ( some_P6772290148444788224nteger @ ( produc1086072967326762835nteger @ X3 @ Y2 ) ) ) )
      = ( V = none_P4506660739021792380nteger ) ) ).

% not_Some_eq2
thf(fact_2868_not__Some__eq2,axiom,
    ! [V: option936205604648967762et_nat] :
      ( ( ! [X3: heap_e7401611519738050253t_unit,Y2: set_nat] :
            ( V
           != ( some_P624177172695371229et_nat @ ( produc7507926704131184380et_nat @ X3 @ Y2 ) ) ) )
      = ( V = none_P533106815845188193et_nat ) ) ).

% not_Some_eq2
thf(fact_2869_not__Some__eq2,axiom,
    ! [V: option2661157926820139483um_num] :
      ( ( ! [X3: num,Y2: num] :
            ( V
           != ( some_P6201964756284913402um_num @ ( product_Pair_num_num @ X3 @ Y2 ) ) ) )
      = ( V = none_P4394680061957285238um_num ) ) ).

% not_Some_eq2
thf(fact_2870_not__Some__eq2,axiom,
    ! [V: option4624381673175914239nt_int] :
      ( ( ! [X3: int,Y2: int] :
            ( V
           != ( some_P4184893108420464158nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) ) ) )
      = ( V = none_P2377608414092835994nt_int ) ) ).

% not_Some_eq2
thf(fact_2871_not__Some__eq2,axiom,
    ! [V: option4927543243414619207at_nat] :
      ( ( ! [X3: nat,Y2: nat] :
            ( V
           != ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) ) ) )
      = ( V = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq2
thf(fact_2872_semiring__norm_I7_J,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
      = ( bit1 @ ( plus_plus_num @ M @ N3 ) ) ) ).

% semiring_norm(7)
thf(fact_2873_semiring__norm_I9_J,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N3 ) )
      = ( bit1 @ ( plus_plus_num @ M @ N3 ) ) ) ).

% semiring_norm(9)
thf(fact_2874_semiring__norm_I14_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
      = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N3 ) ) ) ) ).

% semiring_norm(14)
thf(fact_2875_semiring__norm_I15_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N3 ) )
      = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N3 ) ) ) ).

% semiring_norm(15)
thf(fact_2876_semiring__norm_I72_J,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
      = ( ord_less_eq_num @ M @ N3 ) ) ).

% semiring_norm(72)
thf(fact_2877_semiring__norm_I81_J,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N3 ) )
      = ( ord_less_num @ M @ N3 ) ) ).

% semiring_norm(81)
thf(fact_2878_semiring__norm_I70_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).

% semiring_norm(70)
thf(fact_2879_semiring__norm_I77_J,axiom,
    ! [N3: num] : ( ord_less_num @ one @ ( bit1 @ N3 ) ) ).

% semiring_norm(77)
thf(fact_2880_divide__le__0__1__iff,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A2 ) @ zero_zero_real )
      = ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).

% divide_le_0_1_iff
thf(fact_2881_divide__le__0__1__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).

% divide_le_0_1_iff
thf(fact_2882_zero__le__divide__1__iff,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).

% zero_le_divide_1_iff
thf(fact_2883_zero__le__divide__1__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).

% zero_le_divide_1_iff
thf(fact_2884_divide__less__0__1__iff,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A2 ) @ zero_zero_real )
      = ( ord_less_real @ A2 @ zero_zero_real ) ) ).

% divide_less_0_1_iff
thf(fact_2885_divide__less__0__1__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) @ zero_zero_rat )
      = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).

% divide_less_0_1_iff
thf(fact_2886_divide__less__eq__1__neg,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B2 @ A2 ) @ one_one_real )
        = ( ord_less_real @ A2 @ B2 ) ) ) ).

% divide_less_eq_1_neg
thf(fact_2887_divide__less__eq__1__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A2 ) @ one_one_rat )
        = ( ord_less_rat @ A2 @ B2 ) ) ) ).

% divide_less_eq_1_neg
thf(fact_2888_divide__less__eq__1__pos,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ ( divide_divide_real @ B2 @ A2 ) @ one_one_real )
        = ( ord_less_real @ B2 @ A2 ) ) ) ).

% divide_less_eq_1_pos
thf(fact_2889_divide__less__eq__1__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A2 ) @ one_one_rat )
        = ( ord_less_rat @ B2 @ A2 ) ) ) ).

% divide_less_eq_1_pos
thf(fact_2890_less__divide__eq__1__neg,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ zero_zero_real )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A2 ) )
        = ( ord_less_real @ B2 @ A2 ) ) ) ).

% less_divide_eq_1_neg
thf(fact_2891_less__divide__eq__1__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A2 ) )
        = ( ord_less_rat @ B2 @ A2 ) ) ) ).

% less_divide_eq_1_neg
thf(fact_2892_less__divide__eq__1__pos,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A2 ) )
        = ( ord_less_real @ A2 @ B2 ) ) ) ).

% less_divide_eq_1_pos
thf(fact_2893_less__divide__eq__1__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A2 ) )
        = ( ord_less_rat @ A2 @ B2 ) ) ) ).

% less_divide_eq_1_pos
thf(fact_2894_zero__less__divide__1__iff,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A2 ) )
      = ( ord_less_real @ zero_zero_real @ A2 ) ) ).

% zero_less_divide_1_iff
thf(fact_2895_zero__less__divide__1__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) )
      = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).

% zero_less_divide_1_iff
thf(fact_2896_nonzero__divide__mult__cancel__left,axiom,
    ! [A2: complex,B2: complex] :
      ( ( A2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A2 @ ( times_times_complex @ A2 @ B2 ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ B2 ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_2897_nonzero__divide__mult__cancel__left,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 != zero_zero_real )
     => ( ( divide_divide_real @ A2 @ ( times_times_real @ A2 @ B2 ) )
        = ( divide_divide_real @ one_one_real @ B2 ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_2898_nonzero__divide__mult__cancel__left,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 != zero_zero_rat )
     => ( ( divide_divide_rat @ A2 @ ( times_times_rat @ A2 @ B2 ) )
        = ( divide_divide_rat @ one_one_rat @ B2 ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_2899_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: complex,A2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ B2 @ ( times_times_complex @ A2 @ B2 ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ A2 ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_2900_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( ( divide_divide_real @ B2 @ ( times_times_real @ A2 @ B2 ) )
        = ( divide_divide_real @ one_one_real @ A2 ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_2901_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( divide_divide_rat @ B2 @ ( times_times_rat @ A2 @ B2 ) )
        = ( divide_divide_rat @ one_one_rat @ A2 ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_2902_power__strict__increasing__iff,axiom,
    ! [B2: code_integer,X: nat,Y: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B2 )
     => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B2 @ X ) @ ( power_8256067586552552935nteger @ B2 @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_2903_power__strict__increasing__iff,axiom,
    ! [B2: real,X: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ ( power_power_real @ B2 @ X ) @ ( power_power_real @ B2 @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_2904_power__strict__increasing__iff,axiom,
    ! [B2: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B2 )
     => ( ( ord_less_rat @ ( power_power_rat @ B2 @ X ) @ ( power_power_rat @ B2 @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_2905_power__strict__increasing__iff,axiom,
    ! [B2: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B2 )
     => ( ( ord_less_nat @ ( power_power_nat @ B2 @ X ) @ ( power_power_nat @ B2 @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_2906_power__strict__increasing__iff,axiom,
    ! [B2: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B2 )
     => ( ( ord_less_int @ ( power_power_int @ B2 @ X ) @ ( power_power_int @ B2 @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_2907_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_2908_semiring__norm_I10_J,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
      = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N3 ) @ one ) ) ) ).

% semiring_norm(10)
thf(fact_2909_semiring__norm_I8_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ one )
      = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).

% semiring_norm(8)
thf(fact_2910_semiring__norm_I5_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ one )
      = ( bit1 @ M ) ) ).

% semiring_norm(5)
thf(fact_2911_semiring__norm_I4_J,axiom,
    ! [N3: num] :
      ( ( plus_plus_num @ one @ ( bit1 @ N3 ) )
      = ( bit0 @ ( plus_plus_num @ N3 @ one ) ) ) ).

% semiring_norm(4)
thf(fact_2912_semiring__norm_I3_J,axiom,
    ! [N3: num] :
      ( ( plus_plus_num @ one @ ( bit0 @ N3 ) )
      = ( bit1 @ N3 ) ) ).

% semiring_norm(3)
thf(fact_2913_semiring__norm_I16_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
      = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N3 ) @ ( bit0 @ ( times_times_num @ M @ N3 ) ) ) ) ) ).

% semiring_norm(16)
thf(fact_2914_semiring__norm_I79_J,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
      = ( ord_less_eq_num @ M @ N3 ) ) ).

% semiring_norm(79)
thf(fact_2915_semiring__norm_I74_J,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N3 ) )
      = ( ord_less_num @ M @ N3 ) ) ).

% semiring_norm(74)
thf(fact_2916_divide__le__eq__1__neg,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A2 ) @ one_one_real )
        = ( ord_less_eq_real @ A2 @ B2 ) ) ) ).

% divide_le_eq_1_neg
thf(fact_2917_divide__le__eq__1__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A2 ) @ one_one_rat )
        = ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).

% divide_le_eq_1_neg
thf(fact_2918_divide__le__eq__1__pos,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A2 ) @ one_one_real )
        = ( ord_less_eq_real @ B2 @ A2 ) ) ) ).

% divide_le_eq_1_pos
thf(fact_2919_divide__le__eq__1__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A2 ) @ one_one_rat )
        = ( ord_less_eq_rat @ B2 @ A2 ) ) ) ).

% divide_le_eq_1_pos
thf(fact_2920_le__divide__eq__1__neg,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A2 ) )
        = ( ord_less_eq_real @ B2 @ A2 ) ) ) ).

% le_divide_eq_1_neg
thf(fact_2921_le__divide__eq__1__neg,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A2 ) )
        = ( ord_less_eq_rat @ B2 @ A2 ) ) ) ).

% le_divide_eq_1_neg
thf(fact_2922_le__divide__eq__1__pos,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A2 ) )
        = ( ord_less_eq_real @ A2 @ B2 ) ) ) ).

% le_divide_eq_1_pos
thf(fact_2923_le__divide__eq__1__pos,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A2 ) )
        = ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).

% le_divide_eq_1_pos
thf(fact_2924_one__add__one,axiom,
    ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2925_one__add__one,axiom,
    ( ( plus_plus_real @ one_one_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2926_one__add__one,axiom,
    ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2927_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2928_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2929_power__strict__decreasing__iff,axiom,
    ! [B2: code_integer,M: nat,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
     => ( ( ord_le6747313008572928689nteger @ B2 @ one_one_Code_integer )
       => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B2 @ M ) @ ( power_8256067586552552935nteger @ B2 @ N3 ) )
          = ( ord_less_nat @ N3 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2930_power__strict__decreasing__iff,axiom,
    ! [B2: real,M: nat,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ B2 )
     => ( ( ord_less_real @ B2 @ one_one_real )
       => ( ( ord_less_real @ ( power_power_real @ B2 @ M ) @ ( power_power_real @ B2 @ N3 ) )
          = ( ord_less_nat @ N3 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2931_power__strict__decreasing__iff,axiom,
    ! [B2: rat,M: nat,N3: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B2 )
     => ( ( ord_less_rat @ B2 @ one_one_rat )
       => ( ( ord_less_rat @ ( power_power_rat @ B2 @ M ) @ ( power_power_rat @ B2 @ N3 ) )
          = ( ord_less_nat @ N3 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2932_power__strict__decreasing__iff,axiom,
    ! [B2: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_nat @ B2 @ one_one_nat )
       => ( ( ord_less_nat @ ( power_power_nat @ B2 @ M ) @ ( power_power_nat @ B2 @ N3 ) )
          = ( ord_less_nat @ N3 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2933_power__strict__decreasing__iff,axiom,
    ! [B2: int,M: nat,N3: nat] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ B2 @ one_one_int )
       => ( ( ord_less_int @ ( power_power_int @ B2 @ M ) @ ( power_power_int @ B2 @ N3 ) )
          = ( ord_less_nat @ N3 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2934_power__increasing__iff,axiom,
    ! [B2: code_integer,X: nat,Y: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B2 )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B2 @ X ) @ ( power_8256067586552552935nteger @ B2 @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2935_power__increasing__iff,axiom,
    ! [B2: real,X: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_eq_real @ ( power_power_real @ B2 @ X ) @ ( power_power_real @ B2 @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2936_power__increasing__iff,axiom,
    ! [B2: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B2 )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ B2 @ X ) @ ( power_power_rat @ B2 @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2937_power__increasing__iff,axiom,
    ! [B2: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B2 )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ X ) @ ( power_power_nat @ B2 @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2938_power__increasing__iff,axiom,
    ! [B2: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B2 )
     => ( ( ord_less_eq_int @ ( power_power_int @ B2 @ X ) @ ( power_power_int @ B2 @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2939_numeral__plus__one,axiom,
    ! [N3: num] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N3 ) @ one_one_complex )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ N3 @ one ) ) ) ).

% numeral_plus_one
thf(fact_2940_numeral__plus__one,axiom,
    ! [N3: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ N3 ) @ one_one_real )
      = ( numeral_numeral_real @ ( plus_plus_num @ N3 @ one ) ) ) ).

% numeral_plus_one
thf(fact_2941_numeral__plus__one,axiom,
    ! [N3: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ N3 ) @ one_one_rat )
      = ( numeral_numeral_rat @ ( plus_plus_num @ N3 @ one ) ) ) ).

% numeral_plus_one
thf(fact_2942_numeral__plus__one,axiom,
    ! [N3: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ N3 ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N3 @ one ) ) ) ).

% numeral_plus_one
thf(fact_2943_numeral__plus__one,axiom,
    ! [N3: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ N3 ) @ one_one_int )
      = ( numeral_numeral_int @ ( plus_plus_num @ N3 @ one ) ) ) ).

% numeral_plus_one
thf(fact_2944_one__plus__numeral,axiom,
    ! [N3: num] :
      ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N3 ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N3 ) ) ) ).

% one_plus_numeral
thf(fact_2945_one__plus__numeral,axiom,
    ! [N3: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N3 ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N3 ) ) ) ).

% one_plus_numeral
thf(fact_2946_one__plus__numeral,axiom,
    ! [N3: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N3 ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N3 ) ) ) ).

% one_plus_numeral
thf(fact_2947_one__plus__numeral,axiom,
    ! [N3: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N3 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N3 ) ) ) ).

% one_plus_numeral
thf(fact_2948_one__plus__numeral,axiom,
    ! [N3: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N3 ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N3 ) ) ) ).

% one_plus_numeral
thf(fact_2949_numeral__le__one__iff,axiom,
    ! [N3: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N3 ) @ one_one_real )
      = ( ord_less_eq_num @ N3 @ one ) ) ).

% numeral_le_one_iff
thf(fact_2950_numeral__le__one__iff,axiom,
    ! [N3: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N3 ) @ one_one_rat )
      = ( ord_less_eq_num @ N3 @ one ) ) ).

% numeral_le_one_iff
thf(fact_2951_numeral__le__one__iff,axiom,
    ! [N3: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N3 ) @ one_one_nat )
      = ( ord_less_eq_num @ N3 @ one ) ) ).

% numeral_le_one_iff
thf(fact_2952_numeral__le__one__iff,axiom,
    ! [N3: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N3 ) @ one_one_int )
      = ( ord_less_eq_num @ N3 @ one ) ) ).

% numeral_le_one_iff
thf(fact_2953_one__less__numeral__iff,axiom,
    ! [N3: num] :
      ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N3 ) )
      = ( ord_less_num @ one @ N3 ) ) ).

% one_less_numeral_iff
thf(fact_2954_one__less__numeral__iff,axiom,
    ! [N3: num] :
      ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N3 ) )
      = ( ord_less_num @ one @ N3 ) ) ).

% one_less_numeral_iff
thf(fact_2955_one__less__numeral__iff,axiom,
    ! [N3: num] :
      ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N3 ) )
      = ( ord_less_num @ one @ N3 ) ) ).

% one_less_numeral_iff
thf(fact_2956_one__less__numeral__iff,axiom,
    ! [N3: num] :
      ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N3 ) )
      = ( ord_less_num @ one @ N3 ) ) ).

% one_less_numeral_iff
thf(fact_2957_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_2958_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_2959_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_2960_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_2961_power__decreasing__iff,axiom,
    ! [B2: code_integer,M: nat,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
     => ( ( ord_le6747313008572928689nteger @ B2 @ one_one_Code_integer )
       => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B2 @ M ) @ ( power_8256067586552552935nteger @ B2 @ N3 ) )
          = ( ord_less_eq_nat @ N3 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2962_power__decreasing__iff,axiom,
    ! [B2: real,M: nat,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ B2 )
     => ( ( ord_less_real @ B2 @ one_one_real )
       => ( ( ord_less_eq_real @ ( power_power_real @ B2 @ M ) @ ( power_power_real @ B2 @ N3 ) )
          = ( ord_less_eq_nat @ N3 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2963_power__decreasing__iff,axiom,
    ! [B2: rat,M: nat,N3: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B2 )
     => ( ( ord_less_rat @ B2 @ one_one_rat )
       => ( ( ord_less_eq_rat @ ( power_power_rat @ B2 @ M ) @ ( power_power_rat @ B2 @ N3 ) )
          = ( ord_less_eq_nat @ N3 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2964_power__decreasing__iff,axiom,
    ! [B2: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_nat @ B2 @ one_one_nat )
       => ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ M ) @ ( power_power_nat @ B2 @ N3 ) )
          = ( ord_less_eq_nat @ N3 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2965_power__decreasing__iff,axiom,
    ! [B2: int,M: nat,N3: nat] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ B2 @ one_one_int )
       => ( ( ord_less_eq_int @ ( power_power_int @ B2 @ M ) @ ( power_power_int @ B2 @ N3 ) )
          = ( ord_less_eq_nat @ N3 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2966_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N3: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N3 ) ) ) )
      = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N3 ) ) ) ).

% div_Suc_eq_div_add3
thf(fact_2967_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_2968_one__reorient,axiom,
    ! [X: assn] :
      ( ( one_one_assn = X )
      = ( X = one_one_assn ) ) ).

% one_reorient
thf(fact_2969_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
      = ( X = one_one_real ) ) ).

% one_reorient
thf(fact_2970_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
      = ( X = one_one_rat ) ) ).

% one_reorient
thf(fact_2971_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_2972_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
      = ( X = one_one_int ) ) ).

% one_reorient
thf(fact_2973_numeral__Bit1,axiom,
    ! [N3: num] :
      ( ( numera6690914467698888265omplex @ ( bit1 @ N3 ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N3 ) @ ( numera6690914467698888265omplex @ N3 ) ) @ one_one_complex ) ) ).

% numeral_Bit1
thf(fact_2974_numeral__Bit1,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N3 ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N3 ) @ ( numeral_numeral_real @ N3 ) ) @ one_one_real ) ) ).

% numeral_Bit1
thf(fact_2975_numeral__Bit1,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N3 ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N3 ) @ ( numeral_numeral_rat @ N3 ) ) @ one_one_rat ) ) ).

% numeral_Bit1
thf(fact_2976_numeral__Bit1,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N3 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N3 ) @ ( numeral_numeral_nat @ N3 ) ) @ one_one_nat ) ) ).

% numeral_Bit1
thf(fact_2977_numeral__Bit1,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N3 ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N3 ) @ ( numeral_numeral_int @ N3 ) ) @ one_one_int ) ) ).

% numeral_Bit1
thf(fact_2978_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
thf(fact_2979_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_2980_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_2981_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_2982_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_2983_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_2984_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_2985_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_2986_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_2987_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ one_one_real ) ).

% less_numeral_extra(4)
thf(fact_2988_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_2989_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_2990_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_2991_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: real] :
      ( ( times_times_real @ one_one_real @ A2 )
      = A2 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_2992_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ one_one_rat @ A2 )
      = A2 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_2993_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ one_one_nat @ A2 )
      = A2 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_2994_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: int] :
      ( ( times_times_int @ one_one_int @ A2 )
      = A2 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_2995_comm__monoid__mult__class_Omult__1,axiom,
    ! [A2: assn] :
      ( ( times_times_assn @ one_one_assn @ A2 )
      = A2 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_2996_mult_Ocomm__neutral,axiom,
    ! [A2: real] :
      ( ( times_times_real @ A2 @ one_one_real )
      = A2 ) ).

% mult.comm_neutral
thf(fact_2997_mult_Ocomm__neutral,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ A2 @ one_one_rat )
      = A2 ) ).

% mult.comm_neutral
thf(fact_2998_mult_Ocomm__neutral,axiom,
    ! [A2: nat] :
      ( ( times_times_nat @ A2 @ one_one_nat )
      = A2 ) ).

% mult.comm_neutral
thf(fact_2999_mult_Ocomm__neutral,axiom,
    ! [A2: int] :
      ( ( times_times_int @ A2 @ one_one_int )
      = A2 ) ).

% mult.comm_neutral
thf(fact_3000_mult_Ocomm__neutral,axiom,
    ! [A2: assn] :
      ( ( times_times_assn @ A2 @ one_one_assn )
      = A2 ) ).

% mult.comm_neutral
thf(fact_3001_power__eq__if,axiom,
    ( power_power_complex
    = ( ^ [P4: complex,M5: nat] : ( if_complex @ ( M5 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P4 @ ( power_power_complex @ P4 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3002_power__eq__if,axiom,
    ( power_8256067586552552935nteger
    = ( ^ [P4: code_integer,M5: nat] : ( if_Code_integer @ ( M5 = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ P4 @ ( power_8256067586552552935nteger @ P4 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3003_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P4: real,M5: nat] : ( if_real @ ( M5 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P4 @ ( power_power_real @ P4 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3004_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P4: rat,M5: nat] : ( if_rat @ ( M5 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P4 @ ( power_power_rat @ P4 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3005_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P4: nat,M5: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P4 @ ( power_power_nat @ P4 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3006_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P4: int,M5: nat] : ( if_int @ ( M5 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P4 @ ( power_power_int @ P4 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3007_power__eq__if,axiom,
    ( power_power_assn
    = ( ^ [P4: assn,M5: nat] : ( if_assn @ ( M5 = zero_zero_nat ) @ one_one_assn @ ( times_times_assn @ P4 @ ( power_power_assn @ P4 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3008_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X23: num] :
            ( Y
           != ( bit0 @ X23 ) )
       => ~ ! [X32: num] :
              ( Y
             != ( bit1 @ X32 ) ) ) ) ).

% num.exhaust
thf(fact_3009_xor__num_Ocases,axiom,
    ! [X: product_prod_num_num] :
      ( ( X
       != ( product_Pair_num_num @ one @ one ) )
     => ( ! [N: num] :
            ( X
           != ( product_Pair_num_num @ one @ ( bit0 @ N ) ) )
       => ( ! [N: num] :
              ( X
             != ( product_Pair_num_num @ one @ ( bit1 @ N ) ) )
         => ( ! [M4: num] :
                ( X
               != ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) )
           => ( ! [M4: num,N: num] :
                  ( X
                 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N ) ) )
             => ( ! [M4: num,N: num] :
                    ( X
                   != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N ) ) )
               => ( ! [M4: num] :
                      ( X
                     != ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) )
                 => ( ! [M4: num,N: num] :
                        ( X
                       != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N ) ) )
                   => ~ ! [M4: num,N: num] :
                          ( X
                         != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.cases
thf(fact_3010_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A7: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A7 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_3011_discrete,axiom,
    ( ord_less_int
    = ( ^ [A7: int] : ( ord_less_eq_int @ ( plus_plus_int @ A7 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_3012_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_3013_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_3014_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_3015_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_3016_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_3017_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_3018_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_3019_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_3020_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_3021_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_3022_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_3023_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_3024_less__numeral__extra_I1_J,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% less_numeral_extra(1)
thf(fact_3025_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_3026_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_3027_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_3028_zero__less__one,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% zero_less_one
thf(fact_3029_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_3030_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_3031_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_3032_not__one__less__zero,axiom,
    ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).

% not_one_less_zero
thf(fact_3033_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_3034_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_3035_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_3036_one__le__numeral,axiom,
    ! [N3: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N3 ) ) ).

% one_le_numeral
thf(fact_3037_one__le__numeral,axiom,
    ! [N3: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N3 ) ) ).

% one_le_numeral
thf(fact_3038_one__le__numeral,axiom,
    ! [N3: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N3 ) ) ).

% one_le_numeral
thf(fact_3039_one__le__numeral,axiom,
    ! [N3: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N3 ) ) ).

% one_le_numeral
thf(fact_3040_not__numeral__less__one,axiom,
    ! [N3: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N3 ) @ one_one_real ) ).

% not_numeral_less_one
thf(fact_3041_not__numeral__less__one,axiom,
    ! [N3: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N3 ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_3042_not__numeral__less__one,axiom,
    ! [N3: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N3 ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_3043_not__numeral__less__one,axiom,
    ! [N3: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N3 ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_3044_numeral__One,axiom,
    ( ( numera6690914467698888265omplex @ one )
    = one_one_complex ) ).

% numeral_One
thf(fact_3045_numeral__One,axiom,
    ( ( numeral_numeral_real @ one )
    = one_one_real ) ).

% numeral_One
thf(fact_3046_numeral__One,axiom,
    ( ( numeral_numeral_rat @ one )
    = one_one_rat ) ).

% numeral_One
thf(fact_3047_numeral__One,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numeral_One
thf(fact_3048_numeral__One,axiom,
    ( ( numeral_numeral_int @ one )
    = one_one_int ) ).

% numeral_One
thf(fact_3049_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_3050_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_3051_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_3052_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_3053_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_3054_less__add__one,axiom,
    ! [A2: real] : ( ord_less_real @ A2 @ ( plus_plus_real @ A2 @ one_one_real ) ) ).

% less_add_one
thf(fact_3055_less__add__one,axiom,
    ! [A2: rat] : ( ord_less_rat @ A2 @ ( plus_plus_rat @ A2 @ one_one_rat ) ) ).

% less_add_one
thf(fact_3056_less__add__one,axiom,
    ! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).

% less_add_one
thf(fact_3057_less__add__one,axiom,
    ! [A2: int] : ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ one_one_int ) ) ).

% less_add_one
thf(fact_3058_add__mono1,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ord_less_real @ ( plus_plus_real @ A2 @ one_one_real ) @ ( plus_plus_real @ B2 @ one_one_real ) ) ) ).

% add_mono1
thf(fact_3059_add__mono1,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ord_less_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( plus_plus_rat @ B2 @ one_one_rat ) ) ) ).

% add_mono1
thf(fact_3060_add__mono1,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B2 @ one_one_nat ) ) ) ).

% add_mono1
thf(fact_3061_add__mono1,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ord_less_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( plus_plus_int @ B2 @ one_one_int ) ) ) ).

% add_mono1
thf(fact_3062_less__1__mult,axiom,
    ! [M: real,N3: real] :
      ( ( ord_less_real @ one_one_real @ M )
     => ( ( ord_less_real @ one_one_real @ N3 )
       => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N3 ) ) ) ) ).

% less_1_mult
thf(fact_3063_less__1__mult,axiom,
    ! [M: rat,N3: rat] :
      ( ( ord_less_rat @ one_one_rat @ M )
     => ( ( ord_less_rat @ one_one_rat @ N3 )
       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N3 ) ) ) ) ).

% less_1_mult
thf(fact_3064_less__1__mult,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ one_one_nat @ M )
     => ( ( ord_less_nat @ one_one_nat @ N3 )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N3 ) ) ) ) ).

% less_1_mult
thf(fact_3065_less__1__mult,axiom,
    ! [M: int,N3: int] :
      ( ( ord_less_int @ one_one_int @ M )
     => ( ( ord_less_int @ one_one_int @ N3 )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N3 ) ) ) ) ).

% less_1_mult
thf(fact_3066_right__inverse__eq,axiom,
    ! [B2: complex,A2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( ( divide1717551699836669952omplex @ A2 @ B2 )
          = one_one_complex )
        = ( A2 = B2 ) ) ) ).

% right_inverse_eq
thf(fact_3067_right__inverse__eq,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( ( ( divide_divide_real @ A2 @ B2 )
          = one_one_real )
        = ( A2 = B2 ) ) ) ).

% right_inverse_eq
thf(fact_3068_right__inverse__eq,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( ( divide_divide_rat @ A2 @ B2 )
          = one_one_rat )
        = ( A2 = B2 ) ) ) ).

% right_inverse_eq
thf(fact_3069_one__le__power,axiom,
    ! [A2: real,N3: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A2 )
     => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A2 @ N3 ) ) ) ).

% one_le_power
thf(fact_3070_one__le__power,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 )
     => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ).

% one_le_power
thf(fact_3071_one__le__power,axiom,
    ! [A2: rat,N3: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A2 )
     => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A2 @ N3 ) ) ) ).

% one_le_power
thf(fact_3072_one__le__power,axiom,
    ! [A2: nat,N3: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A2 )
     => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A2 @ N3 ) ) ) ).

% one_le_power
thf(fact_3073_one__le__power,axiom,
    ! [A2: int,N3: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A2 )
     => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A2 @ N3 ) ) ) ).

% one_le_power
thf(fact_3074_left__right__inverse__power,axiom,
    ! [X: complex,Y: complex,N3: nat] :
      ( ( ( times_times_complex @ X @ Y )
        = one_one_complex )
     => ( ( times_times_complex @ ( power_power_complex @ X @ N3 ) @ ( power_power_complex @ Y @ N3 ) )
        = one_one_complex ) ) ).

% left_right_inverse_power
thf(fact_3075_left__right__inverse__power,axiom,
    ! [X: code_integer,Y: code_integer,N3: nat] :
      ( ( ( times_3573771949741848930nteger @ X @ Y )
        = one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N3 ) @ ( power_8256067586552552935nteger @ Y @ N3 ) )
        = one_one_Code_integer ) ) ).

% left_right_inverse_power
thf(fact_3076_left__right__inverse__power,axiom,
    ! [X: real,Y: real,N3: nat] :
      ( ( ( times_times_real @ X @ Y )
        = one_one_real )
     => ( ( times_times_real @ ( power_power_real @ X @ N3 ) @ ( power_power_real @ Y @ N3 ) )
        = one_one_real ) ) ).

% left_right_inverse_power
thf(fact_3077_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N3: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = one_one_rat )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N3 ) @ ( power_power_rat @ Y @ N3 ) )
        = one_one_rat ) ) ).

% left_right_inverse_power
thf(fact_3078_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N3: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N3 ) @ ( power_power_nat @ Y @ N3 ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_3079_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N3: nat] :
      ( ( ( times_times_int @ X @ Y )
        = one_one_int )
     => ( ( times_times_int @ ( power_power_int @ X @ N3 ) @ ( power_power_int @ Y @ N3 ) )
        = one_one_int ) ) ).

% left_right_inverse_power
thf(fact_3080_left__right__inverse__power,axiom,
    ! [X: assn,Y: assn,N3: nat] :
      ( ( ( times_times_assn @ X @ Y )
        = one_one_assn )
     => ( ( times_times_assn @ ( power_power_assn @ X @ N3 ) @ ( power_power_assn @ Y @ N3 ) )
        = one_one_assn ) ) ).

% left_right_inverse_power
thf(fact_3081_power__one__over,axiom,
    ! [A2: complex,N3: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A2 ) @ N3 )
      = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A2 @ N3 ) ) ) ).

% power_one_over
thf(fact_3082_power__one__over,axiom,
    ! [A2: real,N3: nat] :
      ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A2 ) @ N3 )
      = ( divide_divide_real @ one_one_real @ ( power_power_real @ A2 @ N3 ) ) ) ).

% power_one_over
thf(fact_3083_power__one__over,axiom,
    ! [A2: rat,N3: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) @ N3 )
      = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A2 @ N3 ) ) ) ).

% power_one_over
thf(fact_3084_power__0,axiom,
    ! [A2: assn] :
      ( ( power_power_assn @ A2 @ zero_zero_nat )
      = one_one_assn ) ).

% power_0
thf(fact_3085_power__0,axiom,
    ! [A2: rat] :
      ( ( power_power_rat @ A2 @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_3086_power__0,axiom,
    ! [A2: nat] :
      ( ( power_power_nat @ A2 @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_3087_power__0,axiom,
    ! [A2: real] :
      ( ( power_power_real @ A2 @ zero_zero_nat )
      = one_one_real ) ).

% power_0
thf(fact_3088_power__0,axiom,
    ! [A2: int] :
      ( ( power_power_int @ A2 @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_3089_power__0,axiom,
    ! [A2: complex] :
      ( ( power_power_complex @ A2 @ zero_zero_nat )
      = one_one_complex ) ).

% power_0
thf(fact_3090_power__0,axiom,
    ! [A2: code_integer] :
      ( ( power_8256067586552552935nteger @ A2 @ zero_zero_nat )
      = one_one_Code_integer ) ).

% power_0
thf(fact_3091_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ? [N: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N ) ) ) ).

% real_arch_pow
thf(fact_3092_eval__nat__numeral_I3_J,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N3 ) )
      = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N3 ) ) ) ) ).

% eval_nat_numeral(3)
thf(fact_3093_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_3094_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_3095_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_3096_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_3097_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_3098_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_3099_mult__le__one,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ A2 @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ( ord_less_eq_real @ B2 @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ A2 @ B2 ) @ one_one_real ) ) ) ) ).

% mult_le_one
thf(fact_3100_mult__le__one,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ( ord_less_eq_rat @ B2 @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B2 ) @ one_one_rat ) ) ) ) ).

% mult_le_one
thf(fact_3101_mult__le__one,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
       => ( ( ord_less_eq_nat @ B2 @ one_one_nat )
         => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ) ).

% mult_le_one
thf(fact_3102_mult__le__one,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ( ord_less_eq_int @ B2 @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ one_one_int ) ) ) ) ).

% mult_le_one
thf(fact_3103_mult__left__le,axiom,
    ! [C2: real,A2: real] :
      ( ( ord_less_eq_real @ C2 @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
       => ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ A2 ) ) ) ).

% mult_left_le
thf(fact_3104_mult__left__le,axiom,
    ! [C2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ C2 @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ A2 ) ) ) ).

% mult_left_le
thf(fact_3105_mult__left__le,axiom,
    ! [C2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ C2 @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C2 ) @ A2 ) ) ) ).

% mult_left_le
thf(fact_3106_mult__left__le,axiom,
    ! [C2: int,A2: int] :
      ( ( ord_less_eq_int @ C2 @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
       => ( ord_less_eq_int @ ( times_times_int @ A2 @ C2 ) @ A2 ) ) ) ).

% mult_left_le
thf(fact_3107_zero__less__two,axiom,
    ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).

% zero_less_two
thf(fact_3108_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_3109_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_3110_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_3111_divide__less__eq__1,axiom,
    ! [B2: real,A2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ A2 ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A2 )
          & ( ord_less_real @ B2 @ A2 ) )
        | ( ( ord_less_real @ A2 @ zero_zero_real )
          & ( ord_less_real @ A2 @ B2 ) )
        | ( A2 = zero_zero_real ) ) ) ).

% divide_less_eq_1
thf(fact_3112_divide__less__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ A2 ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
          & ( ord_less_rat @ B2 @ A2 ) )
        | ( ( ord_less_rat @ A2 @ zero_zero_rat )
          & ( ord_less_rat @ A2 @ B2 ) )
        | ( A2 = zero_zero_rat ) ) ) ).

% divide_less_eq_1
thf(fact_3113_less__divide__eq__1,axiom,
    ! [B2: real,A2: real] :
      ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B2 @ A2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A2 )
          & ( ord_less_real @ A2 @ B2 ) )
        | ( ( ord_less_real @ A2 @ zero_zero_real )
          & ( ord_less_real @ B2 @ A2 ) ) ) ) ).

% less_divide_eq_1
thf(fact_3114_less__divide__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
          & ( ord_less_rat @ A2 @ B2 ) )
        | ( ( ord_less_rat @ A2 @ zero_zero_rat )
          & ( ord_less_rat @ B2 @ A2 ) ) ) ) ).

% less_divide_eq_1
thf(fact_3115_power__le__one,axiom,
    ! [A2: real,N3: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ A2 @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A2 @ N3 ) @ one_one_real ) ) ) ).

% power_le_one
thf(fact_3116_power__le__one,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ one_one_Code_integer ) ) ) ).

% power_le_one
thf(fact_3117_power__le__one,axiom,
    ! [A2: rat,N3: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ A2 @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N3 ) @ one_one_rat ) ) ) ).

% power_le_one
thf(fact_3118_power__le__one,axiom,
    ! [A2: nat,N3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ A2 @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N3 ) @ one_one_nat ) ) ) ).

% power_le_one
thf(fact_3119_power__le__one,axiom,
    ! [A2: int,N3: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ A2 @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A2 @ N3 ) @ one_one_int ) ) ) ).

% power_le_one
thf(fact_3120_div__add__self1,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
        = ( plus_plus_nat @ ( divide_divide_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ).

% div_add_self1
thf(fact_3121_div__add__self1,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
        = ( plus_plus_int @ ( divide_divide_int @ A2 @ B2 ) @ one_one_int ) ) ) ).

% div_add_self1
thf(fact_3122_div__add__self2,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
        = ( plus_plus_nat @ ( divide_divide_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ).

% div_add_self2
thf(fact_3123_div__add__self2,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
        = ( plus_plus_int @ ( divide_divide_int @ A2 @ B2 ) @ one_one_int ) ) ) ).

% div_add_self2
thf(fact_3124_gt__half__sum,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A2 @ B2 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B2 ) ) ).

% gt_half_sum
thf(fact_3125_gt__half__sum,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A2 @ B2 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B2 ) ) ).

% gt_half_sum
thf(fact_3126_less__half__sum,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ord_less_real @ A2 @ ( divide_divide_real @ ( plus_plus_real @ A2 @ B2 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).

% less_half_sum
thf(fact_3127_less__half__sum,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ord_less_rat @ A2 @ ( divide_divide_rat @ ( plus_plus_rat @ A2 @ B2 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).

% less_half_sum
thf(fact_3128_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_3129_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_3130_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_3131_power__gt1__lemma,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
     => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ) ).

% power_gt1_lemma
thf(fact_3132_power__gt1__lemma,axiom,
    ! [A2: real,N3: nat] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ord_less_real @ one_one_real @ ( times_times_real @ A2 @ ( power_power_real @ A2 @ N3 ) ) ) ) ).

% power_gt1_lemma
thf(fact_3133_power__gt1__lemma,axiom,
    ! [A2: rat,N3: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N3 ) ) ) ) ).

% power_gt1_lemma
thf(fact_3134_power__gt1__lemma,axiom,
    ! [A2: nat,N3: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N3 ) ) ) ) ).

% power_gt1_lemma
thf(fact_3135_power__gt1__lemma,axiom,
    ! [A2: int,N3: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ord_less_int @ one_one_int @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N3 ) ) ) ) ).

% power_gt1_lemma
thf(fact_3136_power__less__power__Suc,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
     => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ) ).

% power_less_power_Suc
thf(fact_3137_power__less__power__Suc,axiom,
    ! [A2: real,N3: nat] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ord_less_real @ ( power_power_real @ A2 @ N3 ) @ ( times_times_real @ A2 @ ( power_power_real @ A2 @ N3 ) ) ) ) ).

% power_less_power_Suc
thf(fact_3138_power__less__power__Suc,axiom,
    ! [A2: rat,N3: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ord_less_rat @ ( power_power_rat @ A2 @ N3 ) @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N3 ) ) ) ) ).

% power_less_power_Suc
thf(fact_3139_power__less__power__Suc,axiom,
    ! [A2: nat,N3: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ord_less_nat @ ( power_power_nat @ A2 @ N3 ) @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N3 ) ) ) ) ).

% power_less_power_Suc
thf(fact_3140_power__less__power__Suc,axiom,
    ! [A2: int,N3: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ord_less_int @ ( power_power_int @ A2 @ N3 ) @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N3 ) ) ) ) ).

% power_less_power_Suc
thf(fact_3141_power__gt1,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
     => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A2 @ ( suc @ N3 ) ) ) ) ).

% power_gt1
thf(fact_3142_power__gt1,axiom,
    ! [A2: real,N3: nat] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ord_less_real @ one_one_real @ ( power_power_real @ A2 @ ( suc @ N3 ) ) ) ) ).

% power_gt1
thf(fact_3143_power__gt1,axiom,
    ! [A2: rat,N3: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A2 @ ( suc @ N3 ) ) ) ) ).

% power_gt1
thf(fact_3144_power__gt1,axiom,
    ! [A2: nat,N3: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ ( suc @ N3 ) ) ) ) ).

% power_gt1
thf(fact_3145_power__gt1,axiom,
    ! [A2: int,N3: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ord_less_int @ one_one_int @ ( power_power_int @ A2 @ ( suc @ N3 ) ) ) ) ).

% power_gt1
thf(fact_3146_power__0__left,axiom,
    ! [N3: nat] :
      ( ( ( N3 = zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N3 )
          = one_one_rat ) )
      & ( ( N3 != zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N3 )
          = zero_zero_rat ) ) ) ).

% power_0_left
thf(fact_3147_power__0__left,axiom,
    ! [N3: nat] :
      ( ( ( N3 = zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N3 )
          = one_one_nat ) )
      & ( ( N3 != zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N3 )
          = zero_zero_nat ) ) ) ).

% power_0_left
thf(fact_3148_power__0__left,axiom,
    ! [N3: nat] :
      ( ( ( N3 = zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N3 )
          = one_one_real ) )
      & ( ( N3 != zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N3 )
          = zero_zero_real ) ) ) ).

% power_0_left
thf(fact_3149_power__0__left,axiom,
    ! [N3: nat] :
      ( ( ( N3 = zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N3 )
          = one_one_int ) )
      & ( ( N3 != zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N3 )
          = zero_zero_int ) ) ) ).

% power_0_left
thf(fact_3150_power__0__left,axiom,
    ! [N3: nat] :
      ( ( ( N3 = zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N3 )
          = one_one_complex ) )
      & ( ( N3 != zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N3 )
          = zero_zero_complex ) ) ) ).

% power_0_left
thf(fact_3151_power__0__left,axiom,
    ! [N3: nat] :
      ( ( ( N3 = zero_zero_nat )
       => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N3 )
          = one_one_Code_integer ) )
      & ( ( N3 != zero_zero_nat )
       => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N3 )
          = zero_z3403309356797280102nteger ) ) ) ).

% power_0_left
thf(fact_3152_power__less__imp__less__exp,axiom,
    ! [A2: code_integer,M: nat,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
     => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ M ) @ ( power_8256067586552552935nteger @ A2 @ N3 ) )
       => ( ord_less_nat @ M @ N3 ) ) ) ).

% power_less_imp_less_exp
thf(fact_3153_power__less__imp__less__exp,axiom,
    ! [A2: real,M: nat,N3: nat] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_real @ ( power_power_real @ A2 @ M ) @ ( power_power_real @ A2 @ N3 ) )
       => ( ord_less_nat @ M @ N3 ) ) ) ).

% power_less_imp_less_exp
thf(fact_3154_power__less__imp__less__exp,axiom,
    ! [A2: rat,M: nat,N3: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ( ord_less_rat @ ( power_power_rat @ A2 @ M ) @ ( power_power_rat @ A2 @ N3 ) )
       => ( ord_less_nat @ M @ N3 ) ) ) ).

% power_less_imp_less_exp
thf(fact_3155_power__less__imp__less__exp,axiom,
    ! [A2: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ( ord_less_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N3 ) )
       => ( ord_less_nat @ M @ N3 ) ) ) ).

% power_less_imp_less_exp
thf(fact_3156_power__less__imp__less__exp,axiom,
    ! [A2: int,M: nat,N3: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ( ord_less_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N3 ) )
       => ( ord_less_nat @ M @ N3 ) ) ) ).

% power_less_imp_less_exp
thf(fact_3157_power__strict__increasing,axiom,
    ! [N3: nat,N7: nat,A2: code_integer] :
      ( ( ord_less_nat @ N3 @ N7 )
     => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
       => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ ( power_8256067586552552935nteger @ A2 @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_3158_power__strict__increasing,axiom,
    ! [N3: nat,N7: nat,A2: real] :
      ( ( ord_less_nat @ N3 @ N7 )
     => ( ( ord_less_real @ one_one_real @ A2 )
       => ( ord_less_real @ ( power_power_real @ A2 @ N3 ) @ ( power_power_real @ A2 @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_3159_power__strict__increasing,axiom,
    ! [N3: nat,N7: nat,A2: rat] :
      ( ( ord_less_nat @ N3 @ N7 )
     => ( ( ord_less_rat @ one_one_rat @ A2 )
       => ( ord_less_rat @ ( power_power_rat @ A2 @ N3 ) @ ( power_power_rat @ A2 @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_3160_power__strict__increasing,axiom,
    ! [N3: nat,N7: nat,A2: nat] :
      ( ( ord_less_nat @ N3 @ N7 )
     => ( ( ord_less_nat @ one_one_nat @ A2 )
       => ( ord_less_nat @ ( power_power_nat @ A2 @ N3 ) @ ( power_power_nat @ A2 @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_3161_power__strict__increasing,axiom,
    ! [N3: nat,N7: nat,A2: int] :
      ( ( ord_less_nat @ N3 @ N7 )
     => ( ( ord_less_int @ one_one_int @ A2 )
       => ( ord_less_int @ ( power_power_int @ A2 @ N3 ) @ ( power_power_int @ A2 @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_3162_power__increasing,axiom,
    ! [N3: nat,N7: nat,A2: real] :
      ( ( ord_less_eq_nat @ N3 @ N7 )
     => ( ( ord_less_eq_real @ one_one_real @ A2 )
       => ( ord_less_eq_real @ ( power_power_real @ A2 @ N3 ) @ ( power_power_real @ A2 @ N7 ) ) ) ) ).

% power_increasing
thf(fact_3163_power__increasing,axiom,
    ! [N3: nat,N7: nat,A2: code_integer] :
      ( ( ord_less_eq_nat @ N3 @ N7 )
     => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ ( power_8256067586552552935nteger @ A2 @ N7 ) ) ) ) ).

% power_increasing
thf(fact_3164_power__increasing,axiom,
    ! [N3: nat,N7: nat,A2: rat] :
      ( ( ord_less_eq_nat @ N3 @ N7 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A2 )
       => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N3 ) @ ( power_power_rat @ A2 @ N7 ) ) ) ) ).

% power_increasing
thf(fact_3165_power__increasing,axiom,
    ! [N3: nat,N7: nat,A2: nat] :
      ( ( ord_less_eq_nat @ N3 @ N7 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A2 )
       => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N3 ) @ ( power_power_nat @ A2 @ N7 ) ) ) ) ).

% power_increasing
thf(fact_3166_power__increasing,axiom,
    ! [N3: nat,N7: nat,A2: int] :
      ( ( ord_less_eq_nat @ N3 @ N7 )
     => ( ( ord_less_eq_int @ one_one_int @ A2 )
       => ( ord_less_eq_int @ ( power_power_int @ A2 @ N3 ) @ ( power_power_int @ A2 @ N7 ) ) ) ) ).

% power_increasing
thf(fact_3167_real__arch__pow__inv,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ X @ one_one_real )
       => ? [N: nat] : ( ord_less_real @ ( power_power_real @ X @ N ) @ Y ) ) ) ).

% real_arch_pow_inv
thf(fact_3168_numeral__Bit1__div__2,axiom,
    ! [N3: num] :
      ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( numeral_numeral_nat @ N3 ) ) ).

% numeral_Bit1_div_2
thf(fact_3169_numeral__Bit1__div__2,axiom,
    ! [N3: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( numeral_numeral_int @ N3 ) ) ).

% numeral_Bit1_div_2
thf(fact_3170_power3__eq__cube,axiom,
    ! [A2: complex] :
      ( ( power_power_complex @ A2 @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_complex @ ( times_times_complex @ A2 @ A2 ) @ A2 ) ) ).

% power3_eq_cube
thf(fact_3171_power3__eq__cube,axiom,
    ! [A2: code_integer] :
      ( ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A2 @ A2 ) @ A2 ) ) ).

% power3_eq_cube
thf(fact_3172_power3__eq__cube,axiom,
    ! [A2: real] :
      ( ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_real @ ( times_times_real @ A2 @ A2 ) @ A2 ) ) ).

% power3_eq_cube
thf(fact_3173_power3__eq__cube,axiom,
    ! [A2: rat] :
      ( ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_rat @ ( times_times_rat @ A2 @ A2 ) @ A2 ) ) ).

% power3_eq_cube
thf(fact_3174_power3__eq__cube,axiom,
    ! [A2: nat] :
      ( ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_nat @ ( times_times_nat @ A2 @ A2 ) @ A2 ) ) ).

% power3_eq_cube
thf(fact_3175_power3__eq__cube,axiom,
    ! [A2: int] :
      ( ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_int @ ( times_times_int @ A2 @ A2 ) @ A2 ) ) ).

% power3_eq_cube
thf(fact_3176_power3__eq__cube,axiom,
    ! [A2: assn] :
      ( ( power_power_assn @ A2 @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_assn @ ( times_times_assn @ A2 @ A2 ) @ A2 ) ) ).

% power3_eq_cube
thf(fact_3177_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_3178_Suc3__eq__add__3,axiom,
    ! [N3: nat] :
      ( ( suc @ ( suc @ ( suc @ N3 ) ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N3 ) ) ).

% Suc3_eq_add_3
thf(fact_3179_mult__le__cancel__left1,axiom,
    ! [C2: real,B2: real] :
      ( ( ord_less_eq_real @ C2 @ ( times_times_real @ C2 @ B2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ one_one_real @ B2 ) )
        & ( ( ord_less_real @ C2 @ zero_zero_real )
         => ( ord_less_eq_real @ B2 @ one_one_real ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_3180_mult__le__cancel__left1,axiom,
    ! [C2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ C2 @ ( times_times_rat @ C2 @ B2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ one_one_rat @ B2 ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ one_one_rat ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_3181_mult__le__cancel__left1,axiom,
    ! [C2: int,B2: int] :
      ( ( ord_less_eq_int @ C2 @ ( times_times_int @ C2 @ B2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ one_one_int @ B2 ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_3182_mult__le__cancel__left2,axiom,
    ! [C2: real,A2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A2 ) @ C2 )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ A2 @ one_one_real ) )
        & ( ( ord_less_real @ C2 @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A2 ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_3183_mult__le__cancel__left2,axiom,
    ! [C2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A2 ) @ C2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ A2 @ one_one_rat ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A2 ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_3184_mult__le__cancel__left2,axiom,
    ! [C2: int,A2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A2 ) @ C2 )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ A2 @ one_one_int ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_3185_mult__le__cancel__right1,axiom,
    ! [C2: real,B2: real] :
      ( ( ord_less_eq_real @ C2 @ ( times_times_real @ B2 @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ one_one_real @ B2 ) )
        & ( ( ord_less_real @ C2 @ zero_zero_real )
         => ( ord_less_eq_real @ B2 @ one_one_real ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_3186_mult__le__cancel__right1,axiom,
    ! [C2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ C2 @ ( times_times_rat @ B2 @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ one_one_rat @ B2 ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ B2 @ one_one_rat ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_3187_mult__le__cancel__right1,axiom,
    ! [C2: int,B2: int] :
      ( ( ord_less_eq_int @ C2 @ ( times_times_int @ B2 @ C2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ one_one_int @ B2 ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ B2 @ one_one_int ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_3188_mult__le__cancel__right2,axiom,
    ! [A2: real,C2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ C2 )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ A2 @ one_one_real ) )
        & ( ( ord_less_real @ C2 @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A2 ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_3189_mult__le__cancel__right2,axiom,
    ! [A2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ C2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ A2 @ one_one_rat ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A2 ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_3190_mult__le__cancel__right2,axiom,
    ! [A2: int,C2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C2 ) @ C2 )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ A2 @ one_one_int ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_3191_mult__less__cancel__left1,axiom,
    ! [C2: real,B2: real] :
      ( ( ord_less_real @ C2 @ ( times_times_real @ C2 @ B2 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ one_one_real @ B2 ) )
        & ( ( ord_less_eq_real @ C2 @ zero_zero_real )
         => ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_3192_mult__less__cancel__left1,axiom,
    ! [C2: rat,B2: rat] :
      ( ( ord_less_rat @ C2 @ ( times_times_rat @ C2 @ B2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ one_one_rat @ B2 ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ one_one_rat ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_3193_mult__less__cancel__left1,axiom,
    ! [C2: int,B2: int] :
      ( ( ord_less_int @ C2 @ ( times_times_int @ C2 @ B2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ one_one_int @ B2 ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_3194_mult__less__cancel__left2,axiom,
    ! [C2: real,A2: real] :
      ( ( ord_less_real @ ( times_times_real @ C2 @ A2 ) @ C2 )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ A2 @ one_one_real ) )
        & ( ( ord_less_eq_real @ C2 @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A2 ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_3195_mult__less__cancel__left2,axiom,
    ! [C2: rat,A2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C2 @ A2 ) @ C2 )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ A2 @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A2 ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_3196_mult__less__cancel__left2,axiom,
    ! [C2: int,A2: int] :
      ( ( ord_less_int @ ( times_times_int @ C2 @ A2 ) @ C2 )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ A2 @ one_one_int ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_3197_mult__less__cancel__right1,axiom,
    ! [C2: real,B2: real] :
      ( ( ord_less_real @ C2 @ ( times_times_real @ B2 @ C2 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ one_one_real @ B2 ) )
        & ( ( ord_less_eq_real @ C2 @ zero_zero_real )
         => ( ord_less_real @ B2 @ one_one_real ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_3198_mult__less__cancel__right1,axiom,
    ! [C2: rat,B2: rat] :
      ( ( ord_less_rat @ C2 @ ( times_times_rat @ B2 @ C2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ one_one_rat @ B2 ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ B2 @ one_one_rat ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_3199_mult__less__cancel__right1,axiom,
    ! [C2: int,B2: int] :
      ( ( ord_less_int @ C2 @ ( times_times_int @ B2 @ C2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ one_one_int @ B2 ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ B2 @ one_one_int ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_3200_mult__less__cancel__right2,axiom,
    ! [A2: real,C2: real] :
      ( ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ C2 )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ A2 @ one_one_real ) )
        & ( ( ord_less_eq_real @ C2 @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A2 ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_3201_mult__less__cancel__right2,axiom,
    ! [A2: rat,C2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ C2 )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ A2 @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A2 ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_3202_mult__less__cancel__right2,axiom,
    ! [A2: int,C2: int] :
      ( ( ord_less_int @ ( times_times_int @ A2 @ C2 ) @ C2 )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ A2 @ one_one_int ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_3203_field__le__mult__one__interval,axiom,
    ! [X: real,Y: real] :
      ( ! [Z2: real] :
          ( ( ord_less_real @ zero_zero_real @ Z2 )
         => ( ( ord_less_real @ Z2 @ one_one_real )
           => ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ Y ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_3204_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ Z2 )
         => ( ( ord_less_rat @ Z2 @ one_one_rat )
           => ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X ) @ Y ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_3205_convex__bound__le,axiom,
    ! [X: real,A2: real,Y: real,U: real,V: real] :
      ( ( ord_less_eq_real @ X @ A2 )
     => ( ( ord_less_eq_real @ Y @ A2 )
       => ( ( 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 ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3206_convex__bound__le,axiom,
    ! [X: rat,A2: rat,Y: rat,U: rat,V: rat] :
      ( ( ord_less_eq_rat @ X @ A2 )
     => ( ( ord_less_eq_rat @ Y @ A2 )
       => ( ( 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 ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3207_convex__bound__le,axiom,
    ! [X: int,A2: int,Y: int,U: int,V: int] :
      ( ( ord_less_eq_int @ X @ A2 )
     => ( ( ord_less_eq_int @ Y @ A2 )
       => ( ( 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 ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3208_divide__le__eq__1,axiom,
    ! [B2: real,A2: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A2 ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A2 )
          & ( ord_less_eq_real @ B2 @ A2 ) )
        | ( ( ord_less_real @ A2 @ zero_zero_real )
          & ( ord_less_eq_real @ A2 @ B2 ) )
        | ( A2 = zero_zero_real ) ) ) ).

% divide_le_eq_1
thf(fact_3209_divide__le__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ A2 ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ B2 @ A2 ) )
        | ( ( ord_less_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ A2 @ B2 ) )
        | ( A2 = zero_zero_rat ) ) ) ).

% divide_le_eq_1
thf(fact_3210_le__divide__eq__1,axiom,
    ! [B2: real,A2: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A2 )
          & ( ord_less_eq_real @ A2 @ B2 ) )
        | ( ( ord_less_real @ A2 @ zero_zero_real )
          & ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).

% le_divide_eq_1
thf(fact_3211_le__divide__eq__1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B2 @ A2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
          & ( ord_less_eq_rat @ A2 @ B2 ) )
        | ( ( ord_less_rat @ A2 @ zero_zero_rat )
          & ( ord_less_eq_rat @ B2 @ A2 ) ) ) ) ).

% le_divide_eq_1
thf(fact_3212_power__Suc__less,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ) ).

% power_Suc_less
thf(fact_3213_power__Suc__less,axiom,
    ! [A2: real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ A2 @ one_one_real )
       => ( ord_less_real @ ( times_times_real @ A2 @ ( power_power_real @ A2 @ N3 ) ) @ ( power_power_real @ A2 @ N3 ) ) ) ) ).

% power_Suc_less
thf(fact_3214_power__Suc__less,axiom,
    ! [A2: rat,N3: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ A2 @ one_one_rat )
       => ( ord_less_rat @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N3 ) ) @ ( power_power_rat @ A2 @ N3 ) ) ) ) ).

% power_Suc_less
thf(fact_3215_power__Suc__less,axiom,
    ! [A2: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ A2 @ one_one_nat )
       => ( ord_less_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N3 ) ) @ ( power_power_nat @ A2 @ N3 ) ) ) ) ).

% power_Suc_less
thf(fact_3216_power__Suc__less,axiom,
    ! [A2: int,N3: nat] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ A2 @ one_one_int )
       => ( ord_less_int @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N3 ) ) @ ( power_power_int @ A2 @ N3 ) ) ) ) ).

% power_Suc_less
thf(fact_3217_power__Suc__le__self,axiom,
    ! [A2: real,N3: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ A2 @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A2 @ ( suc @ N3 ) ) @ A2 ) ) ) ).

% power_Suc_le_self
thf(fact_3218_power__Suc__le__self,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ ( suc @ N3 ) ) @ A2 ) ) ) ).

% power_Suc_le_self
thf(fact_3219_power__Suc__le__self,axiom,
    ! [A2: rat,N3: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ A2 @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ ( suc @ N3 ) ) @ A2 ) ) ) ).

% power_Suc_le_self
thf(fact_3220_power__Suc__le__self,axiom,
    ! [A2: nat,N3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ A2 @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ ( suc @ N3 ) ) @ A2 ) ) ) ).

% power_Suc_le_self
thf(fact_3221_power__Suc__le__self,axiom,
    ! [A2: int,N3: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ A2 @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A2 @ ( suc @ N3 ) ) @ A2 ) ) ) ).

% power_Suc_le_self
thf(fact_3222_power__Suc__less__one,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer )
       => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ ( suc @ N3 ) ) @ one_one_Code_integer ) ) ) ).

% power_Suc_less_one
thf(fact_3223_power__Suc__less__one,axiom,
    ! [A2: real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_real @ A2 @ one_one_real )
       => ( ord_less_real @ ( power_power_real @ A2 @ ( suc @ N3 ) ) @ one_one_real ) ) ) ).

% power_Suc_less_one
thf(fact_3224_power__Suc__less__one,axiom,
    ! [A2: rat,N3: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_rat @ A2 @ one_one_rat )
       => ( ord_less_rat @ ( power_power_rat @ A2 @ ( suc @ N3 ) ) @ one_one_rat ) ) ) ).

% power_Suc_less_one
thf(fact_3225_power__Suc__less__one,axiom,
    ! [A2: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ A2 @ one_one_nat )
       => ( ord_less_nat @ ( power_power_nat @ A2 @ ( suc @ N3 ) ) @ one_one_nat ) ) ) ).

% power_Suc_less_one
thf(fact_3226_power__Suc__less__one,axiom,
    ! [A2: int,N3: nat] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ A2 @ one_one_int )
       => ( ord_less_int @ ( power_power_int @ A2 @ ( suc @ N3 ) ) @ one_one_int ) ) ) ).

% power_Suc_less_one
thf(fact_3227_power__strict__decreasing,axiom,
    ! [N3: nat,N7: nat,A2: code_integer] :
      ( ( ord_less_nat @ N3 @ N7 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
       => ( ( ord_le6747313008572928689nteger @ A2 @ one_one_Code_integer )
         => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ N7 ) @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3228_power__strict__decreasing,axiom,
    ! [N3: nat,N7: nat,A2: real] :
      ( ( ord_less_nat @ N3 @ N7 )
     => ( ( ord_less_real @ zero_zero_real @ A2 )
       => ( ( ord_less_real @ A2 @ one_one_real )
         => ( ord_less_real @ ( power_power_real @ A2 @ N7 ) @ ( power_power_real @ A2 @ N3 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3229_power__strict__decreasing,axiom,
    ! [N3: nat,N7: nat,A2: rat] :
      ( ( ord_less_nat @ N3 @ N7 )
     => ( ( ord_less_rat @ zero_zero_rat @ A2 )
       => ( ( ord_less_rat @ A2 @ one_one_rat )
         => ( ord_less_rat @ ( power_power_rat @ A2 @ N7 ) @ ( power_power_rat @ A2 @ N3 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3230_power__strict__decreasing,axiom,
    ! [N3: nat,N7: nat,A2: nat] :
      ( ( ord_less_nat @ N3 @ N7 )
     => ( ( ord_less_nat @ zero_zero_nat @ A2 )
       => ( ( ord_less_nat @ A2 @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A2 @ N7 ) @ ( power_power_nat @ A2 @ N3 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3231_power__strict__decreasing,axiom,
    ! [N3: nat,N7: nat,A2: int] :
      ( ( ord_less_nat @ N3 @ N7 )
     => ( ( ord_less_int @ zero_zero_int @ A2 )
       => ( ( ord_less_int @ A2 @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A2 @ N7 ) @ ( power_power_int @ A2 @ N3 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3232_one__power2,axiom,
    ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_rat ) ).

% one_power2
thf(fact_3233_one__power2,axiom,
    ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_power2
thf(fact_3234_one__power2,axiom,
    ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_real ) ).

% one_power2
thf(fact_3235_one__power2,axiom,
    ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_power2
thf(fact_3236_one__power2,axiom,
    ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_complex ) ).

% one_power2
thf(fact_3237_one__power2,axiom,
    ( ( power_8256067586552552935nteger @ one_one_Code_integer @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% one_power2
thf(fact_3238_power__decreasing,axiom,
    ! [N3: nat,N7: nat,A2: real] :
      ( ( ord_less_eq_nat @ N3 @ N7 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
       => ( ( ord_less_eq_real @ A2 @ one_one_real )
         => ( ord_less_eq_real @ ( power_power_real @ A2 @ N7 ) @ ( power_power_real @ A2 @ N3 ) ) ) ) ) ).

% power_decreasing
thf(fact_3239_power__decreasing,axiom,
    ! [N3: nat,N7: nat,A2: code_integer] :
      ( ( ord_less_eq_nat @ N3 @ N7 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
       => ( ( ord_le3102999989581377725nteger @ A2 @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N7 ) @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ) ) ).

% power_decreasing
thf(fact_3240_power__decreasing,axiom,
    ! [N3: nat,N7: nat,A2: rat] :
      ( ( ord_less_eq_nat @ N3 @ N7 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
       => ( ( ord_less_eq_rat @ A2 @ one_one_rat )
         => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N7 ) @ ( power_power_rat @ A2 @ N3 ) ) ) ) ) ).

% power_decreasing
thf(fact_3241_power__decreasing,axiom,
    ! [N3: nat,N7: nat,A2: nat] :
      ( ( ord_less_eq_nat @ N3 @ N7 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
       => ( ( ord_less_eq_nat @ A2 @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N7 ) @ ( power_power_nat @ A2 @ N3 ) ) ) ) ) ).

% power_decreasing
thf(fact_3242_power__decreasing,axiom,
    ! [N3: nat,N7: nat,A2: int] :
      ( ( ord_less_eq_nat @ N3 @ N7 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
       => ( ( ord_less_eq_int @ A2 @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A2 @ N7 ) @ ( power_power_int @ A2 @ N3 ) ) ) ) ) ).

% power_decreasing
thf(fact_3243_power__le__imp__le__exp,axiom,
    ! [A2: code_integer,M: nat,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ M ) @ ( power_8256067586552552935nteger @ A2 @ N3 ) )
       => ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% power_le_imp_le_exp
thf(fact_3244_power__le__imp__le__exp,axiom,
    ! [A2: real,M: nat,N3: nat] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_eq_real @ ( power_power_real @ A2 @ M ) @ ( power_power_real @ A2 @ N3 ) )
       => ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% power_le_imp_le_exp
thf(fact_3245_power__le__imp__le__exp,axiom,
    ! [A2: rat,M: nat,N3: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ M ) @ ( power_power_rat @ A2 @ N3 ) )
       => ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% power_le_imp_le_exp
thf(fact_3246_power__le__imp__le__exp,axiom,
    ! [A2: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N3 ) )
       => ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% power_le_imp_le_exp
thf(fact_3247_power__le__imp__le__exp,axiom,
    ! [A2: int,M: nat,N3: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ( ord_less_eq_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N3 ) )
       => ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% power_le_imp_le_exp
thf(fact_3248_self__le__power,axiom,
    ! [A2: real,N3: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_eq_real @ A2 @ ( power_power_real @ A2 @ N3 ) ) ) ) ).

% self_le_power
thf(fact_3249_self__le__power,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_le3102999989581377725nteger @ A2 @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ) ).

% self_le_power
thf(fact_3250_self__le__power,axiom,
    ! [A2: rat,N3: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_eq_rat @ A2 @ ( power_power_rat @ A2 @ N3 ) ) ) ) ).

% self_le_power
thf(fact_3251_self__le__power,axiom,
    ! [A2: nat,N3: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_eq_nat @ A2 @ ( power_power_nat @ A2 @ N3 ) ) ) ) ).

% self_le_power
thf(fact_3252_self__le__power,axiom,
    ! [A2: int,N3: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_eq_int @ A2 @ ( power_power_int @ A2 @ N3 ) ) ) ) ).

% self_le_power
thf(fact_3253_one__less__power,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ) ).

% one_less_power
thf(fact_3254_one__less__power,axiom,
    ! [A2: real,N3: nat] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_real @ one_one_real @ ( power_power_real @ A2 @ N3 ) ) ) ) ).

% one_less_power
thf(fact_3255_one__less__power,axiom,
    ! [A2: rat,N3: nat] :
      ( ( ord_less_rat @ one_one_rat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A2 @ N3 ) ) ) ) ).

% one_less_power
thf(fact_3256_one__less__power,axiom,
    ! [A2: nat,N3: nat] :
      ( ( ord_less_nat @ one_one_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ N3 ) ) ) ) ).

% one_less_power
thf(fact_3257_one__less__power,axiom,
    ! [A2: int,N3: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_int @ one_one_int @ ( power_power_int @ A2 @ N3 ) ) ) ) ).

% one_less_power
thf(fact_3258_num_Osize_I6_J,axiom,
    ! [X33: num] :
      ( ( size_size_num @ ( bit1 @ X33 ) )
      = ( plus_plus_nat @ ( size_size_num @ X33 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(6)
thf(fact_3259_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N3: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N3 )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N3 ) ) ).

% Suc_div_eq_add3_div
thf(fact_3260_convex__bound__lt,axiom,
    ! [X: real,A2: real,Y: real,U: real,V: real] :
      ( ( ord_less_real @ X @ A2 )
     => ( ( ord_less_real @ Y @ A2 )
       => ( ( 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 ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3261_convex__bound__lt,axiom,
    ! [X: rat,A2: rat,Y: rat,U: rat,V: rat] :
      ( ( ord_less_rat @ X @ A2 )
     => ( ( ord_less_rat @ Y @ A2 )
       => ( ( 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 ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3262_convex__bound__lt,axiom,
    ! [X: int,A2: int,Y: int,U: int,V: int] :
      ( ( ord_less_int @ X @ A2 )
     => ( ( ord_less_int @ Y @ A2 )
       => ( ( 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 ) ) @ A2 ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3263_power__diff__power__eq,axiom,
    ! [A2: code_integer,N3: nat,M: nat] :
      ( ( A2 != zero_z3403309356797280102nteger )
     => ( ( ( ord_less_eq_nat @ N3 @ M )
         => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A2 @ M ) @ ( power_8256067586552552935nteger @ A2 @ N3 ) )
            = ( power_8256067586552552935nteger @ A2 @ ( minus_minus_nat @ M @ N3 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N3 @ M )
         => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A2 @ M ) @ ( power_8256067586552552935nteger @ A2 @ N3 ) )
            = ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A2 @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3264_power__diff__power__eq,axiom,
    ! [A2: nat,N3: nat,M: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( ( ord_less_eq_nat @ N3 @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N3 ) )
            = ( power_power_nat @ A2 @ ( minus_minus_nat @ M @ N3 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N3 @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N3 ) )
            = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A2 @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3265_power__diff__power__eq,axiom,
    ! [A2: int,N3: nat,M: nat] :
      ( ( A2 != zero_zero_int )
     => ( ( ( ord_less_eq_nat @ N3 @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N3 ) )
            = ( power_power_int @ A2 @ ( minus_minus_nat @ M @ N3 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N3 @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N3 ) )
            = ( divide_divide_int @ one_one_int @ ( power_power_int @ A2 @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3266_two__realpow__ge__one,axiom,
    ! [N3: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) ) ).

% two_realpow_ge_one
thf(fact_3267_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 ) )
       => ( ! [A3: nat,B3: nat] :
              ( ( X
                = ( some_nat @ A3 ) )
             => ( ( Y
                  = ( some_nat @ B3 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_3268_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 ) )
       => ( ! [A3: nat,B3: product_prod_nat_nat] :
              ( ( X
                = ( some_nat @ A3 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B3 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_3269_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 ) )
       => ( ! [A3: product_prod_nat_nat,B3: nat] :
              ( ( X
                = ( some_P7363390416028606310at_nat @ A3 ) )
             => ( ( Y
                  = ( some_nat @ B3 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_3270_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 ) )
       => ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( ( X
                = ( some_P7363390416028606310at_nat @ A3 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B3 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_3271_split__option__all,axiom,
    ( ( ^ [P5: option_nat > $o] :
        ! [X9: option_nat] : ( P5 @ X9 ) )
    = ( ^ [P3: option_nat > $o] :
          ( ( P3 @ none_nat )
          & ! [X3: nat] : ( P3 @ ( some_nat @ X3 ) ) ) ) ) ).

% split_option_all
thf(fact_3272_split__option__all,axiom,
    ( ( ^ [P5: option4927543243414619207at_nat > $o] :
        ! [X9: option4927543243414619207at_nat] : ( P5 @ X9 ) )
    = ( ^ [P3: option4927543243414619207at_nat > $o] :
          ( ( P3 @ none_P5556105721700978146at_nat )
          & ! [X3: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X3 ) ) ) ) ) ).

% split_option_all
thf(fact_3273_split__option__ex,axiom,
    ( ( ^ [P5: option_nat > $o] :
        ? [X9: option_nat] : ( P5 @ X9 ) )
    = ( ^ [P3: option_nat > $o] :
          ( ( P3 @ none_nat )
          | ? [X3: nat] : ( P3 @ ( some_nat @ X3 ) ) ) ) ) ).

% split_option_ex
thf(fact_3274_split__option__ex,axiom,
    ( ( ^ [P5: option4927543243414619207at_nat > $o] :
        ? [X9: option4927543243414619207at_nat] : ( P5 @ X9 ) )
    = ( ^ [P3: option4927543243414619207at_nat > $o] :
          ( ( P3 @ none_P5556105721700978146at_nat )
          | ? [X3: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X3 ) ) ) ) ) ).

% split_option_ex
thf(fact_3275_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X23: nat] :
            ( Y
           != ( some_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_3276_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] :
            ( Y
           != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_3277_option_OdiscI,axiom,
    ! [Option: option_nat,X22: nat] :
      ( ( Option
        = ( some_nat @ X22 ) )
     => ( Option != none_nat ) ) ).

% option.discI
thf(fact_3278_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
      ( ( Option
        = ( some_P7363390416028606310at_nat @ X22 ) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
thf(fact_3279_option_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( none_nat
     != ( some_nat @ X22 ) ) ).

% option.distinct(1)
thf(fact_3280_option_Odistinct_I1_J,axiom,
    ! [X22: product_prod_nat_nat] :
      ( none_P5556105721700978146at_nat
     != ( some_P7363390416028606310at_nat @ X22 ) ) ).

% option.distinct(1)
thf(fact_3281_option_Osel,axiom,
    ! [X22: nat] :
      ( ( the_nat @ ( some_nat @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_3282_option_Osel,axiom,
    ! [X22: product_prod_nat_nat] :
      ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_3283_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_3284_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_3285_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: vEBT_VEBT > vEBT_VEBT > assn,Xs2: list_VEBT_VEBT,I: nat,Xsi: list_VEBT_VEBT,I3: set_nat,F: assn,C2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L3204528365124325536T_VEBT @ I3 @ A @ Xs2 @ Xsi ) @ F ) @ C2 @ Q )
           => ( hoare_hoare_triple_o @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_3286_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: vEBT_VEBT > nat > assn,Xs2: list_VEBT_VEBT,I: nat,Xsi: list_nat,I3: set_nat,F: assn,C2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8650695023172932196BT_nat @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L8650695023172932196BT_nat @ I3 @ A @ Xs2 @ Xsi ) @ F ) @ C2 @ Q )
           => ( hoare_hoare_triple_o @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_3287_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: real > vEBT_VEBT > assn,Xs2: list_real,I: nat,Xsi: list_VEBT_VEBT,I3: set_nat,F: assn,C2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_real @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3095048238742455910T_VEBT @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L3095048238742455910T_VEBT @ I3 @ A @ Xs2 @ Xsi ) @ F ) @ C2 @ Q )
           => ( hoare_hoare_triple_o @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_3288_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: real > vEBT_VEBTi > assn,Xs2: list_real,I: nat,Xsi: list_VEBT_VEBTi,I3: set_nat,F: assn,C2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_real @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L7851252805511451907_VEBTi @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L7851252805511451907_VEBTi @ I3 @ A @ Xs2 @ Xsi ) @ F ) @ C2 @ Q )
           => ( hoare_hoare_triple_o @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_3289_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: real > nat > assn,Xs2: list_real,I: nat,Xsi: list_nat,I3: set_nat,F: assn,C2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_real @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L234762979517870878al_nat @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L234762979517870878al_nat @ I3 @ A @ Xs2 @ Xsi ) @ F ) @ C2 @ Q )
           => ( hoare_hoare_triple_o @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_3290_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: $o > vEBT_VEBT > assn,Xs2: list_o,I: nat,Xsi: list_VEBT_VEBT,I3: set_nat,F: assn,C2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_o @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L1319876754960170684T_VEBT @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L1319876754960170684T_VEBT @ I3 @ A @ Xs2 @ Xsi ) @ F ) @ C2 @ Q )
           => ( hoare_hoare_triple_o @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_3291_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: $o > vEBT_VEBTi > assn,Xs2: list_o,I: nat,Xsi: list_VEBT_VEBTi,I3: set_nat,F: assn,C2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_o @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L6286945158656146733_VEBTi @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L6286945158656146733_VEBTi @ I3 @ A @ Xs2 @ Xsi ) @ F ) @ C2 @ Q )
           => ( hoare_hoare_triple_o @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_3292_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: $o > nat > assn,Xs2: list_o,I: nat,Xsi: list_nat,I3: set_nat,F: assn,C2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_o @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L2281750874075065672_o_nat @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L2281750874075065672_o_nat @ I3 @ A @ Xs2 @ Xsi ) @ F ) @ C2 @ Q )
           => ( hoare_hoare_triple_o @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_3293_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: nat > vEBT_VEBT > assn,Xs2: list_nat,I: nat,Xsi: list_VEBT_VEBT,I3: set_nat,F: assn,C2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_nat @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L8511957252848910786T_VEBT @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L8511957252848910786T_VEBT @ I3 @ A @ Xs2 @ Xsi ) @ F ) @ C2 @ Q )
           => ( hoare_hoare_triple_o @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_3294_listI__assn__reinsert_H,axiom,
    ! [P: assn,A: nat > vEBT_VEBTi > assn,Xs2: list_nat,I: nat,Xsi: list_VEBT_VEBTi,I3: set_nat,F: assn,C2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_nat @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L7489483478785760935_VEBTi @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L7489483478785760935_VEBTi @ I3 @ A @ Xs2 @ Xsi ) @ F ) @ C2 @ Q )
           => ( hoare_hoare_triple_o @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_reinsert'
thf(fact_3295_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: vEBT_VEBT > vEBT_VEBT > assn,X: vEBT_VEBT,Xi: vEBT_VEBT,I3: set_nat,I: nat,Xs2: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F: assn,C2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ X @ Xi ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L3204528365124325536T_VEBT @ I3 @ A @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F ) @ C2 @ Q )
           => ( hoare_hoare_triple_o @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_3296_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: real > vEBT_VEBTi > assn,X: real,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs2: list_real,Xsi: list_VEBT_VEBTi,F: assn,C2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ X @ Xi ) @ ( vEBT_L7851252805511451907_VEBTi @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L7851252805511451907_VEBTi @ I3 @ A @ ( list_update_real @ Xs2 @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F ) @ C2 @ Q )
           => ( hoare_hoare_triple_o @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_3297_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: real > vEBT_VEBT > assn,X: real,Xi: vEBT_VEBT,I3: set_nat,I: nat,Xs2: list_real,Xsi: list_VEBT_VEBT,F: assn,C2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ X @ Xi ) @ ( vEBT_L3095048238742455910T_VEBT @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L3095048238742455910T_VEBT @ I3 @ A @ ( list_update_real @ Xs2 @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F ) @ C2 @ Q )
           => ( hoare_hoare_triple_o @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_3298_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: $o > vEBT_VEBTi > assn,X: $o,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs2: list_o,Xsi: list_VEBT_VEBTi,F: assn,C2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ X @ Xi ) @ ( vEBT_L6286945158656146733_VEBTi @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L6286945158656146733_VEBTi @ I3 @ A @ ( list_update_o @ Xs2 @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F ) @ C2 @ Q )
           => ( hoare_hoare_triple_o @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_3299_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: $o > vEBT_VEBT > assn,X: $o,Xi: vEBT_VEBT,I3: set_nat,I: nat,Xs2: list_o,Xsi: list_VEBT_VEBT,F: assn,C2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ X @ Xi ) @ ( vEBT_L1319876754960170684T_VEBT @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L1319876754960170684T_VEBT @ I3 @ A @ ( list_update_o @ Xs2 @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F ) @ C2 @ Q )
           => ( hoare_hoare_triple_o @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_3300_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: nat > vEBT_VEBTi > assn,X: nat,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs2: list_nat,Xsi: list_VEBT_VEBTi,F: assn,C2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ X @ Xi ) @ ( vEBT_L7489483478785760935_VEBTi @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L7489483478785760935_VEBTi @ I3 @ A @ ( list_update_nat @ Xs2 @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F ) @ C2 @ Q )
           => ( hoare_hoare_triple_o @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_3301_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: nat > vEBT_VEBT > assn,X: nat,Xi: vEBT_VEBT,I3: set_nat,I: nat,Xs2: list_nat,Xsi: list_VEBT_VEBT,F: assn,C2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ X @ Xi ) @ ( vEBT_L8511957252848910786T_VEBT @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L8511957252848910786T_VEBT @ I3 @ A @ ( list_update_nat @ Xs2 @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F ) @ C2 @ Q )
           => ( hoare_hoare_triple_o @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_3302_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: vEBT_VEBTi > vEBT_VEBTi > assn,X: vEBT_VEBTi,Xi: vEBT_VEBTi,I3: set_nat,I: nat,Xs2: list_VEBT_VEBTi,Xsi: list_VEBT_VEBTi,F: assn,C2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ X @ Xi ) @ ( vEBT_L886525131989349516_VEBTi @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L886525131989349516_VEBTi @ I3 @ A @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X ) @ ( list_u6098035379799741383_VEBTi @ Xsi @ I @ Xi ) ) @ F ) @ C2 @ Q )
           => ( hoare_hoare_triple_o @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_3303_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: vEBT_VEBTi > vEBT_VEBT > assn,X: vEBT_VEBTi,Xi: vEBT_VEBT,I3: set_nat,I: nat,Xs2: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,F: assn,C2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ X @ Xi ) @ ( vEBT_L2497118539674116125T_VEBT @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( vEBT_L2497118539674116125T_VEBT @ I3 @ A @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F ) @ C2 @ Q )
           => ( hoare_hoare_triple_o @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_3304_listI__assn__reinsert__upd_H,axiom,
    ! [P: assn,A: vEBT_VEBT > vEBT_VEBT > assn,X: vEBT_VEBT,Xi: vEBT_VEBT,I3: set_nat,I: nat,Xs2: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F: assn,C2: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( times_times_assn @ ( A @ X @ Xi ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ I3 @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( member_nat @ I @ I3 )
         => ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ ( vEBT_L3204528365124325536T_VEBT @ I3 @ A @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ ( list_u1324408373059187874T_VEBT @ Xsi @ I @ Xi ) ) @ F ) @ C2 @ Q )
           => ( hoare_1429296392585015714_VEBTi @ P @ C2 @ Q ) ) ) ) ) ).

% listI_assn_reinsert_upd'
thf(fact_3305_option_Osize_I4_J,axiom,
    ! [X22: nat] :
      ( ( size_size_option_nat @ ( some_nat @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_3306_option_Osize_I4_J,axiom,
    ! [X22: product_prod_nat_nat] :
      ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_3307_option_Osize_I3_J,axiom,
    ( ( size_size_option_nat @ none_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_3308_option_Osize_I3_J,axiom,
    ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_3309_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option
        = ( some_nat @ ( the_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_3310_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option
        = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_3311_div__neg__pos__less0,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ ( divide_divide_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% div_neg_pos_less0
thf(fact_3312_neg__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ B2 @ zero_zero_int )
     => ( ( ord_less_int @ ( divide_divide_int @ A2 @ B2 ) @ zero_zero_int )
        = ( ord_less_int @ zero_zero_int @ A2 ) ) ) ).

% neg_imp_zdiv_neg_iff
thf(fact_3313_pos__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ ( divide_divide_int @ A2 @ B2 ) @ zero_zero_int )
        = ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).

% pos_imp_zdiv_neg_iff
thf(fact_3314_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ A2 @ B2 )
       => ( ( divide_divide_nat @ A2 @ B2 )
          = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_3315_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ A2 @ B2 )
       => ( ( divide_divide_int @ A2 @ B2 )
          = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_3316_div__positive,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ A2 )
       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).

% div_positive
thf(fact_3317_div__positive,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_eq_int @ B2 @ A2 )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% div_positive
thf(fact_3318_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
     => ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3319_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C2 )
     => ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ C2 ) )
        = ( divide_divide_int @ ( divide_divide_int @ A2 @ B2 ) @ C2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3320_zdiv__mono1,axiom,
    ! [A2: int,A4: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ A4 )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B2 ) @ ( divide_divide_int @ A4 @ B2 ) ) ) ) ).

% zdiv_mono1
thf(fact_3321_zdiv__mono2,axiom,
    ! [A2: int,B4: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B4 )
       => ( ( ord_less_eq_int @ B4 @ B2 )
         => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B2 ) @ ( divide_divide_int @ A2 @ B4 ) ) ) ) ) ).

% zdiv_mono2
thf(fact_3322_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( ( divide_divide_int @ I @ K )
        = zero_zero_int )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I )
          & ( ord_less_int @ I @ K ) )
        | ( ( ord_less_eq_int @ I @ zero_zero_int )
          & ( ord_less_int @ K @ I ) ) ) ) ).

% zdiv_eq_0_iff
thf(fact_3323_zdiv__mono1__neg,axiom,
    ! [A2: int,A4: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ A4 )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B2 ) @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% zdiv_mono1_neg
thf(fact_3324_zdiv__mono2__neg,axiom,
    ! [A2: int,B4: int,B2: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B4 )
       => ( ( ord_less_eq_int @ B4 @ B2 )
         => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B4 ) @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ) ).

% zdiv_mono2_neg
thf(fact_3325_div__int__pos__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) )
      = ( ( K = zero_zero_int )
        | ( L2 = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ K )
          & ( ord_less_eq_int @ zero_zero_int @ L2 ) )
        | ( ( ord_less_int @ K @ zero_zero_int )
          & ( ord_less_int @ L2 @ zero_zero_int ) ) ) ) ).

% div_int_pos_iff
thf(fact_3326_div__positive__int,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_eq_int @ L2 @ K )
     => ( ( ord_less_int @ zero_zero_int @ L2 )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) ) ) ) ).

% div_positive_int
thf(fact_3327_div__nonneg__neg__le0,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% div_nonneg_neg_le0
thf(fact_3328_div__nonpos__pos__le0,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% div_nonpos_pos_le0
thf(fact_3329_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
        = ( ord_less_eq_int @ K @ I ) ) ) ).

% pos_imp_zdiv_pos_iff
thf(fact_3330_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ B2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B2 ) )
        = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ) ).

% neg_imp_zdiv_nonneg_iff
thf(fact_3331_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B2 ) )
        = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ) ).

% pos_imp_zdiv_nonneg_iff
thf(fact_3332_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B2 ) )
        = ( ( ord_less_eq_int @ B2 @ A2 )
          & ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
thf(fact_3333_div__geq,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ~ ( ord_less_nat @ M @ N3 )
       => ( ( divide_divide_nat @ M @ N3 )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N3 ) @ N3 ) ) ) ) ) ).

% div_geq
thf(fact_3334_q__pos__lemma,axiom,
    ! [B4: int,Q7: int,R4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q7 ) @ R4 ) )
     => ( ( ord_less_int @ R4 @ B4 )
       => ( ( ord_less_int @ zero_zero_int @ B4 )
         => ( ord_less_eq_int @ zero_zero_int @ Q7 ) ) ) ) ).

% q_pos_lemma
thf(fact_3335_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q3: int,R3: int,B4: int,Q7: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B2 @ Q3 ) @ R3 )
        = ( plus_plus_int @ ( times_times_int @ B4 @ Q7 ) @ R4 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q7 ) @ R4 ) )
       => ( ( ord_less_int @ R4 @ B4 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
           => ( ( ord_less_int @ zero_zero_int @ B4 )
             => ( ( ord_less_eq_int @ B4 @ B2 )
               => ( ord_less_eq_int @ Q3 @ Q7 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_3336_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q3: int,R3: int,B4: int,Q7: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B2 @ Q3 ) @ R3 )
        = ( plus_plus_int @ ( times_times_int @ B4 @ Q7 ) @ R4 ) )
     => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q7 ) @ R4 ) @ zero_zero_int )
       => ( ( ord_less_int @ R3 @ B2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
           => ( ( ord_less_int @ zero_zero_int @ B4 )
             => ( ( ord_less_eq_int @ B4 @ B2 )
               => ( ord_less_eq_int @ Q7 @ Q3 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_3337_unique__quotient__lemma,axiom,
    ! [B2: int,Q7: int,R4: int,Q3: int,R3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q7 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B2 @ Q3 ) @ R3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
       => ( ( ord_less_int @ R4 @ B2 )
         => ( ( ord_less_int @ R3 @ B2 )
           => ( ord_less_eq_int @ Q7 @ Q3 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_3338_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q7: int,R4: int,Q3: int,R3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q7 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B2 @ Q3 ) @ R3 ) )
     => ( ( ord_less_eq_int @ R3 @ zero_zero_int )
       => ( ( ord_less_int @ B2 @ R3 )
         => ( ( ord_less_int @ B2 @ R4 )
           => ( ord_less_eq_int @ Q3 @ Q7 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_3339_split__zdiv,axiom,
    ! [P: int > $o,N3: int,K: int] :
      ( ( P @ ( divide_divide_int @ N3 @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ zero_zero_int ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I2: int,J: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J )
                & ( ord_less_int @ J @ K )
                & ( N3
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J ) ) )
             => ( P @ I2 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I2: int,J: int] :
              ( ( ( ord_less_int @ K @ J )
                & ( ord_less_eq_int @ J @ zero_zero_int )
                & ( N3
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J ) ) )
             => ( P @ I2 ) ) ) ) ) ).

% split_zdiv
thf(fact_3340_int__div__neg__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R3: int] :
      ( ( A2
        = ( plus_plus_int @ ( times_times_int @ B2 @ Q3 ) @ R3 ) )
     => ( ( ord_less_eq_int @ R3 @ zero_zero_int )
       => ( ( ord_less_int @ B2 @ R3 )
         => ( ( divide_divide_int @ A2 @ B2 )
            = Q3 ) ) ) ) ).

% int_div_neg_eq
thf(fact_3341_int__div__pos__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R3: int] :
      ( ( A2
        = ( plus_plus_int @ ( times_times_int @ B2 @ Q3 ) @ R3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
       => ( ( ord_less_int @ R3 @ B2 )
         => ( ( divide_divide_int @ A2 @ B2 )
            = Q3 ) ) ) ) ).

% int_div_pos_eq
thf(fact_3342_space__bound,axiom,
    ! [T: vEBT_VEBT,N3: nat,U: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( U
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
       => ( ord_less_eq_nat @ ( vEBT_VEBT_space @ T ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ U ) ) ) ) ).

% space_bound
thf(fact_3343_space_H__bound,axiom,
    ! [T: vEBT_VEBT,N3: nat,U: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( U
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
       => ( ord_less_eq_nat @ ( vEBT_VEBT_space2 @ T ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ U ) ) ) ) ).

% space'_bound
thf(fact_3344_in__children__def,axiom,
    ( vEBT_V5917875025757280293ildren
    = ( ^ [N2: nat,TreeList4: list_VEBT_VEBT,X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ X3 @ N2 ) ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) ) ) ).

% in_children_def
thf(fact_3345_minNull__delete__time__bound,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T @ X ) )
       => ( ord_less_eq_nat @ ( vEBT_T_d_e_l_e_t_e @ T @ X ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% minNull_delete_time_bound
thf(fact_3346_tdeletemimi,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ord_less_eq_nat @ ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Mi ) ) @ Deg @ TreeList @ Summary ) @ X ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).

% tdeletemimi
thf(fact_3347_vebt__buildup__bound,axiom,
    ! [U: nat,N3: nat] :
      ( ( U
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
     => ( ord_less_eq_nat @ ( vEBT_V8346862874174094_d_u_p @ N3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) @ U ) ) ) ).

% vebt_buildup_bound
thf(fact_3348_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_3349_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_3350_T__vebt__buildupi__univ,axiom,
    ! [U: nat,N3: nat] :
      ( ( U
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
     => ( ord_less_eq_nat @ ( vEBT_V441764108873111860ildupi @ N3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ U ) ) ) ).

% T_vebt_buildupi_univ
thf(fact_3351_T__vebt__buildupi__gq__0,axiom,
    ! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( vEBT_V441764108873111860ildupi @ N3 ) ) ).

% T_vebt_buildupi_gq_0
thf(fact_3352_space__space_H,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_nat @ ( vEBT_VEBT_space @ T ) @ ( vEBT_VEBT_space2 @ T ) ) ).

% space_space'
thf(fact_3353_pure__true,axiom,
    ( ( pure_assn @ $true )
    = one_one_assn ) ).

% pure_true
thf(fact_3354_pure__assn__eq__emp__iff,axiom,
    ! [P: $o] :
      ( ( ( pure_assn @ P )
        = one_one_assn )
      = P ) ).

% pure_assn_eq_emp_iff
thf(fact_3355_height__compose__summary,axiom,
    ! [Summary: vEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ Summary ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ).

% height_compose_summary
thf(fact_3356_power__one__right,axiom,
    ! [A2: nat] :
      ( ( power_power_nat @ A2 @ one_one_nat )
      = A2 ) ).

% power_one_right
thf(fact_3357_power__one__right,axiom,
    ! [A2: real] :
      ( ( power_power_real @ A2 @ one_one_nat )
      = A2 ) ).

% power_one_right
thf(fact_3358_power__one__right,axiom,
    ! [A2: int] :
      ( ( power_power_int @ A2 @ one_one_nat )
      = A2 ) ).

% power_one_right
thf(fact_3359_power__one__right,axiom,
    ! [A2: complex] :
      ( ( power_power_complex @ A2 @ one_one_nat )
      = A2 ) ).

% power_one_right
thf(fact_3360_power__one__right,axiom,
    ! [A2: code_integer] :
      ( ( power_8256067586552552935nteger @ A2 @ one_one_nat )
      = A2 ) ).

% power_one_right
thf(fact_3361_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( times_times_nat @ M @ N3 )
        = one_one_nat )
      = ( ( M = one_one_nat )
        & ( N3 = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_3362_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M @ N3 ) )
      = ( ( M = one_one_nat )
        & ( N3 = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_3363_ent__pure__pre__iff__sng,axiom,
    ! [B2: $o,Q: assn] :
      ( ( entails @ ( pure_assn @ B2 ) @ Q )
      = ( B2
       => ( entails @ one_one_assn @ Q ) ) ) ).

% ent_pure_pre_iff_sng
thf(fact_3364_delete__bound__height,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_nat @ ( vEBT_T_d_e_l_e_t_e @ T @ X ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% delete_bound_height
thf(fact_3365_less__one,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ N3 @ one_one_nat )
      = ( N3 = zero_zero_nat ) ) ).

% less_one
thf(fact_3366_diff__Suc__1,axiom,
    ! [N3: nat] :
      ( ( minus_minus_nat @ ( suc @ N3 ) @ one_one_nat )
      = N3 ) ).

% diff_Suc_1
thf(fact_3367_int__div__same__is__1,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( ( divide_divide_int @ A2 @ B2 )
          = A2 )
        = ( B2 = one_one_int ) ) ) ).

% int_div_same_is_1
thf(fact_3368_div__eq__dividend__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ( divide_divide_nat @ M @ N3 )
          = M )
        = ( N3 = one_one_nat ) ) ) ).

% div_eq_dividend_iff
thf(fact_3369_Suc__diff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( ord_less_eq_nat @ one_one_nat @ M )
       => ( ( suc @ ( minus_minus_nat @ N3 @ M ) )
          = ( minus_minus_nat @ N3 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ).

% Suc_diff
thf(fact_3370_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_3371_Suc__diff__1,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( suc @ ( minus_minus_nat @ N3 @ one_one_nat ) )
        = N3 ) ) ).

% Suc_diff_1
thf(fact_3372_nat__mult__1__right,axiom,
    ! [N3: nat] :
      ( ( times_times_nat @ N3 @ one_one_nat )
      = N3 ) ).

% nat_mult_1_right
thf(fact_3373_nat__mult__1,axiom,
    ! [N3: nat] :
      ( ( times_times_nat @ one_one_nat @ N3 )
      = N3 ) ).

% nat_mult_1
thf(fact_3374_assn__one__left,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ one_one_assn @ P )
      = P ) ).

% assn_one_left
thf(fact_3375_numerals_I1_J,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numerals(1)
thf(fact_3376_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_3377_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_3378_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_3379_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_3380_nat__geq__1__eq__neqz,axiom,
    ! [X: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ X )
      = ( X != zero_zero_nat ) ) ).

% nat_geq_1_eq_neqz
thf(fact_3381_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N3: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N3 ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N3 ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_3382_mult__eq__self__implies__10,axiom,
    ! [M: nat,N3: nat] :
      ( ( M
        = ( times_times_nat @ M @ N3 ) )
     => ( ( N3 = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_3383_nat__induct__non__zero,axiom,
    ! [N3: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( P @ one_one_nat )
       => ( ! [N: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N )
             => ( ( P @ N )
               => ( P @ ( suc @ N ) ) ) )
         => ( P @ N3 ) ) ) ) ).

% nat_induct_non_zero
thf(fact_3384_div__less__dividend,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ one_one_nat @ N3 )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( divide_divide_nat @ M @ N3 ) @ M ) ) ) ).

% div_less_dividend
thf(fact_3385_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_3386_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_3387_Suc__diff__eq__diff__pred,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N3 )
        = ( minus_minus_nat @ M @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_3388_Suc__pred_H,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( N3
        = ( suc @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_3389_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M5: nat,N2: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N2 ) ) ) ) ) ).

% add_eq_if
thf(fact_3390_Suc__n__minus__m__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( ord_less_nat @ one_one_nat @ M )
       => ( ( suc @ ( minus_minus_nat @ N3 @ M ) )
          = ( minus_minus_nat @ N3 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ).

% Suc_n_minus_m_eq
thf(fact_3391_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M5: nat,N2: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N2 ) ) ) ) ) ).

% mult_eq_if
thf(fact_3392_power__minus__mult,axiom,
    ! [N3: nat,A2: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_times_complex @ ( power_power_complex @ A2 @ ( minus_minus_nat @ N3 @ one_one_nat ) ) @ A2 )
        = ( power_power_complex @ A2 @ N3 ) ) ) ).

% power_minus_mult
thf(fact_3393_power__minus__mult,axiom,
    ! [N3: nat,A2: code_integer] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A2 @ ( minus_minus_nat @ N3 @ one_one_nat ) ) @ A2 )
        = ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ).

% power_minus_mult
thf(fact_3394_power__minus__mult,axiom,
    ! [N3: nat,A2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_times_real @ ( power_power_real @ A2 @ ( minus_minus_nat @ N3 @ one_one_nat ) ) @ A2 )
        = ( power_power_real @ A2 @ N3 ) ) ) ).

% power_minus_mult
thf(fact_3395_power__minus__mult,axiom,
    ! [N3: nat,A2: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_times_rat @ ( power_power_rat @ A2 @ ( minus_minus_nat @ N3 @ one_one_nat ) ) @ A2 )
        = ( power_power_rat @ A2 @ N3 ) ) ) ).

% power_minus_mult
thf(fact_3396_power__minus__mult,axiom,
    ! [N3: nat,A2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_times_nat @ ( power_power_nat @ A2 @ ( minus_minus_nat @ N3 @ one_one_nat ) ) @ A2 )
        = ( power_power_nat @ A2 @ N3 ) ) ) ).

% power_minus_mult
thf(fact_3397_power__minus__mult,axiom,
    ! [N3: nat,A2: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_times_int @ ( power_power_int @ A2 @ ( minus_minus_nat @ N3 @ one_one_nat ) ) @ A2 )
        = ( power_power_int @ A2 @ N3 ) ) ) ).

% power_minus_mult
thf(fact_3398_power__minus__mult,axiom,
    ! [N3: nat,A2: assn] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_times_assn @ ( power_power_assn @ A2 @ ( minus_minus_nat @ N3 @ one_one_nat ) ) @ A2 )
        = ( power_power_assn @ A2 @ N3 ) ) ) ).

% power_minus_mult
thf(fact_3399_div__pos__geq,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L2 )
     => ( ( ord_less_eq_int @ L2 @ K )
       => ( ( divide_divide_int @ K @ L2 )
          = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L2 ) @ L2 ) @ one_one_int ) ) ) ) ).

% div_pos_geq
thf(fact_3400_axxdiv2,axiom,
    ! [X: int] :
      ( ( ( divide_divide_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ X ) @ X ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = X )
      & ( ( divide_divide_int @ ( plus_plus_int @ ( plus_plus_int @ zero_zero_int @ X ) @ X ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = X ) ) ).

% axxdiv2
thf(fact_3401_z1pdiv2,axiom,
    ! [B2: int] :
      ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = B2 ) ).

% z1pdiv2
thf(fact_3402_ex__power__ivl2,axiom,
    ! [B2: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
       => ? [N: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B2 @ N ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_3403_ex__power__ivl1,axiom,
    ! [B2: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
     => ( ( ord_less_eq_nat @ one_one_nat @ K )
       => ? [N: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_3404_small__powers__of__2,axiom,
    ! [X: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ X )
     => ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ X @ one_one_nat ) ) ) ) ).

% small_powers_of_2
thf(fact_3405_power__2__mult__step__le,axiom,
    ! [N6: nat,N3: nat,K4: nat,K: nat] :
      ( ( ord_less_eq_nat @ N6 @ N3 )
     => ( ( ord_less_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ K4 ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ K ) )
       => ( ord_less_eq_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ ( plus_plus_nat @ K4 @ one_one_nat ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ K ) ) ) ) ).

% power_2_mult_step_le
thf(fact_3406_pos__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
        = ( divide_divide_int @ B2 @ A2 ) ) ) ).

% pos_zdiv_mult_2
thf(fact_3407_neg__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
        = ( divide_divide_int @ ( plus_plus_int @ B2 @ one_one_int ) @ A2 ) ) ) ).

% neg_zdiv_mult_2
thf(fact_3408_tdeletemimi_H,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ord_less_eq_nat @ ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Mi ) ) @ Deg @ TreeList @ Summary ) @ X ) @ one_one_nat ) ) ).

% tdeletemimi'
thf(fact_3409_Tb__T__vebt__buildupi_H_H,axiom,
    ! [N3: nat] : ( ord_less_eq_nat @ ( vEBT_V441764108873111860ildupi @ N3 ) @ ( minus_minus_nat @ ( vEBT_VEBT_Tb2 @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Tb_T_vebt_buildupi''
thf(fact_3410_minNull__delete__time__bound_H,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T @ X ) )
       => ( ord_less_eq_nat @ ( vEBT_V1232361888498592333_e_t_e @ T @ X ) @ one_one_nat ) ) ) ).

% minNull_delete_time_bound'
thf(fact_3411_delete__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_nat @ ( vEBT_V1232361888498592333_e_t_e @ T @ X ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% delete_bound_height'
thf(fact_3412_divmod__step__eq,axiom,
    ! [L2: num,R3: nat,Q3: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R3 )
       => ( ( unique5026877609467782581ep_nat @ L2 @ ( product_Pair_nat_nat @ Q3 @ R3 ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ one_one_nat ) @ ( minus_minus_nat @ R3 @ ( numeral_numeral_nat @ L2 ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R3 )
       => ( ( unique5026877609467782581ep_nat @ L2 @ ( product_Pair_nat_nat @ Q3 @ R3 ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ R3 ) ) ) ) ).

% divmod_step_eq
thf(fact_3413_divmod__step__eq,axiom,
    ! [L2: num,R3: int,Q3: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R3 )
       => ( ( unique5024387138958732305ep_int @ L2 @ ( product_Pair_int_int @ Q3 @ R3 ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ one_one_int ) @ ( minus_minus_int @ R3 @ ( numeral_numeral_int @ L2 ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R3 )
       => ( ( unique5024387138958732305ep_int @ L2 @ ( product_Pair_int_int @ Q3 @ R3 ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ R3 ) ) ) ) ).

% divmod_step_eq
thf(fact_3414_divmod__step__eq,axiom,
    ! [L2: num,R3: code_integer,Q3: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R3 )
       => ( ( unique4921790084139445826nteger @ L2 @ ( produc1086072967326762835nteger @ Q3 @ R3 ) )
          = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R3 @ ( numera6620942414471956472nteger @ L2 ) ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R3 )
       => ( ( unique4921790084139445826nteger @ L2 @ ( produc1086072967326762835nteger @ Q3 @ R3 ) )
          = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ R3 ) ) ) ) ).

% divmod_step_eq
thf(fact_3415_norm__pre__pure__iff,axiom,
    ! [P: assn,B2: $o,F2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( hoare_hoare_triple_o @ ( times_times_assn @ P @ ( pure_assn @ B2 ) ) @ F2 @ Q )
      = ( B2
       => ( hoare_hoare_triple_o @ P @ F2 @ Q ) ) ) ).

% norm_pre_pure_iff
thf(fact_3416_norm__pre__pure__iff,axiom,
    ! [P: assn,B2: $o,F2: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ P @ ( pure_assn @ B2 ) ) @ F2 @ Q )
      = ( B2
       => ( hoare_1429296392585015714_VEBTi @ P @ F2 @ Q ) ) ) ).

% norm_pre_pure_iff
thf(fact_3417_norm__pre__pure__iff,axiom,
    ! [P: assn,B2: $o,F2: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ( hoare_7629718768684598413on_nat @ ( times_times_assn @ P @ ( pure_assn @ B2 ) ) @ F2 @ Q )
      = ( B2
       => ( hoare_7629718768684598413on_nat @ P @ F2 @ Q ) ) ) ).

% norm_pre_pure_iff
thf(fact_3418_norm__pre__pure__iff,axiom,
    ! [P: assn,B2: $o,F2: heap_Time_Heap_nat,Q: nat > assn] :
      ( ( hoare_3067605981109127869le_nat @ ( times_times_assn @ P @ ( pure_assn @ B2 ) ) @ F2 @ Q )
      = ( B2
       => ( hoare_3067605981109127869le_nat @ P @ F2 @ Q ) ) ) ).

% norm_pre_pure_iff
thf(fact_3419_space__2__pow__bound,axiom,
    ! [T: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_VEBT_space2 @ T ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( minus_minus_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) @ one_one_real ) ) ) ) ).

% space_2_pow_bound
thf(fact_3420_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% zle_add1_eq_le
thf(fact_3421_of__nat__eq__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( semiri5074537144036343181t_real @ N3 ) )
      = ( M = N3 ) ) ).

% of_nat_eq_iff
thf(fact_3422_of__nat__eq__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N3 ) )
      = ( M = N3 ) ) ).

% of_nat_eq_iff
thf(fact_3423_of__nat__eq__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = ( semiri1316708129612266289at_nat @ N3 ) )
      = ( M = N3 ) ) ).

% of_nat_eq_iff
thf(fact_3424_of__nat__eq__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( semiri8010041392384452111omplex @ M )
        = ( semiri8010041392384452111omplex @ N3 ) )
      = ( M = N3 ) ) ).

% of_nat_eq_iff
thf(fact_3425_of__nat__eq__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( semiri4939895301339042750nteger @ M )
        = ( semiri4939895301339042750nteger @ N3 ) )
      = ( M = N3 ) ) ).

% of_nat_eq_iff
thf(fact_3426_two__realpow__ge__two,axiom,
    ! [N3: nat] :
      ( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N3 ) )
     => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% two_realpow_ge_two
thf(fact_3427_double__eq__0__iff,axiom,
    ! [A2: real] :
      ( ( ( plus_plus_real @ A2 @ A2 )
        = zero_zero_real )
      = ( A2 = zero_zero_real ) ) ).

% double_eq_0_iff
thf(fact_3428_double__eq__0__iff,axiom,
    ! [A2: rat] :
      ( ( ( plus_plus_rat @ A2 @ A2 )
        = zero_zero_rat )
      = ( A2 = zero_zero_rat ) ) ).

% double_eq_0_iff
thf(fact_3429_double__eq__0__iff,axiom,
    ! [A2: int] :
      ( ( ( plus_plus_int @ A2 @ A2 )
        = zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% double_eq_0_iff
thf(fact_3430_count__buildup_H,axiom,
    ! [N3: nat] : ( ord_less_eq_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N3 ) ) @ ( semiri5074537144036343181t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% count_buildup'
thf(fact_3431_space__cnt,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_VEBT_space2 @ T ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_cnt @ T ) ) ) ).

% space_cnt
thf(fact_3432_T__vebt__buildupi__cnt_H,axiom,
    ! [N3: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V441764108873111860ildupi @ N3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N3 ) ) ) ) ).

% T_vebt_buildupi_cnt'
thf(fact_3433_t__buildup__cnt,axiom,
    ! [N3: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8346862874174094_d_u_p @ N3 ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N3 ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ).

% t_buildup_cnt
thf(fact_3434_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% semiring_1_class.of_nat_0
thf(fact_3435_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% semiring_1_class.of_nat_0
thf(fact_3436_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% semiring_1_class.of_nat_0
thf(fact_3437_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% semiring_1_class.of_nat_0
thf(fact_3438_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri8010041392384452111omplex @ zero_zero_nat )
    = zero_zero_complex ) ).

% semiring_1_class.of_nat_0
thf(fact_3439_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri4939895301339042750nteger @ zero_zero_nat )
    = zero_z3403309356797280102nteger ) ).

% semiring_1_class.of_nat_0
thf(fact_3440_of__nat__0__eq__iff,axiom,
    ! [N3: nat] :
      ( ( zero_zero_rat
        = ( semiri681578069525770553at_rat @ N3 ) )
      = ( zero_zero_nat = N3 ) ) ).

% of_nat_0_eq_iff
thf(fact_3441_of__nat__0__eq__iff,axiom,
    ! [N3: nat] :
      ( ( zero_zero_real
        = ( semiri5074537144036343181t_real @ N3 ) )
      = ( zero_zero_nat = N3 ) ) ).

% of_nat_0_eq_iff
thf(fact_3442_of__nat__0__eq__iff,axiom,
    ! [N3: nat] :
      ( ( zero_zero_int
        = ( semiri1314217659103216013at_int @ N3 ) )
      = ( zero_zero_nat = N3 ) ) ).

% of_nat_0_eq_iff
thf(fact_3443_of__nat__0__eq__iff,axiom,
    ! [N3: nat] :
      ( ( zero_zero_nat
        = ( semiri1316708129612266289at_nat @ N3 ) )
      = ( zero_zero_nat = N3 ) ) ).

% of_nat_0_eq_iff
thf(fact_3444_of__nat__0__eq__iff,axiom,
    ! [N3: nat] :
      ( ( zero_zero_complex
        = ( semiri8010041392384452111omplex @ N3 ) )
      = ( zero_zero_nat = N3 ) ) ).

% of_nat_0_eq_iff
thf(fact_3445_of__nat__0__eq__iff,axiom,
    ! [N3: nat] :
      ( ( zero_z3403309356797280102nteger
        = ( semiri4939895301339042750nteger @ N3 ) )
      = ( zero_zero_nat = N3 ) ) ).

% of_nat_0_eq_iff
thf(fact_3446_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_3447_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_3448_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_3449_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_3450_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri8010041392384452111omplex @ M )
        = zero_zero_complex )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_3451_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri4939895301339042750nteger @ M )
        = zero_z3403309356797280102nteger )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_3452_of__nat__numeral,axiom,
    ! [N3: num] :
      ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N3 ) )
      = ( numeral_numeral_rat @ N3 ) ) ).

% of_nat_numeral
thf(fact_3453_of__nat__numeral,axiom,
    ! [N3: num] :
      ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N3 ) )
      = ( numeral_numeral_real @ N3 ) ) ).

% of_nat_numeral
thf(fact_3454_of__nat__numeral,axiom,
    ! [N3: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N3 ) )
      = ( numeral_numeral_int @ N3 ) ) ).

% of_nat_numeral
thf(fact_3455_of__nat__numeral,axiom,
    ! [N3: num] :
      ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N3 ) )
      = ( numeral_numeral_nat @ N3 ) ) ).

% of_nat_numeral
thf(fact_3456_of__nat__numeral,axiom,
    ! [N3: num] :
      ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N3 ) )
      = ( numera6690914467698888265omplex @ N3 ) ) ).

% of_nat_numeral
thf(fact_3457_of__nat__numeral,axiom,
    ! [N3: num] :
      ( ( semiri4939895301339042750nteger @ ( numeral_numeral_nat @ N3 ) )
      = ( numera6620942414471956472nteger @ N3 ) ) ).

% of_nat_numeral
thf(fact_3458_of__nat__less__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N3 ) )
      = ( ord_less_nat @ M @ N3 ) ) ).

% of_nat_less_iff
thf(fact_3459_of__nat__less__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N3 ) )
      = ( ord_less_nat @ M @ N3 ) ) ).

% of_nat_less_iff
thf(fact_3460_of__nat__less__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) )
      = ( ord_less_nat @ M @ N3 ) ) ).

% of_nat_less_iff
thf(fact_3461_of__nat__less__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N3 ) )
      = ( ord_less_nat @ M @ N3 ) ) ).

% of_nat_less_iff
thf(fact_3462_of__nat__less__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N3 ) )
      = ( ord_less_nat @ M @ N3 ) ) ).

% of_nat_less_iff
thf(fact_3463_of__nat__le__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N3 ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% of_nat_le_iff
thf(fact_3464_of__nat__le__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N3 ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% of_nat_le_iff
thf(fact_3465_of__nat__le__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N3 ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% of_nat_le_iff
thf(fact_3466_of__nat__le__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N3 ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% of_nat_le_iff
thf(fact_3467_of__nat__le__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% of_nat_le_iff
thf(fact_3468_of__nat__add,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N3 ) )
      = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N3 ) ) ) ).

% of_nat_add
thf(fact_3469_of__nat__add,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N3 ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ).

% of_nat_add
thf(fact_3470_of__nat__add,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N3 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% of_nat_add
thf(fact_3471_of__nat__add,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N3 ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N3 ) ) ) ).

% of_nat_add
thf(fact_3472_of__nat__add,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N3 ) )
      = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N3 ) ) ) ).

% of_nat_add
thf(fact_3473_of__nat__add,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri4939895301339042750nteger @ ( plus_plus_nat @ M @ N3 ) )
      = ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N3 ) ) ) ).

% of_nat_add
thf(fact_3474_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_3475_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_3476_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_3477_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_3478_of__nat__1,axiom,
    ( ( semiri8010041392384452111omplex @ one_one_nat )
    = one_one_complex ) ).

% of_nat_1
thf(fact_3479_of__nat__1,axiom,
    ( ( semiri4939895301339042750nteger @ one_one_nat )
    = one_one_Code_integer ) ).

% of_nat_1
thf(fact_3480_of__nat__1__eq__iff,axiom,
    ! [N3: nat] :
      ( ( one_one_rat
        = ( semiri681578069525770553at_rat @ N3 ) )
      = ( N3 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_3481_of__nat__1__eq__iff,axiom,
    ! [N3: nat] :
      ( ( one_one_real
        = ( semiri5074537144036343181t_real @ N3 ) )
      = ( N3 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_3482_of__nat__1__eq__iff,axiom,
    ! [N3: nat] :
      ( ( one_one_int
        = ( semiri1314217659103216013at_int @ N3 ) )
      = ( N3 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_3483_of__nat__1__eq__iff,axiom,
    ! [N3: nat] :
      ( ( one_one_nat
        = ( semiri1316708129612266289at_nat @ N3 ) )
      = ( N3 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_3484_of__nat__1__eq__iff,axiom,
    ! [N3: nat] :
      ( ( one_one_complex
        = ( semiri8010041392384452111omplex @ N3 ) )
      = ( N3 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_3485_of__nat__1__eq__iff,axiom,
    ! [N3: nat] :
      ( ( one_one_Code_integer
        = ( semiri4939895301339042750nteger @ N3 ) )
      = ( N3 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_3486_of__nat__eq__1__iff,axiom,
    ! [N3: nat] :
      ( ( ( semiri681578069525770553at_rat @ N3 )
        = one_one_rat )
      = ( N3 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_3487_of__nat__eq__1__iff,axiom,
    ! [N3: nat] :
      ( ( ( semiri5074537144036343181t_real @ N3 )
        = one_one_real )
      = ( N3 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_3488_of__nat__eq__1__iff,axiom,
    ! [N3: nat] :
      ( ( ( semiri1314217659103216013at_int @ N3 )
        = one_one_int )
      = ( N3 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_3489_of__nat__eq__1__iff,axiom,
    ! [N3: nat] :
      ( ( ( semiri1316708129612266289at_nat @ N3 )
        = one_one_nat )
      = ( N3 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_3490_of__nat__eq__1__iff,axiom,
    ! [N3: nat] :
      ( ( ( semiri8010041392384452111omplex @ N3 )
        = one_one_complex )
      = ( N3 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_3491_of__nat__eq__1__iff,axiom,
    ! [N3: nat] :
      ( ( ( semiri4939895301339042750nteger @ N3 )
        = one_one_Code_integer )
      = ( N3 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_3492_of__nat__mult,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N3 ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N3 ) ) ) ).

% of_nat_mult
thf(fact_3493_of__nat__mult,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N3 ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ).

% of_nat_mult
thf(fact_3494_of__nat__mult,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N3 ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% of_nat_mult
thf(fact_3495_of__nat__mult,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N3 ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N3 ) ) ) ).

% of_nat_mult
thf(fact_3496_of__nat__mult,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N3 ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N3 ) ) ) ).

% of_nat_mult
thf(fact_3497_of__nat__mult,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri4939895301339042750nteger @ ( times_times_nat @ M @ N3 ) )
      = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N3 ) ) ) ).

% of_nat_mult
thf(fact_3498_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N3 ) )
      = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N3 ) ) ).

% semiring_1_class.of_nat_power
thf(fact_3499_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N3 ) )
      = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N3 ) ) ).

% semiring_1_class.of_nat_power
thf(fact_3500_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N3 ) )
      = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N3 ) ) ).

% semiring_1_class.of_nat_power
thf(fact_3501_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N3 ) )
      = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N3 ) ) ).

% semiring_1_class.of_nat_power
thf(fact_3502_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri4939895301339042750nteger @ ( power_power_nat @ M @ N3 ) )
      = ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ M ) @ N3 ) ) ).

% semiring_1_class.of_nat_power
thf(fact_3503_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X: nat] :
      ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W )
        = ( semiri5074537144036343181t_real @ X ) )
      = ( ( power_power_nat @ B2 @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_3504_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X: nat] :
      ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W )
        = ( semiri1314217659103216013at_int @ X ) )
      = ( ( power_power_nat @ B2 @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_3505_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X: nat] :
      ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W )
        = ( semiri1316708129612266289at_nat @ X ) )
      = ( ( power_power_nat @ B2 @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_3506_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X: nat] :
      ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B2 ) @ W )
        = ( semiri8010041392384452111omplex @ X ) )
      = ( ( power_power_nat @ B2 @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_3507_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X: nat] :
      ( ( ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B2 ) @ W )
        = ( semiri4939895301339042750nteger @ X ) )
      = ( ( power_power_nat @ B2 @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_3508_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W: nat] :
      ( ( ( semiri5074537144036343181t_real @ X )
        = ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
      = ( X
        = ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_3509_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W: nat] :
      ( ( ( semiri1314217659103216013at_int @ X )
        = ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
      = ( X
        = ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_3510_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W: nat] :
      ( ( ( semiri1316708129612266289at_nat @ X )
        = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
      = ( X
        = ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_3511_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W: nat] :
      ( ( ( semiri8010041392384452111omplex @ X )
        = ( power_power_complex @ ( semiri8010041392384452111omplex @ B2 ) @ W ) )
      = ( X
        = ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_3512_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W: nat] :
      ( ( ( semiri4939895301339042750nteger @ X )
        = ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B2 ) @ W ) )
      = ( X
        = ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_3513_norm__pre__pure__iff__sng,axiom,
    ! [B2: $o,F2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( hoare_hoare_triple_o @ ( pure_assn @ B2 ) @ F2 @ Q )
      = ( B2
       => ( hoare_hoare_triple_o @ one_one_assn @ F2 @ Q ) ) ) ).

% norm_pre_pure_iff_sng
thf(fact_3514_norm__pre__pure__iff__sng,axiom,
    ! [B2: $o,F2: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( hoare_1429296392585015714_VEBTi @ ( pure_assn @ B2 ) @ F2 @ Q )
      = ( B2
       => ( hoare_1429296392585015714_VEBTi @ one_one_assn @ F2 @ Q ) ) ) ).

% norm_pre_pure_iff_sng
thf(fact_3515_norm__pre__pure__iff__sng,axiom,
    ! [B2: $o,F2: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ( hoare_7629718768684598413on_nat @ ( pure_assn @ B2 ) @ F2 @ Q )
      = ( B2
       => ( hoare_7629718768684598413on_nat @ one_one_assn @ F2 @ Q ) ) ) ).

% norm_pre_pure_iff_sng
thf(fact_3516_norm__pre__pure__iff__sng,axiom,
    ! [B2: $o,F2: heap_Time_Heap_nat,Q: nat > assn] :
      ( ( hoare_3067605981109127869le_nat @ ( pure_assn @ B2 ) @ F2 @ Q )
      = ( B2
       => ( hoare_3067605981109127869le_nat @ one_one_assn @ F2 @ Q ) ) ) ).

% norm_pre_pure_iff_sng
thf(fact_3517_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_3518_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_3519_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_3520_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_3521_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_3522_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).

% of_nat_Suc
thf(fact_3523_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).

% of_nat_Suc
thf(fact_3524_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% of_nat_Suc
thf(fact_3525_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).

% of_nat_Suc
thf(fact_3526_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
      = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).

% of_nat_Suc
thf(fact_3527_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri4939895301339042750nteger @ ( suc @ M ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( semiri4939895301339042750nteger @ M ) ) ) ).

% of_nat_Suc
thf(fact_3528_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
      = ( ord_less_int @ W @ Z ) ) ).

% zle_diff1_eq
thf(fact_3529_of__nat__0__less__iff,axiom,
    ! [N3: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N3 ) )
      = ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).

% of_nat_0_less_iff
thf(fact_3530_of__nat__0__less__iff,axiom,
    ! [N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N3 ) )
      = ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).

% of_nat_0_less_iff
thf(fact_3531_of__nat__0__less__iff,axiom,
    ! [N3: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N3 ) )
      = ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).

% of_nat_0_less_iff
thf(fact_3532_of__nat__0__less__iff,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N3 ) )
      = ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).

% of_nat_0_less_iff
thf(fact_3533_of__nat__0__less__iff,axiom,
    ! [N3: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( semiri4939895301339042750nteger @ N3 ) )
      = ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).

% of_nat_0_less_iff
thf(fact_3534_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: nat] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N3 )
        = ( semiri681578069525770553at_rat @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_3535_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: nat] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N3 )
        = ( semiri5074537144036343181t_real @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_3536_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 )
        = ( semiri1314217659103216013at_int @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_3537_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 )
        = ( semiri1316708129612266289at_nat @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_3538_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: nat] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N3 )
        = ( semiri8010041392384452111omplex @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_3539_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N3 )
        = ( semiri4939895301339042750nteger @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_3540_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N3: nat] :
      ( ( ( semiri681578069525770553at_rat @ Y )
        = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N3 ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_3541_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N3: nat] :
      ( ( ( semiri5074537144036343181t_real @ Y )
        = ( power_power_real @ ( numeral_numeral_real @ X ) @ N3 ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_3542_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N3: nat] :
      ( ( ( semiri1314217659103216013at_int @ Y )
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_3543_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N3: nat] :
      ( ( ( semiri1316708129612266289at_nat @ Y )
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_3544_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N3: nat] :
      ( ( ( semiri8010041392384452111omplex @ Y )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N3 ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_3545_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N3: nat] :
      ( ( ( semiri4939895301339042750nteger @ Y )
        = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N3 ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_3546_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_3547_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_3548_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_3549_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_3550_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B2 ) @ W ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B2 @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_3551_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_3552_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_3553_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_3554_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_3555_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B2 ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_3556_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_3557_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B2 ) @ W ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_3558_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_3559_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_3560_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B2 @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_3561_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B2 ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_3562_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B2 ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_3563_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B2 ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_3564_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B2 ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_3565_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B2: nat,W: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B2 ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B2 @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_3566_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,N3: nat] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N3 ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N3 ) ) ).

% numeral_less_real_of_nat_iff
thf(fact_3567_real__of__nat__less__numeral__iff,axiom,
    ! [N3: nat,W: num] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( numeral_numeral_real @ W ) )
      = ( ord_less_nat @ N3 @ ( numeral_numeral_nat @ W ) ) ) ).

% real_of_nat_less_numeral_iff
thf(fact_3568_numeral__le__real__of__nat__iff,axiom,
    ! [N3: num,M: nat] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N3 ) @ ( semiri5074537144036343181t_real @ M ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N3 ) @ M ) ) ).

% numeral_le_real_of_nat_iff
thf(fact_3569_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N3: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N3 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N3 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_3570_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N3 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N3 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_3571_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N3: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N3 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N3 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_3572_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N3 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N3 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_3573_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ X ) @ N3 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N3 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_3574_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N3: nat,X: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N3 ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_3575_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N3: nat,X: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N3 ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_3576_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N3: nat,X: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N3 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_3577_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N3: nat,X: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_3578_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N3: nat,X: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N3 ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_3579_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N3: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N3 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_3580_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N3: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N3 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_3581_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N3: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N3 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_3582_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N3: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_3583_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N3 ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_3584_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N3: nat,X: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N3 ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_3585_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N3: nat,X: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N3 ) @ ( semiri4939895301339042750nteger @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_3586_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N3: nat,X: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N3 ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_3587_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N3: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_3588_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N3: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N3 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_3589_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N3: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N3 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_3590_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N3 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_3591_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N3: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N3 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_3592_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N3: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_3593_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N3: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N3 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N3 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_3594_mult__of__nat__commute,axiom,
    ! [X: nat,Y: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y )
      = ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_3595_mult__of__nat__commute,axiom,
    ! [X: nat,Y: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
      = ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_3596_mult__of__nat__commute,axiom,
    ! [X: nat,Y: int] :
      ( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
      = ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_3597_mult__of__nat__commute,axiom,
    ! [X: nat,Y: nat] :
      ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
      = ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_3598_mult__of__nat__commute,axiom,
    ! [X: nat,Y: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X ) @ Y )
      = ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_3599_mult__of__nat__commute,axiom,
    ! [X: nat,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ X ) @ Y )
      = ( times_3573771949741848930nteger @ Y @ ( semiri4939895301339042750nteger @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_3600_of__nat__0__le__iff,axiom,
    ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N3 ) ) ).

% of_nat_0_le_iff
thf(fact_3601_of__nat__0__le__iff,axiom,
    ! [N3: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( semiri4939895301339042750nteger @ N3 ) ) ).

% of_nat_0_le_iff
thf(fact_3602_of__nat__0__le__iff,axiom,
    ! [N3: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N3 ) ) ).

% of_nat_0_le_iff
thf(fact_3603_of__nat__0__le__iff,axiom,
    ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N3 ) ) ).

% of_nat_0_le_iff
thf(fact_3604_of__nat__0__le__iff,axiom,
    ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N3 ) ) ).

% of_nat_0_le_iff
thf(fact_3605_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).

% of_nat_less_0_iff
thf(fact_3606_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).

% of_nat_less_0_iff
thf(fact_3607_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_3608_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_3609_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ zero_z3403309356797280102nteger ) ).

% of_nat_less_0_iff
thf(fact_3610_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N3: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ N3 ) )
     != zero_zero_rat ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_3611_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N3: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ N3 ) )
     != zero_zero_real ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_3612_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N3: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N3 ) )
     != zero_zero_int ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_3613_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N3: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N3 ) )
     != zero_zero_nat ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_3614_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N3: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ N3 ) )
     != zero_zero_complex ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_3615_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N3: nat] :
      ( ( semiri4939895301339042750nteger @ ( suc @ N3 ) )
     != zero_z3403309356797280102nteger ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_3616_less__imp__of__nat__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N3 ) ) ) ).

% less_imp_of_nat_less
thf(fact_3617_less__imp__of__nat__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ).

% less_imp_of_nat_less
thf(fact_3618_less__imp__of__nat__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% less_imp_of_nat_less
thf(fact_3619_less__imp__of__nat__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N3 ) ) ) ).

% less_imp_of_nat_less
thf(fact_3620_less__imp__of__nat__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N3 ) ) ) ).

% less_imp_of_nat_less
thf(fact_3621_of__nat__less__imp__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N3 ) )
     => ( ord_less_nat @ M @ N3 ) ) ).

% of_nat_less_imp_less
thf(fact_3622_of__nat__less__imp__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N3 ) )
     => ( ord_less_nat @ M @ N3 ) ) ).

% of_nat_less_imp_less
thf(fact_3623_of__nat__less__imp__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) )
     => ( ord_less_nat @ M @ N3 ) ) ).

% of_nat_less_imp_less
thf(fact_3624_of__nat__less__imp__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N3 ) )
     => ( ord_less_nat @ M @ N3 ) ) ).

% of_nat_less_imp_less
thf(fact_3625_of__nat__less__imp__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N3 ) )
     => ( ord_less_nat @ M @ N3 ) ) ).

% of_nat_less_imp_less
thf(fact_3626_div__mult2__eq_H,axiom,
    ! [A2: int,M: nat,N3: nat] :
      ( ( divide_divide_int @ A2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) ) )
      = ( divide_divide_int @ ( divide_divide_int @ A2 @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% div_mult2_eq'
thf(fact_3627_div__mult2__eq_H,axiom,
    ! [A2: nat,M: nat,N3: nat] :
      ( ( divide_divide_nat @ A2 @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N3 ) ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ A2 @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N3 ) ) ) ).

% div_mult2_eq'
thf(fact_3628_div__mult2__eq_H,axiom,
    ! [A2: code_integer,M: nat,N3: nat] :
      ( ( divide6298287555418463151nteger @ A2 @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N3 ) ) )
      = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A2 @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N3 ) ) ) ).

% div_mult2_eq'
thf(fact_3629_of__nat__mono,axiom,
    ! [I: nat,J2: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J2 ) ) ) ).

% of_nat_mono
thf(fact_3630_of__nat__mono,axiom,
    ! [I: nat,J2: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ I ) @ ( semiri4939895301339042750nteger @ J2 ) ) ) ).

% of_nat_mono
thf(fact_3631_of__nat__mono,axiom,
    ! [I: nat,J2: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J2 ) ) ) ).

% of_nat_mono
thf(fact_3632_of__nat__mono,axiom,
    ! [I: nat,J2: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J2 ) ) ) ).

% of_nat_mono
thf(fact_3633_of__nat__mono,axiom,
    ! [I: nat,J2: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J2 ) ) ) ).

% of_nat_mono
thf(fact_3634_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N3 ) )
      = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_3635_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N3 ) )
      = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N3 ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_3636_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri4939895301339042750nteger @ ( divide_divide_nat @ M @ N3 ) )
      = ( divide6298287555418463151nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N3 ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_3637_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ Y ) ) ) ).

% of_nat_max
thf(fact_3638_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).

% of_nat_max
thf(fact_3639_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).

% of_nat_max
thf(fact_3640_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri4939895301339042750nteger @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_Code_integer @ ( semiri4939895301339042750nteger @ X ) @ ( semiri4939895301339042750nteger @ Y ) ) ) ).

% of_nat_max
thf(fact_3641_of__nat__diff,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N3 ) )
        = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N3 ) ) ) ) ).

% of_nat_diff
thf(fact_3642_of__nat__diff,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N3 ) )
        = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ).

% of_nat_diff
thf(fact_3643_of__nat__diff,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N3 ) )
        = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).

% of_nat_diff
thf(fact_3644_of__nat__diff,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N3 ) )
        = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N3 ) ) ) ) ).

% of_nat_diff
thf(fact_3645_of__nat__diff,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( semiri8010041392384452111omplex @ ( minus_minus_nat @ M @ N3 ) )
        = ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N3 ) ) ) ) ).

% of_nat_diff
thf(fact_3646_of__nat__diff,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( semiri4939895301339042750nteger @ ( minus_minus_nat @ M @ N3 ) )
        = ( minus_8373710615458151222nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N3 ) ) ) ) ).

% of_nat_diff
thf(fact_3647_reals__Archimedean3,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ! [Y4: real] :
        ? [N: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) ) ).

% reals_Archimedean3
thf(fact_3648_real__of__nat__div4,axiom,
    ! [N3: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N3 @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).

% real_of_nat_div4
thf(fact_3649_minus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( minus_minus_int @ K @ zero_zero_int )
      = K ) ).

% minus_int_code(1)
thf(fact_3650_int__le__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_eq_int @ I @ K )
     => ( ( P @ K )
       => ( ! [I5: int] :
              ( ( ord_less_eq_int @ I5 @ K )
             => ( ( P @ I5 )
               => ( P @ ( minus_minus_int @ I5 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_le_induct
thf(fact_3651_int__less__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_int @ I @ K )
     => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
       => ( ! [I5: int] :
              ( ( ord_less_int @ I5 @ K )
             => ( ( P @ I5 )
               => ( P @ ( minus_minus_int @ I5 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_less_induct
thf(fact_3652_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_3653_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_3654_nat__less__real__le,axiom,
    ( ord_less_nat
    = ( ^ [N2: nat,M5: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M5 ) ) ) ) ).

% nat_less_real_le
thf(fact_3655_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N2: nat,M5: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M5 ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_3656_of__nat__less__two__power,axiom,
    ! [N3: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N3 ) ) ).

% of_nat_less_two_power
thf(fact_3657_of__nat__less__two__power,axiom,
    ! [N3: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) ) ).

% of_nat_less_two_power
thf(fact_3658_of__nat__less__two__power,axiom,
    ! [N3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ).

% of_nat_less_two_power
thf(fact_3659_of__nat__less__two__power,axiom,
    ! [N3: nat] : ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ N3 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ).

% of_nat_less_two_power
thf(fact_3660_inverse__of__nat__le,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( N3 != zero_zero_nat )
       => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_3661_inverse__of__nat__le,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( N3 != zero_zero_nat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N3 ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_3662_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
       => ( ! [M4: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ M4 )
             => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X ) @ C2 ) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
thf(fact_3663_real__of__nat__div2,axiom,
    ! [N3: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N3 @ X ) ) ) ) ).

% real_of_nat_div2
thf(fact_3664_real__of__nat__div3,axiom,
    ! [N3: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N3 @ X ) ) ) @ one_one_real ) ).

% real_of_nat_div3
thf(fact_3665_less__int__code_I1_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_int_code(1)
thf(fact_3666_int__induct,axiom,
    ! [P: int > $o,K: int,I: int] :
      ( ( P @ K )
     => ( ! [I5: int] :
            ( ( ord_less_eq_int @ K @ I5 )
           => ( ( P @ I5 )
             => ( P @ ( plus_plus_int @ I5 @ one_one_int ) ) ) )
       => ( ! [I5: int] :
              ( ( ord_less_eq_int @ I5 @ K )
             => ( ( P @ I5 )
               => ( P @ ( minus_minus_int @ I5 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_induct
thf(fact_3667_int__gr__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_int @ K @ I )
     => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
       => ( ! [I5: int] :
              ( ( ord_less_int @ K @ I5 )
             => ( ( P @ I5 )
               => ( P @ ( plus_plus_int @ I5 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_gr_induct
thf(fact_3668_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
      = ( ( ord_less_int @ W @ Z )
        | ( W = Z ) ) ) ).

% zless_add1_eq
thf(fact_3669_cons__post__rule,axiom,
    ! [P: assn,C2: heap_Time_Heap_o,Q: $o > assn,Q2: $o > assn] :
      ( ( hoare_hoare_triple_o @ P @ C2 @ Q )
     => ( ! [X4: $o] : ( entails @ ( Q @ X4 ) @ ( Q2 @ X4 ) )
       => ( hoare_hoare_triple_o @ P @ C2 @ Q2 ) ) ) ).

% cons_post_rule
thf(fact_3670_cons__post__rule,axiom,
    ! [P: assn,C2: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,Q2: vEBT_VEBTi > assn] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C2 @ Q )
     => ( ! [X4: vEBT_VEBTi] : ( entails @ ( Q @ X4 ) @ ( Q2 @ X4 ) )
       => ( hoare_1429296392585015714_VEBTi @ P @ C2 @ Q2 ) ) ) ).

% cons_post_rule
thf(fact_3671_cons__post__rule,axiom,
    ! [P: assn,C2: heap_T2636463487746394924on_nat,Q: option_nat > assn,Q2: option_nat > assn] :
      ( ( hoare_7629718768684598413on_nat @ P @ C2 @ Q )
     => ( ! [X4: option_nat] : ( entails @ ( Q @ X4 ) @ ( Q2 @ X4 ) )
       => ( hoare_7629718768684598413on_nat @ P @ C2 @ Q2 ) ) ) ).

% cons_post_rule
thf(fact_3672_cons__post__rule,axiom,
    ! [P: assn,C2: heap_Time_Heap_nat,Q: nat > assn,Q2: nat > assn] :
      ( ( hoare_3067605981109127869le_nat @ P @ C2 @ Q )
     => ( ! [X4: nat] : ( entails @ ( Q @ X4 ) @ ( Q2 @ X4 ) )
       => ( hoare_3067605981109127869le_nat @ P @ C2 @ Q2 ) ) ) ).

% cons_post_rule
thf(fact_3673_cons__rule,axiom,
    ! [P: assn,P2: assn,Q: $o > assn,Q2: $o > assn,C2: heap_Time_Heap_o] :
      ( ( entails @ P @ P2 )
     => ( ! [X4: $o] : ( entails @ ( Q @ X4 ) @ ( Q2 @ X4 ) )
       => ( ( hoare_hoare_triple_o @ P2 @ C2 @ Q )
         => ( hoare_hoare_triple_o @ P @ C2 @ Q2 ) ) ) ) ).

% cons_rule
thf(fact_3674_cons__rule,axiom,
    ! [P: assn,P2: assn,Q: vEBT_VEBTi > assn,Q2: vEBT_VEBTi > assn,C2: heap_T8145700208782473153_VEBTi] :
      ( ( entails @ P @ P2 )
     => ( ! [X4: vEBT_VEBTi] : ( entails @ ( Q @ X4 ) @ ( Q2 @ X4 ) )
       => ( ( hoare_1429296392585015714_VEBTi @ P2 @ C2 @ Q )
         => ( hoare_1429296392585015714_VEBTi @ P @ C2 @ Q2 ) ) ) ) ).

% cons_rule
thf(fact_3675_cons__rule,axiom,
    ! [P: assn,P2: assn,Q: option_nat > assn,Q2: option_nat > assn,C2: heap_T2636463487746394924on_nat] :
      ( ( entails @ P @ P2 )
     => ( ! [X4: option_nat] : ( entails @ ( Q @ X4 ) @ ( Q2 @ X4 ) )
       => ( ( hoare_7629718768684598413on_nat @ P2 @ C2 @ Q )
         => ( hoare_7629718768684598413on_nat @ P @ C2 @ Q2 ) ) ) ) ).

% cons_rule
thf(fact_3676_cons__rule,axiom,
    ! [P: assn,P2: assn,Q: nat > assn,Q2: nat > assn,C2: heap_Time_Heap_nat] :
      ( ( entails @ P @ P2 )
     => ( ! [X4: nat] : ( entails @ ( Q @ X4 ) @ ( Q2 @ X4 ) )
       => ( ( hoare_3067605981109127869le_nat @ P2 @ C2 @ Q )
         => ( hoare_3067605981109127869le_nat @ P @ C2 @ Q2 ) ) ) ) ).

% cons_rule
thf(fact_3677_norm__pre__pure__rule2,axiom,
    ! [B2: $o,F2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( B2
       => ( hoare_hoare_triple_o @ one_one_assn @ F2 @ Q ) )
     => ( hoare_hoare_triple_o @ ( pure_assn @ B2 ) @ F2 @ Q ) ) ).

% norm_pre_pure_rule2
thf(fact_3678_norm__pre__pure__rule2,axiom,
    ! [B2: $o,F2: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( B2
       => ( hoare_1429296392585015714_VEBTi @ one_one_assn @ F2 @ Q ) )
     => ( hoare_1429296392585015714_VEBTi @ ( pure_assn @ B2 ) @ F2 @ Q ) ) ).

% norm_pre_pure_rule2
thf(fact_3679_norm__pre__pure__rule2,axiom,
    ! [B2: $o,F2: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ( B2
       => ( hoare_7629718768684598413on_nat @ one_one_assn @ F2 @ Q ) )
     => ( hoare_7629718768684598413on_nat @ ( pure_assn @ B2 ) @ F2 @ Q ) ) ).

% norm_pre_pure_rule2
thf(fact_3680_norm__pre__pure__rule2,axiom,
    ! [B2: $o,F2: heap_Time_Heap_nat,Q: nat > assn] :
      ( ( B2
       => ( hoare_3067605981109127869le_nat @ one_one_assn @ F2 @ Q ) )
     => ( hoare_3067605981109127869le_nat @ ( pure_assn @ B2 ) @ F2 @ Q ) ) ).

% norm_pre_pure_rule2
thf(fact_3681_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ one_one_int @ Z )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% int_one_le_iff_zero_less
thf(fact_3682_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% odd_less_0_iff
thf(fact_3683_zmult__zless__mono2,axiom,
    ! [I: int,J2: int,K: int] :
      ( ( ord_less_int @ I @ J2 )
     => ( ( ord_less_int @ zero_zero_int @ K )
       => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J2 ) ) ) ) ).

% zmult_zless_mono2
thf(fact_3684_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N3: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ( times_times_int @ M @ N3 )
          = one_one_int )
        = ( ( M = one_one_int )
          & ( N3 = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
thf(fact_3685_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
      = ( ord_less_int @ W @ Z ) ) ).

% add1_zle_eq
thf(fact_3686_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ W @ Z )
     => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).

% zless_imp_add1_zle
thf(fact_3687_norm__pre__pure__rule1,axiom,
    ! [B2: $o,P: assn,F2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( B2
       => ( hoare_hoare_triple_o @ P @ F2 @ Q ) )
     => ( hoare_hoare_triple_o @ ( times_times_assn @ P @ ( pure_assn @ B2 ) ) @ F2 @ Q ) ) ).

% norm_pre_pure_rule1
thf(fact_3688_norm__pre__pure__rule1,axiom,
    ! [B2: $o,P: assn,F2: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( B2
       => ( hoare_1429296392585015714_VEBTi @ P @ F2 @ Q ) )
     => ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ P @ ( pure_assn @ B2 ) ) @ F2 @ Q ) ) ).

% norm_pre_pure_rule1
thf(fact_3689_norm__pre__pure__rule1,axiom,
    ! [B2: $o,P: assn,F2: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ( B2
       => ( hoare_7629718768684598413on_nat @ P @ F2 @ Q ) )
     => ( hoare_7629718768684598413on_nat @ ( times_times_assn @ P @ ( pure_assn @ B2 ) ) @ F2 @ Q ) ) ).

% norm_pre_pure_rule1
thf(fact_3690_norm__pre__pure__rule1,axiom,
    ! [B2: $o,P: assn,F2: heap_Time_Heap_nat,Q: nat > assn] :
      ( ( B2
       => ( hoare_3067605981109127869le_nat @ P @ F2 @ Q ) )
     => ( hoare_3067605981109127869le_nat @ ( times_times_assn @ P @ ( pure_assn @ B2 ) ) @ F2 @ Q ) ) ).

% norm_pre_pure_rule1
thf(fact_3691_le__imp__0__less,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).

% le_imp_0_less
thf(fact_3692_Tb_H__cnt,axiom,
    ! [N3: nat] : ( ord_less_eq_nat @ ( vEBT_VEBT_Tb2 @ N3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_cnt2 @ ( vEBT_vebt_buildup @ N3 ) ) ) ) ).

% Tb'_cnt
thf(fact_3693_cnt__cnt__eq,axiom,
    ( vEBT_VEBT_cnt
    = ( ^ [T2: vEBT_VEBT] : ( semiri5074537144036343181t_real @ ( vEBT_VEBT_cnt2 @ T2 ) ) ) ) ).

% cnt_cnt_eq
thf(fact_3694_t__build__cnt,axiom,
    ! [N3: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ N3 ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N3 ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ).

% t_build_cnt
thf(fact_3695_height__node,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N3 )
     => ( ord_less_eq_nat @ one_one_nat @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% height_node
thf(fact_3696_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ TreeList @ Summary ) @ X )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
thf(fact_3697_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ TreeList @ Summary ) @ X )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
thf(fact_3698_linear__plus__1__le__power,axiom,
    ! [X: real,N3: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N3 ) ) ) ).

% linear_plus_1_le_power
thf(fact_3699_delete__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N3: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_d_e_l_e_t_e @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).

% delete_bound_size_univ
thf(fact_3700_buildup__build__time,axiom,
    ! [N3: nat] : ( ord_less_nat @ ( vEBT_V8346862874174094_d_u_p @ N3 ) @ ( vEBT_V8646137997579335489_i_l_d @ N3 ) ) ).

% buildup_build_time
thf(fact_3701_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_3702_delete__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N3: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V1232361888498592333_e_t_e @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ).

% delete_bound_size_univ'
thf(fact_3703_height__double__log__univ__size,axiom,
    ! [U: real,Deg: nat,T: vEBT_VEBT] :
      ( ( U
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Deg ) )
     => ( ( vEBT_invar_vebt @ T @ Deg )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_VEBT_height @ T ) ) @ ( plus_plus_real @ one_one_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ).

% height_double_log_univ_size
thf(fact_3704_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M4: nat,N: nat] :
          ( Z
         != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% int_diff_cases
thf(fact_3705_zle__int,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% zle_int
thf(fact_3706_zadd__int__left,axiom,
    ! [M: nat,N3: nat,Z: int] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N3 ) @ Z ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N3 ) ) @ Z ) ) ).

% zadd_int_left
thf(fact_3707_zdiv__int,axiom,
    ! [A2: nat,B2: nat] :
      ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A2 @ B2 ) )
      = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).

% zdiv_int
thf(fact_3708_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W2: int,Z5: int] :
        ? [N2: nat] :
          ( Z5
          = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_3709_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ? [N: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N )
          & ( K
            = ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_3710_pos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ~ ! [N: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% pos_int_cases
thf(fact_3711_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J2: int,K: nat] :
      ( ( ord_less_int @ I @ J2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J2 ) ) ) ) ).

% zmult_zless_mono2_lemma
thf(fact_3712_Bolzano,axiom,
    ! [A2: real,B2: real,P: real > real > $o] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ! [A3: real,B3: real,C4: real] :
            ( ( P @ A3 @ B3 )
           => ( ( P @ B3 @ C4 )
             => ( ( ord_less_eq_real @ A3 @ B3 )
               => ( ( ord_less_eq_real @ B3 @ C4 )
                 => ( P @ A3 @ C4 ) ) ) ) )
       => ( ! [X4: real] :
              ( ( ord_less_eq_real @ A2 @ X4 )
             => ( ( ord_less_eq_real @ X4 @ B2 )
               => ? [D4: real] :
                    ( ( ord_less_real @ zero_zero_real @ D4 )
                    & ! [A3: real,B3: real] :
                        ( ( ( ord_less_eq_real @ A3 @ X4 )
                          & ( ord_less_eq_real @ X4 @ B3 )
                          & ( ord_less_real @ ( minus_minus_real @ B3 @ A3 ) @ D4 ) )
                       => ( P @ A3 @ B3 ) ) ) ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% Bolzano
thf(fact_3713_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Uu )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
thf(fact_3714_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Uu )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(4)
thf(fact_3715_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TreeList @ Summary ) @ X )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
thf(fact_3716_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TreeList @ Summary ) @ X )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
thf(fact_3717_log__pow__cancel,axiom,
    ! [A2: real,B2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( A2 != one_one_real )
       => ( ( log @ A2 @ ( power_power_real @ A2 @ B2 ) )
          = ( semiri5074537144036343181t_real @ B2 ) ) ) ) ).

% log_pow_cancel
thf(fact_3718_Tb__T__vebt__buildupi,axiom,
    ! [N3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( vEBT_V441764108873111860ildupi @ N3 ) ) @ ( minus_minus_int @ ( vEBT_VEBT_Tb @ N3 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% Tb_T_vebt_buildupi
thf(fact_3719_zero__le__log__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A2 @ X ) )
          = ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).

% zero_le_log_cancel_iff
thf(fact_3720_log__le__zero__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( log @ A2 @ X ) @ zero_zero_real )
          = ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).

% log_le_zero_cancel_iff
thf(fact_3721_one__le__log__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ one_one_real @ ( log @ A2 @ X ) )
          = ( ord_less_eq_real @ A2 @ X ) ) ) ) ).

% one_le_log_cancel_iff
thf(fact_3722_log__le__one__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( log @ A2 @ X ) @ one_one_real )
          = ( ord_less_eq_real @ X @ A2 ) ) ) ) ).

% log_le_one_cancel_iff
thf(fact_3723_log__le__cancel__iff,axiom,
    ! [A2: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ ( log @ A2 @ X ) @ ( log @ A2 @ Y ) )
            = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).

% log_le_cancel_iff
thf(fact_3724_log2__of__power__le,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ).

% log2_of_power_le
thf(fact_3725_Tb__Tb_H,axiom,
    ( vEBT_VEBT_Tb
    = ( ^ [T2: nat] : ( semiri1314217659103216013at_int @ ( vEBT_VEBT_Tb2 @ T2 ) ) ) ) ).

% Tb_Tb'
thf(fact_3726_log__one,axiom,
    ! [A2: real] :
      ( ( log @ A2 @ one_one_real )
      = zero_zero_real ) ).

% log_one
thf(fact_3727_log__eq__one,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( A2 != one_one_real )
       => ( ( log @ A2 @ A2 )
          = one_one_real ) ) ) ).

% log_eq_one
thf(fact_3728_log__less__cancel__iff,axiom,
    ! [A2: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ zero_zero_real @ Y )
         => ( ( ord_less_real @ ( log @ A2 @ X ) @ ( log @ A2 @ Y ) )
            = ( ord_less_real @ X @ Y ) ) ) ) ) ).

% log_less_cancel_iff
thf(fact_3729_log__less__one__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( log @ A2 @ X ) @ one_one_real )
          = ( ord_less_real @ X @ A2 ) ) ) ) ).

% log_less_one_cancel_iff
thf(fact_3730_one__less__log__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ one_one_real @ ( log @ A2 @ X ) )
          = ( ord_less_real @ A2 @ X ) ) ) ) ).

% one_less_log_cancel_iff
thf(fact_3731_log__less__zero__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( log @ A2 @ X ) @ zero_zero_real )
          = ( ord_less_real @ X @ one_one_real ) ) ) ) ).

% log_less_zero_cancel_iff
thf(fact_3732_zero__less__log__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ zero_zero_real @ ( log @ A2 @ X ) )
          = ( ord_less_real @ one_one_real @ X ) ) ) ) ).

% zero_less_log_cancel_iff
thf(fact_3733_log__base__change,axiom,
    ! [A2: real,B2: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( A2 != one_one_real )
       => ( ( log @ B2 @ X )
          = ( divide_divide_real @ ( log @ A2 @ X ) @ ( log @ A2 @ B2 ) ) ) ) ) ).

% log_base_change
thf(fact_3734_less__log__of__power,axiom,
    ! [B2: real,N3: nat,M: real] :
      ( ( ord_less_real @ ( power_power_real @ B2 @ N3 ) @ M )
     => ( ( ord_less_real @ one_one_real @ B2 )
       => ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( log @ B2 @ M ) ) ) ) ).

% less_log_of_power
thf(fact_3735_log__of__power__eq,axiom,
    ! [M: nat,B2: real,N3: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( power_power_real @ B2 @ N3 ) )
     => ( ( ord_less_real @ one_one_real @ B2 )
       => ( ( semiri5074537144036343181t_real @ N3 )
          = ( log @ B2 @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).

% log_of_power_eq
thf(fact_3736_log__mult,axiom,
    ! [A2: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( A2 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ zero_zero_real @ Y )
           => ( ( log @ A2 @ ( times_times_real @ X @ Y ) )
              = ( plus_plus_real @ ( log @ A2 @ X ) @ ( log @ A2 @ Y ) ) ) ) ) ) ) ).

% log_mult
thf(fact_3737_le__log__of__power,axiom,
    ! [B2: real,N3: nat,M: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ B2 @ N3 ) @ M )
     => ( ( ord_less_real @ one_one_real @ B2 )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( log @ B2 @ M ) ) ) ) ).

% le_log_of_power
thf(fact_3738_log__divide,axiom,
    ! [A2: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( A2 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ zero_zero_real @ Y )
           => ( ( log @ A2 @ ( divide_divide_real @ X @ Y ) )
              = ( minus_minus_real @ ( log @ A2 @ X ) @ ( log @ A2 @ Y ) ) ) ) ) ) ) ).

% log_divide
thf(fact_3739_log__base__pow,axiom,
    ! [A2: real,N3: nat,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( log @ ( power_power_real @ A2 @ N3 ) @ X )
        = ( divide_divide_real @ ( log @ A2 @ X ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ).

% log_base_pow
thf(fact_3740_log__nat__power,axiom,
    ! [X: real,B2: real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( log @ B2 @ ( power_power_real @ X @ N3 ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( log @ B2 @ X ) ) ) ) ).

% log_nat_power
thf(fact_3741_log2__of__power__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( M
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
     => ( ( semiri5074537144036343181t_real @ N3 )
        = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% log2_of_power_eq
thf(fact_3742_log__of__power__less,axiom,
    ! [M: nat,B2: real,N3: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B2 @ N3 ) )
     => ( ( ord_less_real @ one_one_real @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_real @ ( log @ B2 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ).

% log_of_power_less
thf(fact_3743_log__of__power__le,axiom,
    ! [M: nat,B2: real,N3: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B2 @ N3 ) )
     => ( ( ord_less_real @ one_one_real @ B2 )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_eq_real @ ( log @ B2 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ).

% log_of_power_le
thf(fact_3744_less__log2__of__power,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ M )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% less_log2_of_power
thf(fact_3745_le__log2__of__power,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ M )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% le_log2_of_power
thf(fact_3746_log2__of__power__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ).

% log2_of_power_less
thf(fact_3747_Tb__T__vebt__buildupi_H,axiom,
    ! [N3: nat] : ( ord_less_eq_int @ ( vEBT_V9176841429113362141ildupi @ N3 ) @ ( minus_minus_int @ ( vEBT_VEBT_Tb @ N3 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% Tb_T_vebt_buildupi'
thf(fact_3748_Tbuildupi__buildupi_H,axiom,
    ! [N3: nat] :
      ( ( semiri1314217659103216013at_int @ ( vEBT_V441764108873111860ildupi @ N3 ) )
      = ( vEBT_V9176841429113362141ildupi @ N3 ) ) ).

% Tbuildupi_buildupi'
thf(fact_3749_arcosh__1,axiom,
    ( ( arcosh_real @ one_one_real )
    = zero_zero_real ) ).

% arcosh_1
thf(fact_3750_setprop,axiom,
    ! [T: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ T @ ( set_VEBT_VEBT2 @ treeList ) )
     => ( vEBT_invar_vebt @ T @ ( divide_divide_nat @ na @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% setprop
thf(fact_3751_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_3752_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_3753_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_3754_Leaf__0__not,axiom,
    ! [A2: $o,B2: $o] :
      ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B2 ) @ zero_zero_nat ) ).

% Leaf_0_not
thf(fact_3755_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
      = ( ? [A7: $o,B7: $o] :
            ( T
            = ( vEBT_Leaf @ A7 @ B7 ) ) ) ) ).

% deg1Leaf
thf(fact_3756_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
     => ? [A3: $o,B3: $o] :
          ( T
          = ( vEBT_Leaf @ A3 @ B3 ) ) ) ).

% deg_1_Leaf
thf(fact_3757_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( N3 = one_one_nat )
       => ? [A3: $o,B3: $o] :
            ( T
            = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ).

% deg_1_Leafy
thf(fact_3758_inthall,axiom,
    ! [Xs2: list_int,P: int > $o,N3: nat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N3 @ ( size_size_list_int @ Xs2 ) )
       => ( P @ ( nth_int @ Xs2 @ N3 ) ) ) ) ).

% inthall
thf(fact_3759_inthall,axiom,
    ! [Xs2: list_complex,P: complex > $o,N3: nat] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ ( set_complex2 @ Xs2 ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N3 @ ( size_s3451745648224563538omplex @ Xs2 ) )
       => ( P @ ( nth_complex @ Xs2 @ N3 ) ) ) ) ).

% inthall
thf(fact_3760_inthall,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,N3: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N3 ) ) ) ) ).

% inthall
thf(fact_3761_inthall,axiom,
    ! [Xs2: list_real,P: real > $o,N3: nat] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N3 @ ( size_size_list_real @ Xs2 ) )
       => ( P @ ( nth_real @ Xs2 @ N3 ) ) ) ) ).

% inthall
thf(fact_3762_inthall,axiom,
    ! [Xs2: list_o,P: $o > $o,N3: nat] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N3 @ ( size_size_list_o @ Xs2 ) )
       => ( P @ ( nth_o @ Xs2 @ N3 ) ) ) ) ).

% inthall
thf(fact_3763_inthall,axiom,
    ! [Xs2: list_nat,P: nat > $o,N3: nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs2 ) )
       => ( P @ ( nth_nat @ Xs2 @ N3 ) ) ) ) ).

% inthall
thf(fact_3764_inthall,axiom,
    ! [Xs2: list_VEBT_VEBTi,P: vEBT_VEBTi > $o,N3: nat] :
      ( ! [X4: vEBT_VEBTi] :
          ( ( member_VEBT_VEBTi @ X4 @ ( set_VEBT_VEBTi2 @ Xs2 ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N3 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( P @ ( nth_VEBT_VEBTi @ Xs2 @ N3 ) ) ) ) ).

% inthall
thf(fact_3765_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X222: $o,Y21: $o,Y222: $o] :
      ( ( ( vEBT_Leaf @ X21 @ X222 )
        = ( vEBT_Leaf @ Y21 @ Y222 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% VEBT.inject(2)
thf(fact_3766_height__compose__child,axiom,
    ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Info: option4927543243414619207at_nat,Deg: nat,Summary: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ T @ ( set_VEBT_VEBT2 @ TreeList ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ) ).

% height_compose_child
thf(fact_3767_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 )
       => ( ! [X5: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
             => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
          & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).

% mi_eq_ma_no_ch
thf(fact_3768_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,N3: nat,M: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X4 @ N3 ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
       => ( ord_less_eq_nat @ one_one_nat @ N3 ) ) ) ).

% set_n_deg_not_0
thf(fact_3769_set__swap,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,J2: nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ord_less_nat @ J2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( nth_VEBT_VEBT @ Xs2 @ J2 ) ) @ J2 @ ( nth_VEBT_VEBT @ Xs2 @ I ) ) )
          = ( set_VEBT_VEBT2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_3770_set__swap,axiom,
    ! [I: nat,Xs2: list_real,J2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
     => ( ( ord_less_nat @ J2 @ ( size_size_list_real @ Xs2 ) )
       => ( ( set_real2 @ ( list_update_real @ ( list_update_real @ Xs2 @ I @ ( nth_real @ Xs2 @ J2 ) ) @ J2 @ ( nth_real @ Xs2 @ I ) ) )
          = ( set_real2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_3771_set__swap,axiom,
    ! [I: nat,Xs2: list_o,J2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( ord_less_nat @ J2 @ ( size_size_list_o @ Xs2 ) )
       => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs2 @ I @ ( nth_o @ Xs2 @ J2 ) ) @ J2 @ ( nth_o @ Xs2 @ I ) ) )
          = ( set_o2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_3772_set__swap,axiom,
    ! [I: nat,Xs2: list_nat,J2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
       => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ J2 ) ) @ J2 @ ( nth_nat @ Xs2 @ I ) ) )
          = ( set_nat2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_3773_set__swap,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,J2: nat] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( ord_less_nat @ J2 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ ( nth_VEBT_VEBTi @ Xs2 @ J2 ) ) @ J2 @ ( nth_VEBT_VEBTi @ Xs2 @ I ) ) )
          = ( set_VEBT_VEBTi2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_3774_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] :
      ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
      = zero_zero_nat ) ).

% VEBT.size(4)
thf(fact_3775_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv: $o,D5: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ D5 ) )
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Deg3 ) ) ) ).

% VEBT_internal.valid'.cases
thf(fact_3776_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
          ( Y
         != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
     => ~ ! [X212: $o,X223: $o] :
            ( Y
           != ( vEBT_Leaf @ X212 @ X223 ) ) ) ).

% VEBT.exhaust
thf(fact_3777_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
      ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
     != ( vEBT_Leaf @ X21 @ X222 ) ) ).

% VEBT.distinct(1)
thf(fact_3778_subset__code_I1_J,axiom,
    ! [Xs2: list_int,B: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ B )
      = ( ! [X3: int] :
            ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
           => ( member_int @ X3 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_3779_subset__code_I1_J,axiom,
    ! [Xs2: list_complex,B: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ B )
      = ( ! [X3: complex] :
            ( ( member_complex @ X3 @ ( set_complex2 @ Xs2 ) )
           => ( member_complex @ X3 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_3780_subset__code_I1_J,axiom,
    ! [Xs2: list_VEBT_VEBT,B: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ B )
      = ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( member_VEBT_VEBT @ X3 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_3781_subset__code_I1_J,axiom,
    ! [Xs2: list_real,B: set_real] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ B )
      = ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
           => ( member_real @ X3 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_3782_subset__code_I1_J,axiom,
    ! [Xs2: list_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
           => ( member_nat @ X3 @ B ) ) ) ) ).

% subset_code(1)
thf(fact_3783_list__assn__cong,axiom,
    ! [Xs2: list_int,Xs4: list_int,Xsi: list_int,Xsi2: list_int,A: int > int > assn,A5: int > int > assn] :
      ( ( Xs2 = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X4: int,Xi2: int] :
              ( ( member_int @ X4 @ ( set_int2 @ Xs4 ) )
             => ( ( member_int @ Xi2 @ ( set_int2 @ Xsi2 ) )
               => ( ( A @ X4 @ Xi2 )
                  = ( A5 @ X4 @ Xi2 ) ) ) )
         => ( ( vEBT_L74593716426352029nt_int @ A @ Xs2 @ Xsi )
            = ( vEBT_L74593716426352029nt_int @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_3784_list__assn__cong,axiom,
    ! [Xs2: list_int,Xs4: list_int,Xsi: list_complex,Xsi2: list_complex,A: int > complex > assn,A5: int > complex > assn] :
      ( ( Xs2 = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X4: int,Xi2: complex] :
              ( ( member_int @ X4 @ ( set_int2 @ Xs4 ) )
             => ( ( member_complex @ Xi2 @ ( set_complex2 @ Xsi2 ) )
               => ( ( A @ X4 @ Xi2 )
                  = ( A5 @ X4 @ Xi2 ) ) ) )
         => ( ( vEBT_L6716599654974302751omplex @ A @ Xs2 @ Xsi )
            = ( vEBT_L6716599654974302751omplex @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_3785_list__assn__cong,axiom,
    ! [Xs2: list_complex,Xs4: list_complex,Xsi: list_int,Xsi2: list_int,A: complex > int > assn,A5: complex > int > assn] :
      ( ( Xs2 = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X4: complex,Xi2: int] :
              ( ( member_complex @ X4 @ ( set_complex2 @ Xs4 ) )
             => ( ( member_int @ Xi2 @ ( set_int2 @ Xsi2 ) )
               => ( ( A @ X4 @ Xi2 )
                  = ( A5 @ X4 @ Xi2 ) ) ) )
         => ( ( vEBT_L134985006839036959ex_int @ A @ Xs2 @ Xsi )
            = ( vEBT_L134985006839036959ex_int @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_3786_list__assn__cong,axiom,
    ! [Xs2: list_complex,Xs4: list_complex,Xsi: list_complex,Xsi2: list_complex,A: complex > complex > assn,A5: complex > complex > assn] :
      ( ( Xs2 = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X4: complex,Xi2: complex] :
              ( ( member_complex @ X4 @ ( set_complex2 @ Xs4 ) )
             => ( ( member_complex @ Xi2 @ ( set_complex2 @ Xsi2 ) )
               => ( ( A @ X4 @ Xi2 )
                  = ( A5 @ X4 @ Xi2 ) ) ) )
         => ( ( vEBT_L4260503343685368993omplex @ A @ Xs2 @ Xsi )
            = ( vEBT_L4260503343685368993omplex @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_3787_list__assn__cong,axiom,
    ! [Xs2: list_int,Xs4: list_int,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A: int > vEBT_VEBT > assn,A5: int > vEBT_VEBT > assn] :
      ( ( Xs2 = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X4: int,Xi2: vEBT_VEBT] :
              ( ( member_int @ X4 @ ( set_int2 @ Xs4 ) )
             => ( ( member_VEBT_VEBT @ Xi2 @ ( set_VEBT_VEBT2 @ Xsi2 ) )
               => ( ( A @ X4 @ Xi2 )
                  = ( A5 @ X4 @ Xi2 ) ) ) )
         => ( ( vEBT_L1664421287176695555T_VEBT @ A @ Xs2 @ Xsi )
            = ( vEBT_L1664421287176695555T_VEBT @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_3788_list__assn__cong,axiom,
    ! [Xs2: list_complex,Xs4: list_complex,Xsi: list_VEBT_VEBT,Xsi2: list_VEBT_VEBT,A: complex > vEBT_VEBT > assn,A5: complex > vEBT_VEBT > assn] :
      ( ( Xs2 = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X4: complex,Xi2: vEBT_VEBT] :
              ( ( member_complex @ X4 @ ( set_complex2 @ Xs4 ) )
             => ( ( member_VEBT_VEBT @ Xi2 @ ( set_VEBT_VEBT2 @ Xsi2 ) )
               => ( ( A @ X4 @ Xi2 )
                  = ( A5 @ X4 @ Xi2 ) ) ) )
         => ( ( vEBT_L8524933119956041985T_VEBT @ A @ Xs2 @ Xsi )
            = ( vEBT_L8524933119956041985T_VEBT @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_3789_list__assn__cong,axiom,
    ! [Xs2: list_int,Xs4: list_int,Xsi: list_nat,Xsi2: list_nat,A: int > nat > assn,A5: int > nat > assn] :
      ( ( Xs2 = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X4: int,Xi2: nat] :
              ( ( member_int @ X4 @ ( set_int2 @ Xs4 ) )
             => ( ( member_nat @ Xi2 @ ( set_nat2 @ Xsi2 ) )
               => ( ( A @ X4 @ Xi2 )
                  = ( A5 @ X4 @ Xi2 ) ) ) )
         => ( ( vEBT_L77084186935402305nt_nat @ A @ Xs2 @ Xsi )
            = ( vEBT_L77084186935402305nt_nat @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_3790_list__assn__cong,axiom,
    ! [Xs2: list_complex,Xs4: list_complex,Xsi: list_nat,Xsi2: list_nat,A: complex > nat > assn,A5: complex > nat > assn] :
      ( ( Xs2 = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X4: complex,Xi2: nat] :
              ( ( member_complex @ X4 @ ( set_complex2 @ Xs4 ) )
             => ( ( member_nat @ Xi2 @ ( set_nat2 @ Xsi2 ) )
               => ( ( A @ X4 @ Xi2 )
                  = ( A5 @ X4 @ Xi2 ) ) ) )
         => ( ( vEBT_L137475477348087235ex_nat @ A @ Xs2 @ Xsi )
            = ( vEBT_L137475477348087235ex_nat @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_3791_list__assn__cong,axiom,
    ! [Xs2: list_int,Xs4: list_int,Xsi: list_real,Xsi2: list_real,A: int > real > assn,A5: int > real > assn] :
      ( ( Xs2 = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X4: int,Xi2: real] :
              ( ( member_int @ X4 @ ( set_int2 @ Xs4 ) )
             => ( ( member_real @ Xi2 @ ( set_real2 @ Xsi2 ) )
               => ( ( A @ X4 @ Xi2 )
                  = ( A5 @ X4 @ Xi2 ) ) ) )
         => ( ( vEBT_L8288995350762215837t_real @ A @ Xs2 @ Xsi )
            = ( vEBT_L8288995350762215837t_real @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_3792_list__assn__cong,axiom,
    ! [Xs2: list_complex,Xs4: list_complex,Xsi: list_real,Xsi2: list_real,A: complex > real > assn,A5: complex > real > assn] :
      ( ( Xs2 = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X4: complex,Xi2: real] :
              ( ( member_complex @ X4 @ ( set_complex2 @ Xs4 ) )
             => ( ( member_real @ Xi2 @ ( set_real2 @ Xsi2 ) )
               => ( ( A @ X4 @ Xi2 )
                  = ( A5 @ X4 @ Xi2 ) ) ) )
         => ( ( vEBT_L2479436891206192927x_real @ A @ Xs2 @ Xsi )
            = ( vEBT_L2479436891206192927x_real @ A5 @ Xs4 @ Xsi2 ) ) ) ) ) ).

% list_assn_cong
thf(fact_3793_set__update__subsetI,axiom,
    ! [Xs2: list_int,A: set_int,X: int,I: nat] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A )
     => ( ( member_int @ X @ A )
       => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I @ X ) ) @ A ) ) ) ).

% set_update_subsetI
thf(fact_3794_set__update__subsetI,axiom,
    ! [Xs2: list_complex,A: set_complex,X: complex,I: nat] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A )
     => ( ( member_complex @ X @ A )
       => ( ord_le211207098394363844omplex @ ( set_complex2 @ ( list_update_complex @ Xs2 @ I @ X ) ) @ A ) ) ) ).

% set_update_subsetI
thf(fact_3795_set__update__subsetI,axiom,
    ! [Xs2: list_real,A: set_real,X: real,I: nat] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ A )
     => ( ( member_real @ X @ A )
       => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs2 @ I @ X ) ) @ A ) ) ) ).

% set_update_subsetI
thf(fact_3796_set__update__subsetI,axiom,
    ! [Xs2: list_VEBT_VEBTi,A: set_VEBT_VEBTi,X: vEBT_VEBTi,I: nat] :
      ( ( ord_le6592769550269828683_VEBTi @ ( set_VEBT_VEBTi2 @ Xs2 ) @ A )
     => ( ( member_VEBT_VEBTi @ X @ A )
       => ( ord_le6592769550269828683_VEBTi @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X ) ) @ A ) ) ) ).

% set_update_subsetI
thf(fact_3797_set__update__subsetI,axiom,
    ! [Xs2: list_VEBT_VEBT,A: set_VEBT_VEBT,X: vEBT_VEBT,I: nat] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A )
     => ( ( member_VEBT_VEBT @ X @ A )
       => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) ) @ A ) ) ) ).

% set_update_subsetI
thf(fact_3798_set__update__subsetI,axiom,
    ! [Xs2: list_nat,A: set_nat,X: nat,I: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A )
     => ( ( member_nat @ X @ A )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X ) ) @ A ) ) ) ).

% set_update_subsetI
thf(fact_3799_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw: $o,N3: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv2 @ Uw ) @ ( suc @ N3 ) )
      = none_nat ) ).

% vebt_succ.simps(2)
thf(fact_3800_vebt__insert_Osimps_I1_J,axiom,
    ! [X: nat,A2: $o,B2: $o] :
      ( ( ( X = zero_zero_nat )
       => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
          = ( vEBT_Leaf @ $true @ B2 ) ) )
      & ( ( X != zero_zero_nat )
       => ( ( ( X = one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
              = ( vEBT_Leaf @ A2 @ $true ) ) )
          & ( ( X != one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
              = ( vEBT_Leaf @ A2 @ B2 ) ) ) ) ) ) ).

% vebt_insert.simps(1)
thf(fact_3801_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv2: $o] :
      ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv2 ) @ zero_zero_nat )
      = none_nat ) ).

% vebt_pred.simps(1)
thf(fact_3802_vebt__buildup_Osimps_I1_J,axiom,
    ( ( vEBT_vebt_buildup @ zero_zero_nat )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(1)
thf(fact_3803_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X4: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X4 ) )
     => ( ! [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw2 ) @ Ux ) )
       => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ X4 ) ) ) ) ).

% VEBT_internal.naive_member.cases
thf(fact_3804_invar__vebt_Ointros_I1_J,axiom,
    ! [A2: $o,B2: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ zero_zero_nat ) ) ).

% invar_vebt.intros(1)
thf(fact_3805_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_3806_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_3807_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_3808_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_3809_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_3810_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_3811_length__pos__if__in__set,axiom,
    ! [X: vEBT_VEBTi,Xs2: list_VEBT_VEBTi] :
      ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_3812_all__set__conv__all__nth,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( P @ X3 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3813_all__set__conv__all__nth,axiom,
    ! [Xs2: list_real,P: real > $o] :
      ( ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
           => ( P @ X3 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs2 ) )
           => ( P @ ( nth_real @ Xs2 @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3814_all__set__conv__all__nth,axiom,
    ! [Xs2: list_o,P: $o > $o] :
      ( ( ! [X3: $o] :
            ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
           => ( P @ X3 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
           => ( P @ ( nth_o @ Xs2 @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3815_all__set__conv__all__nth,axiom,
    ! [Xs2: list_nat,P: nat > $o] :
      ( ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
           => ( P @ X3 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
           => ( P @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3816_all__set__conv__all__nth,axiom,
    ! [Xs2: list_VEBT_VEBTi,P: vEBT_VEBTi > $o] :
      ( ( ! [X3: vEBT_VEBTi] :
            ( ( member_VEBT_VEBTi @ X3 @ ( set_VEBT_VEBTi2 @ Xs2 ) )
           => ( P @ X3 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
           => ( P @ ( nth_VEBT_VEBTi @ Xs2 @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3817_all__nth__imp__all__set,axiom,
    ! [Xs2: list_int,P: int > $o,X: int] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ ( size_size_list_int @ Xs2 ) )
         => ( P @ ( nth_int @ Xs2 @ I5 ) ) )
     => ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_3818_all__nth__imp__all__set,axiom,
    ! [Xs2: list_complex,P: complex > $o,X: complex] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ ( size_s3451745648224563538omplex @ Xs2 ) )
         => ( P @ ( nth_complex @ Xs2 @ I5 ) ) )
     => ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_3819_all__nth__imp__all__set,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
         => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I5 ) ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_3820_all__nth__imp__all__set,axiom,
    ! [Xs2: list_real,P: real > $o,X: real] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ ( size_size_list_real @ Xs2 ) )
         => ( P @ ( nth_real @ Xs2 @ I5 ) ) )
     => ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_3821_all__nth__imp__all__set,axiom,
    ! [Xs2: list_o,P: $o > $o,X: $o] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ ( size_size_list_o @ Xs2 ) )
         => ( P @ ( nth_o @ Xs2 @ I5 ) ) )
     => ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_3822_all__nth__imp__all__set,axiom,
    ! [Xs2: list_nat,P: nat > $o,X: nat] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs2 ) )
         => ( P @ ( nth_nat @ Xs2 @ I5 ) ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_3823_all__nth__imp__all__set,axiom,
    ! [Xs2: list_VEBT_VEBTi,P: vEBT_VEBTi > $o,X: vEBT_VEBTi] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
         => ( P @ ( nth_VEBT_VEBTi @ Xs2 @ I5 ) ) )
     => ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ Xs2 ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_3824_all__set__conv__nth,axiom,
    ! [L2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ L2 ) )
           => ( P @ X3 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ L2 ) )
           => ( P @ ( nth_VEBT_VEBT @ L2 @ I2 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_3825_all__set__conv__nth,axiom,
    ! [L2: list_real,P: real > $o] :
      ( ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_real2 @ L2 ) )
           => ( P @ X3 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_real @ L2 ) )
           => ( P @ ( nth_real @ L2 @ I2 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_3826_all__set__conv__nth,axiom,
    ! [L2: list_o,P: $o > $o] :
      ( ( ! [X3: $o] :
            ( ( member_o @ X3 @ ( set_o2 @ L2 ) )
           => ( P @ X3 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ L2 ) )
           => ( P @ ( nth_o @ L2 @ I2 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_3827_all__set__conv__nth,axiom,
    ! [L2: list_nat,P: nat > $o] :
      ( ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ L2 ) )
           => ( P @ X3 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ L2 ) )
           => ( P @ ( nth_nat @ L2 @ I2 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_3828_all__set__conv__nth,axiom,
    ! [L2: list_VEBT_VEBTi,P: vEBT_VEBTi > $o] :
      ( ( ! [X3: vEBT_VEBTi] :
            ( ( member_VEBT_VEBTi @ X3 @ ( set_VEBT_VEBTi2 @ L2 ) )
           => ( P @ X3 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ L2 ) )
           => ( P @ ( nth_VEBT_VEBTi @ L2 @ I2 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_3829_in__set__conv__nth,axiom,
    ! [X: int,Xs2: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
            & ( ( nth_int @ Xs2 @ I2 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_3830_in__set__conv__nth,axiom,
    ! [X: complex,Xs2: list_complex] :
      ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ Xs2 ) )
            & ( ( nth_complex @ Xs2 @ I2 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_3831_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
            & ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_3832_in__set__conv__nth,axiom,
    ! [X: real,Xs2: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs2 ) )
            & ( ( nth_real @ Xs2 @ I2 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_3833_in__set__conv__nth,axiom,
    ! [X: $o,Xs2: list_o] :
      ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
            & ( ( nth_o @ Xs2 @ I2 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_3834_in__set__conv__nth,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
            & ( ( nth_nat @ Xs2 @ I2 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_3835_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBTi,Xs2: list_VEBT_VEBTi] :
      ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
            & ( ( nth_VEBT_VEBTi @ Xs2 @ I2 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_3836_list__ball__nth,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ord_less_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N3 ) ) ) ) ).

% list_ball_nth
thf(fact_3837_list__ball__nth,axiom,
    ! [N3: nat,Xs2: list_real,P: real > $o] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_real @ Xs2 ) )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_real @ Xs2 @ N3 ) ) ) ) ).

% list_ball_nth
thf(fact_3838_list__ball__nth,axiom,
    ! [N3: nat,Xs2: list_o,P: $o > $o] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_o @ Xs2 ) )
     => ( ! [X4: $o] :
            ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_o @ Xs2 @ N3 ) ) ) ) ).

% list_ball_nth
thf(fact_3839_list__ball__nth,axiom,
    ! [N3: nat,Xs2: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs2 ) )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_nat @ Xs2 @ N3 ) ) ) ) ).

% list_ball_nth
thf(fact_3840_list__ball__nth,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBTi,P: vEBT_VEBTi > $o] :
      ( ( ord_less_nat @ N3 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ! [X4: vEBT_VEBTi] :
            ( ( member_VEBT_VEBTi @ X4 @ ( set_VEBT_VEBTi2 @ Xs2 ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_VEBT_VEBTi @ Xs2 @ N3 ) ) ) ) ).

% list_ball_nth
thf(fact_3841_nth__mem,axiom,
    ! [N3: nat,Xs2: list_int] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_int @ Xs2 ) )
     => ( member_int @ ( nth_int @ Xs2 @ N3 ) @ ( set_int2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_3842_nth__mem,axiom,
    ! [N3: nat,Xs2: list_complex] :
      ( ( ord_less_nat @ N3 @ ( size_s3451745648224563538omplex @ Xs2 ) )
     => ( member_complex @ ( nth_complex @ Xs2 @ N3 ) @ ( set_complex2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_3843_nth__mem,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ N3 ) @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_3844_nth__mem,axiom,
    ! [N3: nat,Xs2: list_real] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_real @ Xs2 ) )
     => ( member_real @ ( nth_real @ Xs2 @ N3 ) @ ( set_real2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_3845_nth__mem,axiom,
    ! [N3: nat,Xs2: list_o] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_o @ Xs2 ) )
     => ( member_o @ ( nth_o @ Xs2 @ N3 ) @ ( set_o2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_3846_nth__mem,axiom,
    ! [N3: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs2 ) )
     => ( member_nat @ ( nth_nat @ Xs2 @ N3 ) @ ( set_nat2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_3847_nth__mem,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ N3 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( member_VEBT_VEBTi @ ( nth_VEBT_VEBTi @ Xs2 @ N3 ) @ ( set_VEBT_VEBTi2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_3848_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ( X
       != ( vEBT_Leaf @ $false @ $false ) )
     => ( ! [Uv: $o] :
            ( X
           != ( vEBT_Leaf @ $true @ Uv ) )
       => ( ! [Uu2: $o] :
              ( X
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ( ! [Uw2: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux @ Uy ) )
           => ~ ! [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                  ( X
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va2 @ Vb @ Vc ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
thf(fact_3849_in__set__upd__eq__aux,axiom,
    ! [I: nat,L2: list_int,X: int,Y: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L2 ) )
     => ( ( member_int @ X @ ( set_int2 @ ( list_update_int @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y2: int] : ( member_int @ X @ ( set_int2 @ ( list_update_int @ L2 @ I @ Y2 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_3850_in__set__upd__eq__aux,axiom,
    ! [I: nat,L2: list_complex,X: complex,Y: complex] :
      ( ( ord_less_nat @ I @ ( size_s3451745648224563538omplex @ L2 ) )
     => ( ( member_complex @ X @ ( set_complex2 @ ( list_update_complex @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y2: complex] : ( member_complex @ X @ ( set_complex2 @ ( list_update_complex @ L2 @ I @ Y2 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_3851_in__set__upd__eq__aux,axiom,
    ! [I: nat,L2: list_VEBT_VEBT,X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y2: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ Y2 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_3852_in__set__upd__eq__aux,axiom,
    ! [I: nat,L2: list_real,X: real,Y: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ L2 ) )
     => ( ( member_real @ X @ ( set_real2 @ ( list_update_real @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y2: real] : ( member_real @ X @ ( set_real2 @ ( list_update_real @ L2 @ I @ Y2 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_3853_in__set__upd__eq__aux,axiom,
    ! [I: nat,L2: list_o,X: $o,Y: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L2 ) )
     => ( ( member_o @ X @ ( set_o2 @ ( list_update_o @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y2: $o] : ( member_o @ X @ ( set_o2 @ ( list_update_o @ L2 @ I @ Y2 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_3854_in__set__upd__eq__aux,axiom,
    ! [I: nat,L2: list_nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L2 ) )
     => ( ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y2: nat] : ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L2 @ I @ Y2 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_3855_in__set__upd__eq__aux,axiom,
    ! [I: nat,L2: list_VEBT_VEBTi,X: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L2 ) )
     => ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ! [Y2: vEBT_VEBTi] : ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L2 @ I @ Y2 ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_3856_in__set__upd__cases,axiom,
    ! [X: int,L2: list_int,I: nat,Y: int] :
      ( ( member_int @ X @ ( set_int2 @ ( list_update_int @ L2 @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_int @ L2 ) )
         => ( X != Y ) )
       => ( member_int @ X @ ( set_int2 @ L2 ) ) ) ) ).

% in_set_upd_cases
thf(fact_3857_in__set__upd__cases,axiom,
    ! [X: complex,L2: list_complex,I: nat,Y: complex] :
      ( ( member_complex @ X @ ( set_complex2 @ ( list_update_complex @ L2 @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_s3451745648224563538omplex @ L2 ) )
         => ( X != Y ) )
       => ( member_complex @ X @ ( set_complex2 @ L2 ) ) ) ) ).

% in_set_upd_cases
thf(fact_3858_in__set__upd__cases,axiom,
    ! [X: vEBT_VEBT,L2: list_VEBT_VEBT,I: nat,Y: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) )
         => ( X != Y ) )
       => ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ L2 ) ) ) ) ).

% in_set_upd_cases
thf(fact_3859_in__set__upd__cases,axiom,
    ! [X: real,L2: list_real,I: nat,Y: real] :
      ( ( member_real @ X @ ( set_real2 @ ( list_update_real @ L2 @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_real @ L2 ) )
         => ( X != Y ) )
       => ( member_real @ X @ ( set_real2 @ L2 ) ) ) ) ).

% in_set_upd_cases
thf(fact_3860_in__set__upd__cases,axiom,
    ! [X: $o,L2: list_o,I: nat,Y: $o] :
      ( ( member_o @ X @ ( set_o2 @ ( list_update_o @ L2 @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_o @ L2 ) )
         => ( X = ~ Y ) )
       => ( member_o @ X @ ( set_o2 @ L2 ) ) ) ) ).

% in_set_upd_cases
thf(fact_3861_in__set__upd__cases,axiom,
    ! [X: nat,L2: list_nat,I: nat,Y: nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L2 @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_nat @ L2 ) )
         => ( X != Y ) )
       => ( member_nat @ X @ ( set_nat2 @ L2 ) ) ) ) ).

% in_set_upd_cases
thf(fact_3862_in__set__upd__cases,axiom,
    ! [X: vEBT_VEBTi,L2: list_VEBT_VEBTi,I: nat,Y: vEBT_VEBTi] :
      ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L2 @ I @ Y ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L2 ) )
         => ( X != Y ) )
       => ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ L2 ) ) ) ) ).

% in_set_upd_cases
thf(fact_3863_set__update__memI,axiom,
    ! [N3: nat,Xs2: list_int,X: int] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_int @ Xs2 ) )
     => ( member_int @ X @ ( set_int2 @ ( list_update_int @ Xs2 @ N3 @ X ) ) ) ) ).

% set_update_memI
thf(fact_3864_set__update__memI,axiom,
    ! [N3: nat,Xs2: list_complex,X: complex] :
      ( ( ord_less_nat @ N3 @ ( size_s3451745648224563538omplex @ Xs2 ) )
     => ( member_complex @ X @ ( set_complex2 @ ( list_update_complex @ Xs2 @ N3 @ X ) ) ) ) ).

% set_update_memI
thf(fact_3865_set__update__memI,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ N3 @ X ) ) ) ) ).

% set_update_memI
thf(fact_3866_set__update__memI,axiom,
    ! [N3: nat,Xs2: list_real,X: real] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_real @ Xs2 ) )
     => ( member_real @ X @ ( set_real2 @ ( list_update_real @ Xs2 @ N3 @ X ) ) ) ) ).

% set_update_memI
thf(fact_3867_set__update__memI,axiom,
    ! [N3: nat,Xs2: list_o,X: $o] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_o @ Xs2 ) )
     => ( member_o @ X @ ( set_o2 @ ( list_update_o @ Xs2 @ N3 @ X ) ) ) ) ).

% set_update_memI
thf(fact_3868_set__update__memI,axiom,
    ! [N3: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs2 ) )
     => ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N3 @ X ) ) ) ) ).

% set_update_memI
thf(fact_3869_set__update__memI,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( ord_less_nat @ N3 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ Xs2 @ N3 @ X ) ) ) ) ).

% set_update_memI
thf(fact_3870_in__set__upd__eq,axiom,
    ! [I: nat,L2: list_int,X: int,Y: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L2 ) )
     => ( ( member_int @ X @ ( set_int2 @ ( list_update_int @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_int @ X @ ( set_int2 @ L2 ) )
            & ! [Y2: int] : ( member_int @ X @ ( set_int2 @ ( list_update_int @ L2 @ I @ Y2 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_3871_in__set__upd__eq,axiom,
    ! [I: nat,L2: list_complex,X: complex,Y: complex] :
      ( ( ord_less_nat @ I @ ( size_s3451745648224563538omplex @ L2 ) )
     => ( ( member_complex @ X @ ( set_complex2 @ ( list_update_complex @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_complex @ X @ ( set_complex2 @ L2 ) )
            & ! [Y2: complex] : ( member_complex @ X @ ( set_complex2 @ ( list_update_complex @ L2 @ I @ Y2 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_3872_in__set__upd__eq,axiom,
    ! [I: nat,L2: list_VEBT_VEBT,X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ L2 ) )
            & ! [Y2: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ Y2 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_3873_in__set__upd__eq,axiom,
    ! [I: nat,L2: list_real,X: real,Y: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ L2 ) )
     => ( ( member_real @ X @ ( set_real2 @ ( list_update_real @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_real @ X @ ( set_real2 @ L2 ) )
            & ! [Y2: real] : ( member_real @ X @ ( set_real2 @ ( list_update_real @ L2 @ I @ Y2 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_3874_in__set__upd__eq,axiom,
    ! [I: nat,L2: list_o,X: $o,Y: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L2 ) )
     => ( ( member_o @ X @ ( set_o2 @ ( list_update_o @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_o @ X @ ( set_o2 @ L2 ) )
            & ! [Y2: $o] : ( member_o @ X @ ( set_o2 @ ( list_update_o @ L2 @ I @ Y2 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_3875_in__set__upd__eq,axiom,
    ! [I: nat,L2: list_nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L2 ) )
     => ( ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_nat @ X @ ( set_nat2 @ L2 ) )
            & ! [Y2: nat] : ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ L2 @ I @ Y2 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_3876_in__set__upd__eq,axiom,
    ! [I: nat,L2: list_VEBT_VEBTi,X: vEBT_VEBTi,Y: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L2 ) )
     => ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L2 @ I @ Y ) ) )
        = ( ( X = Y )
          | ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ L2 ) )
            & ! [Y2: vEBT_VEBTi] : ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L2 @ I @ Y2 ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_3877_vebt__buildup_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(2)
thf(fact_3878_set__update__subset__insert,axiom,
    ! [Xs2: list_int,I: nat,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I @ X ) ) @ ( insert_int @ X @ ( set_int2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_3879_set__update__subset__insert,axiom,
    ! [Xs2: list_o,I: nat,X: $o] : ( ord_less_eq_set_o @ ( set_o2 @ ( list_update_o @ Xs2 @ I @ X ) ) @ ( insert_o @ X @ ( set_o2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_3880_set__update__subset__insert,axiom,
    ! [Xs2: list_real,I: nat,X: real] : ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs2 @ I @ X ) ) @ ( insert_real @ X @ ( set_real2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_3881_set__update__subset__insert,axiom,
    ! [Xs2: list_VEBT_VEBTi,I: nat,X: vEBT_VEBTi] : ( ord_le6592769550269828683_VEBTi @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X ) ) @ ( insert_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_3882_set__update__subset__insert,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) ) @ ( insert_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_3883_set__update__subset__insert,axiom,
    ! [Xs2: list_nat,I: nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X ) ) @ ( insert_nat @ X @ ( set_nat2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_3884_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,N3: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ N3 ) ) )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(3)
thf(fact_3885_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,N3: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ N3 ) ) )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(3)
thf(fact_3886_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(1)
thf(fact_3887_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ zero_zero_nat )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(1)
thf(fact_3888_vebt__pred_Osimps_I2_J,axiom,
    ! [A2: $o,Uw: $o] :
      ( ( A2
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ Uw ) @ ( suc @ zero_zero_nat ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A2
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ Uw ) @ ( suc @ zero_zero_nat ) )
          = none_nat ) ) ) ).

% vebt_pred.simps(2)
thf(fact_3889_vebt__succ_Osimps_I1_J,axiom,
    ! [B2: $o,Uu: $o] :
      ( ( B2
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B2 ) @ zero_zero_nat )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B2
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B2 ) @ zero_zero_nat )
          = none_nat ) ) ) ).

% vebt_succ.simps(1)
thf(fact_3890_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A3: $o,B3: $o] :
          ( X
         != ( vEBT_Leaf @ A3 @ B3 ) )
     => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
            ( X
           != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
thf(fact_3891_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I2_J,axiom,
    ! [A2: $o,B2: $o] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(2)
thf(fact_3892_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I2_J,axiom,
    ! [A2: $o,B2: $o] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(2)
thf(fact_3893_vebt__pred_Osimps_I3_J,axiom,
    ! [B2: $o,A2: $o,Va: nat] :
      ( ( B2
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va ) ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B2
       => ( ( A2
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va ) ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A2
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va ) ) )
              = none_nat ) ) ) ) ) ).

% vebt_pred.simps(3)
thf(fact_3894_insert__swap__set__eq,axiom,
    ! [I: nat,L2: list_int,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L2 ) )
     => ( ( insert_int @ ( nth_int @ L2 @ I ) @ ( set_int2 @ ( list_update_int @ L2 @ I @ X ) ) )
        = ( insert_int @ X @ ( set_int2 @ L2 ) ) ) ) ).

% insert_swap_set_eq
thf(fact_3895_insert__swap__set__eq,axiom,
    ! [I: nat,L2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) )
     => ( ( insert_VEBT_VEBT @ ( nth_VEBT_VEBT @ L2 @ I ) @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ X ) ) )
        = ( insert_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ L2 ) ) ) ) ).

% insert_swap_set_eq
thf(fact_3896_insert__swap__set__eq,axiom,
    ! [I: nat,L2: list_real,X: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ L2 ) )
     => ( ( insert_real @ ( nth_real @ L2 @ I ) @ ( set_real2 @ ( list_update_real @ L2 @ I @ X ) ) )
        = ( insert_real @ X @ ( set_real2 @ L2 ) ) ) ) ).

% insert_swap_set_eq
thf(fact_3897_insert__swap__set__eq,axiom,
    ! [I: nat,L2: list_o,X: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L2 ) )
     => ( ( insert_o @ ( nth_o @ L2 @ I ) @ ( set_o2 @ ( list_update_o @ L2 @ I @ X ) ) )
        = ( insert_o @ X @ ( set_o2 @ L2 ) ) ) ) ).

% insert_swap_set_eq
thf(fact_3898_insert__swap__set__eq,axiom,
    ! [I: nat,L2: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L2 ) )
     => ( ( insert_nat @ ( nth_nat @ L2 @ I ) @ ( set_nat2 @ ( list_update_nat @ L2 @ I @ X ) ) )
        = ( insert_nat @ X @ ( set_nat2 @ L2 ) ) ) ) ).

% insert_swap_set_eq
thf(fact_3899_insert__swap__set__eq,axiom,
    ! [I: nat,L2: list_VEBT_VEBTi,X: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L2 ) )
     => ( ( insert_VEBT_VEBTi @ ( nth_VEBT_VEBTi @ L2 @ I ) @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L2 @ I @ X ) ) )
        = ( insert_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ L2 ) ) ) ) ).

% insert_swap_set_eq
thf(fact_3900_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X4: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X4 ) )
     => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT,X4: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) @ X4 ) )
       => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X4: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) @ X4 ) )
         => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X4: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X4 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ X4 ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
thf(fact_3901_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X4: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X4 ) )
     => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) @ X4 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X4: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) @ X4 ) )
         => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ X4 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ X4 ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
thf(fact_3902_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,B3: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B3 ) @ zero_zero_nat ) )
     => ( ! [Uv: $o,Uw2: $o,N: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw2 ) @ ( suc @ N ) ) )
       => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va2: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va2 ) )
         => ( ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc @ Vd ) @ Ve ) )
           => ( ! [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,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ X4 ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
thf(fact_3903_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ zero_zero_nat ) )
     => ( ! [A3: $o,Uw2: $o] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A3: $o,B3: $o,Va3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va3 ) ) ) )
         => ( ! [Uy: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va2 ) @ Vb ) )
           => ( ! [V2: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd @ Ve ) @ Vf ) )
             => ( ! [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,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
                      ( X
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ X4 ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
thf(fact_3904_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ zero_zero_nat ) )
     => ( ! [A3: $o,B3: $o] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A3: $o,B3: $o,N: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ N ) ) ) )
         => ( ! [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) @ Uu2 ) )
           => ( ! [Mi2: nat,Ma2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList3 @ Summary2 ) @ X4 ) )
             => ( ! [Mi2: nat,Ma2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList3 @ Summary2 ) @ X4 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
                      ( X
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ X4 ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
thf(fact_3905_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv: $o,Uw2: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ Uw2 ) )
     => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) )
       => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,X4: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) @ X4 ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc: vEBT_VEBT,X4: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc ) @ X4 ) )
           => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT,X4: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ X4 ) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
thf(fact_3906_invar__vebt_Osimps,axiom,
    ( vEBT_invar_vebt
    = ( ^ [A1: vEBT_VEBT,A22: nat] :
          ( ( ? [A7: $o,B7: $o] :
                ( A1
                = ( vEBT_Leaf @ A7 @ B7 ) )
            & ( A22
              = ( suc @ zero_zero_nat ) ) )
          | ? [TreeList4: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList4 @ Summary3 ) )
              & ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ( vEBT_invar_vebt @ X3 @ N2 ) )
              & ( vEBT_invar_vebt @ Summary3 @ N2 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
              & ( A22
                = ( plus_plus_nat @ N2 @ N2 ) )
              & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
              & ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
          | ? [TreeList4: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList4 @ Summary3 ) )
              & ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ( vEBT_invar_vebt @ X3 @ N2 ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
              & ( A22
                = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
              & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
              & ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
          | ? [TreeList4: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A1
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList4 @ Summary3 ) )
              & ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ( vEBT_invar_vebt @ X3 @ N2 ) )
              & ( vEBT_invar_vebt @ Summary3 @ N2 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
              & ( A22
                = ( plus_plus_nat @ N2 @ N2 ) )
              & ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I2 ) @ X8 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X3: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
                          = I2 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
                      & ! [X3: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
                              = I2 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X3 )
                            & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) )
          | ? [TreeList4: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A1
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList4 @ Summary3 ) )
              & ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ( vEBT_invar_vebt @ X3 @ N2 ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
              & ( A22
                = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
              & ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I2 ) @ X8 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X3: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X8 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
                          = I2 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
                      & ! [X3: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
                              = I2 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X3 )
                            & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
thf(fact_3907_invar__vebt_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( ( vEBT_invar_vebt @ A12 @ A23 )
     => ( ( ? [A3: $o,B3: $o] :
              ( A12
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( A23
           != ( suc @ zero_zero_nat ) ) )
       => ( ! [TreeList3: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( ( A23 = Deg2 )
               => ( ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                     => ( vEBT_invar_vebt @ X5 @ N ) )
                 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                   => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                     => ( ( M4 = N )
                       => ( ( Deg2
                            = ( plus_plus_nat @ N @ M4 ) )
                         => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
                           => ~ ! [X5: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList3: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
                ( ( A12
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( A23 = Deg2 )
                 => ( ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ( vEBT_invar_vebt @ X5 @ N ) )
                   => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                       => ( ( M4
                            = ( suc @ N ) )
                         => ( ( Deg2
                              = ( plus_plus_nat @ N @ M4 ) )
                           => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
                             => ~ ! [X5: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList3: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A12
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
                 => ( ( A23 = Deg2 )
                   => ( ! [X5: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_invar_vebt @ X5 @ N ) )
                     => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                         => ( ( M4 = N )
                           => ( ( Deg2
                                = ( plus_plus_nat @ N @ M4 ) )
                             => ( ! [I6: nat] :
                                    ( ( ord_less_nat @ I6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                   => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I6 ) @ X8 ) )
                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I6 ) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X5: vEBT_VEBT] :
                                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
                                 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                   => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I6: nat] :
                                              ( ( ord_less_nat @ I6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                             => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N )
                                                    = I6 )
                                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I6 ) @ ( vEBT_VEBT_low @ Ma2 @ N ) ) )
                                                & ! [X5: nat] :
                                                    ( ( ( ( vEBT_VEBT_high @ X5 @ N )
                                                        = I6 )
                                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I6 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
                                                   => ( ( ord_less_nat @ Mi2 @ X5 )
                                                      & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList3: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A12
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
                   => ( ( A23 = Deg2 )
                     => ( ! [X5: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                           => ( vEBT_invar_vebt @ X5 @ N ) )
                       => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                           => ( ( M4
                                = ( suc @ N ) )
                             => ( ( Deg2
                                  = ( plus_plus_nat @ N @ M4 ) )
                               => ( ! [I6: nat] :
                                      ( ( ord_less_nat @ I6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                     => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I6 ) @ X8 ) )
                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I6 ) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X5: vEBT_VEBT] :
                                          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
                                   => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                     => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I6: nat] :
                                                ( ( ord_less_nat @ I6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N )
                                                      = I6 )
                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I6 ) @ ( vEBT_VEBT_low @ Ma2 @ N ) ) )
                                                  & ! [X5: nat] :
                                                      ( ( ( ( vEBT_VEBT_high @ X5 @ N )
                                                          = I6 )
                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I6 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
                                                     => ( ( ord_less_nat @ Mi2 @ X5 )
                                                        & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
thf(fact_3908_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N3: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X4 @ N3 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N3 )
           => ( ( Deg
                = ( plus_plus_nat @ N3 @ M ) )
             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
               => ( ! [X4: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
thf(fact_3909_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N3: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X4 @ N3 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N3 ) )
           => ( ( Deg
                = ( plus_plus_nat @ N3 @ M ) )
             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
               => ( ! [X4: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
thf(fact_3910_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S2 ) @ X )
      = ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S2 ) ) ).

% vebt_insert.simps(2)
thf(fact_3911_is__succ__in__set__def,axiom,
    ( vEBT_is_succ_in_set
    = ( ^ [Xs: set_nat,X3: nat,Y2: nat] :
          ( ( member_nat @ Y2 @ Xs )
          & ( ord_less_nat @ X3 @ Y2 )
          & ! [Z5: nat] :
              ( ( member_nat @ Z5 @ Xs )
             => ( ( ord_less_nat @ X3 @ Z5 )
               => ( ord_less_eq_nat @ Y2 @ Z5 ) ) ) ) ) ) ).

% is_succ_in_set_def
thf(fact_3912_is__pred__in__set__def,axiom,
    ( vEBT_is_pred_in_set
    = ( ^ [Xs: set_nat,X3: nat,Y2: nat] :
          ( ( member_nat @ Y2 @ Xs )
          & ( ord_less_nat @ Y2 @ X3 )
          & ! [Z5: nat] :
              ( ( member_nat @ Z5 @ Xs )
             => ( ( ord_less_nat @ Z5 @ X3 )
               => ( ord_less_eq_nat @ Z5 @ Y2 ) ) ) ) ) ) ).

% is_pred_in_set_def
thf(fact_3913_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ X )
      = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) ) ).

% vebt_insert.simps(3)
thf(fact_3914_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va )
      = none_nat ) ).

% vebt_succ.simps(3)
thf(fact_3915_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va: vEBT_VEBT,Vb2: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va ) @ Vb2 )
      = none_nat ) ).

% vebt_pred.simps(4)
thf(fact_3916_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N3: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X4 @ N3 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N3 )
           => ( ( Deg
                = ( plus_plus_nat @ N3 @ M ) )
             => ( ! [I5: nat] :
                    ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I5 ) @ X8 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I5 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X4: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I5: nat] :
                              ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N3 )
                                    = I5 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I5 ) @ ( vEBT_VEBT_low @ Ma @ N3 ) ) )
                                & ! [X4: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X4 @ N3 )
                                        = I5 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I5 ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) )
                                   => ( ( ord_less_nat @ Mi @ X4 )
                                      & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
thf(fact_3917_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N3: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X4 @ N3 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N3 ) )
           => ( ( Deg
                = ( plus_plus_nat @ N3 @ M ) )
             => ( ! [I5: nat] :
                    ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I5 ) @ X8 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I5 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X4: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I5: nat] :
                              ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N3 )
                                    = I5 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I5 ) @ ( vEBT_VEBT_low @ Ma @ N3 ) ) )
                                & ! [X4: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X4 @ N3 )
                                        = I5 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I5 ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) )
                                   => ( ( ord_less_nat @ Mi @ X4 )
                                      & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
thf(fact_3918_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve2 )
      = none_nat ) ).

% vebt_succ.simps(4)
thf(fact_3919_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Vf2 )
      = none_nat ) ).

% vebt_pred.simps(5)
thf(fact_3920_vebt__maxt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( B3
                 => ( Y
                    = ( some_nat @ one_one_nat ) ) )
                & ( ~ B3
                 => ( ( A3
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A3
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) )
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
               => ( Y
                 != ( some_nat @ Ma2 ) ) ) ) ) ) ).

% vebt_maxt.elims
thf(fact_3921_vebt__mint_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( A3
                 => ( Y
                    = ( some_nat @ zero_zero_nat ) ) )
                & ( ~ A3
                 => ( ( B3
                     => ( Y
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B3
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) )
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
               => ( Y
                 != ( some_nat @ Mi2 ) ) ) ) ) ) ).

% vebt_mint.elims
thf(fact_3922_vebt__maxt_Osimps_I1_J,axiom,
    ! [B2: $o,A2: $o] :
      ( ( B2
       => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B2 ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B2
       => ( ( A2
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B2 ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A2
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B2 ) )
              = none_nat ) ) ) ) ) ).

% vebt_maxt.simps(1)
thf(fact_3923_inrange,axiom,
    ! [T: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( 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 ) ) @ N3 ) @ one_one_nat ) ) ) ) ).

% inrange
thf(fact_3924_greater__shift,axiom,
    ( ord_less_nat
    = ( ^ [Y2: nat,X3: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X3 ) @ ( some_nat @ Y2 ) ) ) ) ).

% greater_shift
thf(fact_3925_less__shift,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y2: nat] : ( vEBT_VEBT_less @ ( some_nat @ X3 ) @ ( some_nat @ Y2 ) ) ) ) ).

% less_shift
thf(fact_3926_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L2: set_nat,U: set_nat] :
      ( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L2 @ U ) )
      = ( ( ord_less_eq_set_nat @ L2 @ I )
        & ( ord_less_eq_set_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3927_atLeastAtMost__iff,axiom,
    ! [I: rat,L2: rat,U: rat] :
      ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L2 @ U ) )
      = ( ( ord_less_eq_rat @ L2 @ I )
        & ( ord_less_eq_rat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3928_atLeastAtMost__iff,axiom,
    ! [I: num,L2: num,U: num] :
      ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L2 @ U ) )
      = ( ( ord_less_eq_num @ L2 @ I )
        & ( ord_less_eq_num @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3929_atLeastAtMost__iff,axiom,
    ! [I: nat,L2: nat,U: nat] :
      ( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L2 @ U ) )
      = ( ( ord_less_eq_nat @ L2 @ I )
        & ( ord_less_eq_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3930_atLeastAtMost__iff,axiom,
    ! [I: int,L2: int,U: int] :
      ( ( member_int @ I @ ( set_or1266510415728281911st_int @ L2 @ U ) )
      = ( ( ord_less_eq_int @ L2 @ I )
        & ( ord_less_eq_int @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3931_atLeastAtMost__iff,axiom,
    ! [I: code_integer,L2: code_integer,U: code_integer] :
      ( ( member_Code_integer @ I @ ( set_or189985376899183464nteger @ L2 @ U ) )
      = ( ( ord_le3102999989581377725nteger @ L2 @ I )
        & ( ord_le3102999989581377725nteger @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3932_atLeastAtMost__iff,axiom,
    ! [I: real,L2: real,U: real] :
      ( ( member_real @ I @ ( set_or1222579329274155063t_real @ L2 @ U ) )
      = ( ( ord_less_eq_real @ L2 @ I )
        & ( ord_less_eq_real @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3933_Icc__eq__Icc,axiom,
    ! [L2: set_nat,H2: set_nat,L3: set_nat,H3: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ L2 @ H2 )
        = ( set_or4548717258645045905et_nat @ L3 @ H3 ) )
      = ( ( ( L2 = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_set_nat @ L2 @ H2 )
          & ~ ( ord_less_eq_set_nat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3934_Icc__eq__Icc,axiom,
    ! [L2: rat,H2: rat,L3: rat,H3: rat] :
      ( ( ( set_or633870826150836451st_rat @ L2 @ H2 )
        = ( set_or633870826150836451st_rat @ L3 @ H3 ) )
      = ( ( ( L2 = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_rat @ L2 @ H2 )
          & ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3935_Icc__eq__Icc,axiom,
    ! [L2: num,H2: num,L3: num,H3: num] :
      ( ( ( set_or7049704709247886629st_num @ L2 @ H2 )
        = ( set_or7049704709247886629st_num @ L3 @ H3 ) )
      = ( ( ( L2 = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_num @ L2 @ H2 )
          & ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3936_Icc__eq__Icc,axiom,
    ! [L2: nat,H2: nat,L3: nat,H3: nat] :
      ( ( ( set_or1269000886237332187st_nat @ L2 @ H2 )
        = ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
      = ( ( ( L2 = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_nat @ L2 @ H2 )
          & ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3937_Icc__eq__Icc,axiom,
    ! [L2: int,H2: int,L3: int,H3: int] :
      ( ( ( set_or1266510415728281911st_int @ L2 @ H2 )
        = ( set_or1266510415728281911st_int @ L3 @ H3 ) )
      = ( ( ( L2 = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_int @ L2 @ H2 )
          & ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3938_Icc__eq__Icc,axiom,
    ! [L2: code_integer,H2: code_integer,L3: code_integer,H3: code_integer] :
      ( ( ( set_or189985376899183464nteger @ L2 @ H2 )
        = ( set_or189985376899183464nteger @ L3 @ H3 ) )
      = ( ( ( L2 = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_le3102999989581377725nteger @ L2 @ H2 )
          & ~ ( ord_le3102999989581377725nteger @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3939_Icc__eq__Icc,axiom,
    ! [L2: real,H2: real,L3: real,H3: real] :
      ( ( ( set_or1222579329274155063t_real @ L2 @ H2 )
        = ( set_or1222579329274155063t_real @ L3 @ H3 ) )
      = ( ( ( L2 = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_real @ L2 @ H2 )
          & ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_3940_atLeastatMost__empty__iff2,axiom,
    ! [A2: $o,B2: $o] :
      ( ( bot_bot_set_o
        = ( set_or8904488021354931149Most_o @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_o @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3941_atLeastatMost__empty__iff2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( bot_bot_set_set_nat
        = ( set_or4548717258645045905et_nat @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3942_atLeastatMost__empty__iff2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( bot_bot_set_rat
        = ( set_or633870826150836451st_rat @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3943_atLeastatMost__empty__iff2,axiom,
    ! [A2: num,B2: num] :
      ( ( bot_bot_set_num
        = ( set_or7049704709247886629st_num @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_num @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3944_atLeastatMost__empty__iff2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( bot_bot_set_nat
        = ( set_or1269000886237332187st_nat @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3945_atLeastatMost__empty__iff2,axiom,
    ! [A2: int,B2: int] :
      ( ( bot_bot_set_int
        = ( set_or1266510415728281911st_int @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_int @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3946_atLeastatMost__empty__iff2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( bot_bo3990330152332043303nteger
        = ( set_or189985376899183464nteger @ A2 @ B2 ) )
      = ( ~ ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3947_atLeastatMost__empty__iff2,axiom,
    ! [A2: real,B2: real] :
      ( ( bot_bot_set_real
        = ( set_or1222579329274155063t_real @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_real @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3948_atLeastatMost__empty__iff,axiom,
    ! [A2: $o,B2: $o] :
      ( ( ( set_or8904488021354931149Most_o @ A2 @ B2 )
        = bot_bot_set_o )
      = ( ~ ( ord_less_eq_o @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3949_atLeastatMost__empty__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ A2 @ B2 )
        = bot_bot_set_set_nat )
      = ( ~ ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3950_atLeastatMost__empty__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( set_or633870826150836451st_rat @ A2 @ B2 )
        = bot_bot_set_rat )
      = ( ~ ( ord_less_eq_rat @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3951_atLeastatMost__empty__iff,axiom,
    ! [A2: num,B2: num] :
      ( ( ( set_or7049704709247886629st_num @ A2 @ B2 )
        = bot_bot_set_num )
      = ( ~ ( ord_less_eq_num @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3952_atLeastatMost__empty__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A2 @ B2 )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3953_atLeastatMost__empty__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ( set_or1266510415728281911st_int @ A2 @ B2 )
        = bot_bot_set_int )
      = ( ~ ( ord_less_eq_int @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3954_atLeastatMost__empty__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( set_or189985376899183464nteger @ A2 @ B2 )
        = bot_bo3990330152332043303nteger )
      = ( ~ ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3955_atLeastatMost__empty__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ( set_or1222579329274155063t_real @ A2 @ B2 )
        = bot_bot_set_real )
      = ( ~ ( ord_less_eq_real @ A2 @ B2 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3956_atLeastatMost__empty,axiom,
    ! [B2: $o,A2: $o] :
      ( ( ord_less_o @ B2 @ A2 )
     => ( ( set_or8904488021354931149Most_o @ A2 @ B2 )
        = bot_bot_set_o ) ) ).

% atLeastatMost_empty
thf(fact_3957_atLeastatMost__empty,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ( set_or633870826150836451st_rat @ A2 @ B2 )
        = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
thf(fact_3958_atLeastatMost__empty,axiom,
    ! [B2: num,A2: num] :
      ( ( ord_less_num @ B2 @ A2 )
     => ( ( set_or7049704709247886629st_num @ A2 @ B2 )
        = bot_bot_set_num ) ) ).

% atLeastatMost_empty
thf(fact_3959_atLeastatMost__empty,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( set_or1269000886237332187st_nat @ A2 @ B2 )
        = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
thf(fact_3960_atLeastatMost__empty,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( set_or1266510415728281911st_int @ A2 @ B2 )
        = bot_bot_set_int ) ) ).

% atLeastatMost_empty
thf(fact_3961_atLeastatMost__empty,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B2 @ A2 )
     => ( ( set_or189985376899183464nteger @ A2 @ B2 )
        = bot_bo3990330152332043303nteger ) ) ).

% atLeastatMost_empty
thf(fact_3962_atLeastatMost__empty,axiom,
    ! [B2: real,A2: real] :
      ( ( ord_less_real @ B2 @ A2 )
     => ( ( set_or1222579329274155063t_real @ A2 @ B2 )
        = bot_bot_set_real ) ) ).

% atLeastatMost_empty
thf(fact_3963_atLeastatMost__subset__iff,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat,D2: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A2 @ B2 ) @ ( set_or4548717258645045905et_nat @ C2 @ D2 ) )
      = ( ~ ( ord_less_eq_set_nat @ A2 @ B2 )
        | ( ( ord_less_eq_set_nat @ C2 @ A2 )
          & ( ord_less_eq_set_nat @ B2 @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3964_atLeastatMost__subset__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A2 @ B2 ) @ ( set_or633870826150836451st_rat @ C2 @ D2 ) )
      = ( ~ ( ord_less_eq_rat @ A2 @ B2 )
        | ( ( ord_less_eq_rat @ C2 @ A2 )
          & ( ord_less_eq_rat @ B2 @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3965_atLeastatMost__subset__iff,axiom,
    ! [A2: num,B2: num,C2: num,D2: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A2 @ B2 ) @ ( set_or7049704709247886629st_num @ C2 @ D2 ) )
      = ( ~ ( ord_less_eq_num @ A2 @ B2 )
        | ( ( ord_less_eq_num @ C2 @ A2 )
          & ( ord_less_eq_num @ B2 @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3966_atLeastatMost__subset__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) @ ( set_or1269000886237332187st_nat @ C2 @ D2 ) )
      = ( ~ ( ord_less_eq_nat @ A2 @ B2 )
        | ( ( ord_less_eq_nat @ C2 @ A2 )
          & ( ord_less_eq_nat @ B2 @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3967_atLeastatMost__subset__iff,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A2 @ B2 ) @ ( set_or1266510415728281911st_int @ C2 @ D2 ) )
      = ( ~ ( ord_less_eq_int @ A2 @ B2 )
        | ( ( ord_less_eq_int @ C2 @ A2 )
          & ( ord_less_eq_int @ B2 @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3968_atLeastatMost__subset__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or189985376899183464nteger @ A2 @ B2 ) @ ( set_or189985376899183464nteger @ C2 @ D2 ) )
      = ( ~ ( ord_le3102999989581377725nteger @ A2 @ B2 )
        | ( ( ord_le3102999989581377725nteger @ C2 @ A2 )
          & ( ord_le3102999989581377725nteger @ B2 @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3969_atLeastatMost__subset__iff,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ ( set_or1222579329274155063t_real @ C2 @ D2 ) )
      = ( ~ ( ord_less_eq_real @ A2 @ B2 )
        | ( ( ord_less_eq_real @ C2 @ A2 )
          & ( ord_less_eq_real @ B2 @ D2 ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3970_atLeastAtMost__singleton,axiom,
    ! [A2: $o] :
      ( ( set_or8904488021354931149Most_o @ A2 @ A2 )
      = ( insert_o @ A2 @ bot_bot_set_o ) ) ).

% atLeastAtMost_singleton
thf(fact_3971_atLeastAtMost__singleton,axiom,
    ! [A2: nat] :
      ( ( set_or1269000886237332187st_nat @ A2 @ A2 )
      = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% atLeastAtMost_singleton
thf(fact_3972_atLeastAtMost__singleton,axiom,
    ! [A2: int] :
      ( ( set_or1266510415728281911st_int @ A2 @ A2 )
      = ( insert_int @ A2 @ bot_bot_set_int ) ) ).

% atLeastAtMost_singleton
thf(fact_3973_atLeastAtMost__singleton,axiom,
    ! [A2: code_integer] :
      ( ( set_or189985376899183464nteger @ A2 @ A2 )
      = ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) ) ).

% atLeastAtMost_singleton
thf(fact_3974_atLeastAtMost__singleton,axiom,
    ! [A2: real] :
      ( ( set_or1222579329274155063t_real @ A2 @ A2 )
      = ( insert_real @ A2 @ bot_bot_set_real ) ) ).

% atLeastAtMost_singleton
thf(fact_3975_atLeastAtMost__singleton__iff,axiom,
    ! [A2: $o,B2: $o,C2: $o] :
      ( ( ( set_or8904488021354931149Most_o @ A2 @ B2 )
        = ( insert_o @ C2 @ bot_bot_set_o ) )
      = ( ( A2 = B2 )
        & ( B2 = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_3976_atLeastAtMost__singleton__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A2 @ B2 )
        = ( insert_nat @ C2 @ bot_bot_set_nat ) )
      = ( ( A2 = B2 )
        & ( B2 = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_3977_atLeastAtMost__singleton__iff,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ( set_or1266510415728281911st_int @ A2 @ B2 )
        = ( insert_int @ C2 @ bot_bot_set_int ) )
      = ( ( A2 = B2 )
        & ( B2 = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_3978_atLeastAtMost__singleton__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( ( set_or189985376899183464nteger @ A2 @ B2 )
        = ( insert_Code_integer @ C2 @ bot_bo3990330152332043303nteger ) )
      = ( ( A2 = B2 )
        & ( B2 = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_3979_atLeastAtMost__singleton__iff,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ( set_or1222579329274155063t_real @ A2 @ B2 )
        = ( insert_real @ C2 @ bot_bot_set_real ) )
      = ( ( A2 = B2 )
        & ( B2 = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_3980_atLeastAtMost__singleton_H,axiom,
    ! [A2: $o,B2: $o] :
      ( ( A2 = B2 )
     => ( ( set_or8904488021354931149Most_o @ A2 @ B2 )
        = ( insert_o @ A2 @ bot_bot_set_o ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_3981_atLeastAtMost__singleton_H,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
     => ( ( set_or1269000886237332187st_nat @ A2 @ B2 )
        = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_3982_atLeastAtMost__singleton_H,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = B2 )
     => ( ( set_or1266510415728281911st_int @ A2 @ B2 )
        = ( insert_int @ A2 @ bot_bot_set_int ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_3983_atLeastAtMost__singleton_H,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2 = B2 )
     => ( ( set_or189985376899183464nteger @ A2 @ B2 )
        = ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_3984_atLeastAtMost__singleton_H,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 = B2 )
     => ( ( set_or1222579329274155063t_real @ A2 @ B2 )
        = ( insert_real @ A2 @ bot_bot_set_real ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_3985_all__nat__less,axiom,
    ! [N3: nat,P: nat > $o] :
      ( ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ M5 @ N3 )
           => ( P @ M5 ) ) )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
           => ( P @ X3 ) ) ) ) ).

% all_nat_less
thf(fact_3986_ex__nat__less,axiom,
    ! [N3: nat,P: nat > $o] :
      ( ( ? [M5: nat] :
            ( ( ord_less_eq_nat @ M5 @ N3 )
            & ( P @ M5 ) ) )
      = ( ? [X3: nat] :
            ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
            & ( P @ X3 ) ) ) ) ).

% ex_nat_less
thf(fact_3987_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L2: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ L2 @ ( suc @ U ) )
      = ( set_or1269000886237332187st_nat @ L2 @ U ) ) ).

% atLeastLessThanSuc_atLeastAtMost
thf(fact_3988_atLeastatMost__psubset__iff,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat,D2: set_nat] :
      ( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A2 @ B2 ) @ ( set_or4548717258645045905et_nat @ C2 @ D2 ) )
      = ( ( ~ ( ord_less_eq_set_nat @ A2 @ B2 )
          | ( ( ord_less_eq_set_nat @ C2 @ A2 )
            & ( ord_less_eq_set_nat @ B2 @ D2 )
            & ( ( ord_less_set_nat @ C2 @ A2 )
              | ( ord_less_set_nat @ B2 @ D2 ) ) ) )
        & ( ord_less_eq_set_nat @ C2 @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3989_atLeastatMost__psubset__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A2 @ B2 ) @ ( set_or633870826150836451st_rat @ C2 @ D2 ) )
      = ( ( ~ ( ord_less_eq_rat @ A2 @ B2 )
          | ( ( ord_less_eq_rat @ C2 @ A2 )
            & ( ord_less_eq_rat @ B2 @ D2 )
            & ( ( ord_less_rat @ C2 @ A2 )
              | ( ord_less_rat @ B2 @ D2 ) ) ) )
        & ( ord_less_eq_rat @ C2 @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3990_atLeastatMost__psubset__iff,axiom,
    ! [A2: num,B2: num,C2: num,D2: num] :
      ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A2 @ B2 ) @ ( set_or7049704709247886629st_num @ C2 @ D2 ) )
      = ( ( ~ ( ord_less_eq_num @ A2 @ B2 )
          | ( ( ord_less_eq_num @ C2 @ A2 )
            & ( ord_less_eq_num @ B2 @ D2 )
            & ( ( ord_less_num @ C2 @ A2 )
              | ( ord_less_num @ B2 @ D2 ) ) ) )
        & ( ord_less_eq_num @ C2 @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3991_atLeastatMost__psubset__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) @ ( set_or1269000886237332187st_nat @ C2 @ D2 ) )
      = ( ( ~ ( ord_less_eq_nat @ A2 @ B2 )
          | ( ( ord_less_eq_nat @ C2 @ A2 )
            & ( ord_less_eq_nat @ B2 @ D2 )
            & ( ( ord_less_nat @ C2 @ A2 )
              | ( ord_less_nat @ B2 @ D2 ) ) ) )
        & ( ord_less_eq_nat @ C2 @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3992_atLeastatMost__psubset__iff,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A2 @ B2 ) @ ( set_or1266510415728281911st_int @ C2 @ D2 ) )
      = ( ( ~ ( ord_less_eq_int @ A2 @ B2 )
          | ( ( ord_less_eq_int @ C2 @ A2 )
            & ( ord_less_eq_int @ B2 @ D2 )
            & ( ( ord_less_int @ C2 @ A2 )
              | ( ord_less_int @ B2 @ D2 ) ) ) )
        & ( ord_less_eq_int @ C2 @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3993_atLeastatMost__psubset__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le1307284697595431911nteger @ ( set_or189985376899183464nteger @ A2 @ B2 ) @ ( set_or189985376899183464nteger @ C2 @ D2 ) )
      = ( ( ~ ( ord_le3102999989581377725nteger @ A2 @ B2 )
          | ( ( ord_le3102999989581377725nteger @ C2 @ A2 )
            & ( ord_le3102999989581377725nteger @ B2 @ D2 )
            & ( ( ord_le6747313008572928689nteger @ C2 @ A2 )
              | ( ord_le6747313008572928689nteger @ B2 @ D2 ) ) ) )
        & ( ord_le3102999989581377725nteger @ C2 @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3994_atLeastatMost__psubset__iff,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ ( set_or1222579329274155063t_real @ C2 @ D2 ) )
      = ( ( ~ ( ord_less_eq_real @ A2 @ B2 )
          | ( ( ord_less_eq_real @ C2 @ A2 )
            & ( ord_less_eq_real @ B2 @ D2 )
            & ( ( ord_less_real @ C2 @ A2 )
              | ( ord_less_real @ B2 @ D2 ) ) ) )
        & ( ord_less_eq_real @ C2 @ D2 ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3995_atLeast0__atMost__Suc,axiom,
    ! [N3: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N3 ) )
      = ( insert_nat @ ( suc @ N3 ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) ) ).

% atLeast0_atMost_Suc
thf(fact_3996_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( set_or1269000886237332187st_nat @ M @ N3 )
        = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N3 ) ) ) ) ).

% Icc_eq_insert_lb_nat
thf(fact_3997_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N3 ) )
     => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N3 ) )
        = ( insert_nat @ ( suc @ N3 ) @ ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ) ).

% atLeastAtMostSuc_conv
thf(fact_3998_atLeastAtMost__insertL,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N3 ) )
        = ( set_or1269000886237332187st_nat @ M @ N3 ) ) ) ).

% atLeastAtMost_insertL
thf(fact_3999_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat,D2: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A2 @ B2 ) @ ( set_or3540276404033026485et_nat @ C2 @ D2 ) )
      = ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( ( ord_less_eq_set_nat @ C2 @ A2 )
          & ( ord_less_set_nat @ B2 @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4000_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A2 @ B2 ) @ ( set_or4029947393144176647an_rat @ C2 @ D2 ) )
      = ( ( ord_less_eq_rat @ A2 @ B2 )
       => ( ( ord_less_eq_rat @ C2 @ A2 )
          & ( ord_less_rat @ B2 @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4001_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A2: num,B2: num,C2: num,D2: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A2 @ B2 ) @ ( set_or1222409239386451017an_num @ C2 @ D2 ) )
      = ( ( ord_less_eq_num @ A2 @ B2 )
       => ( ( ord_less_eq_num @ C2 @ A2 )
          & ( ord_less_num @ B2 @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4002_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ ( set_or66887138388493659n_real @ C2 @ D2 ) )
      = ( ( ord_less_eq_real @ A2 @ B2 )
       => ( ( ord_less_eq_real @ C2 @ A2 )
          & ( ord_less_real @ B2 @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4003_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) @ ( set_or4665077453230672383an_nat @ C2 @ D2 ) )
      = ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( ( ord_less_eq_nat @ C2 @ A2 )
          & ( ord_less_nat @ B2 @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4004_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A2 @ B2 ) @ ( set_or4662586982721622107an_int @ C2 @ D2 ) )
      = ( ( ord_less_eq_int @ A2 @ B2 )
       => ( ( ord_less_eq_int @ C2 @ A2 )
          & ( ord_less_int @ B2 @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4005_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] :
      ( ( ord_le7084787975880047091nteger @ ( set_or189985376899183464nteger @ A2 @ B2 ) @ ( set_or8404916559141939852nteger @ C2 @ D2 ) )
      = ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
       => ( ( ord_le3102999989581377725nteger @ C2 @ A2 )
          & ( ord_le6747313008572928689nteger @ B2 @ D2 ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4006_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or4029947393144176647an_rat @ A2 @ B2 ) @ ( set_or633870826150836451st_rat @ C2 @ D2 ) )
      = ( ( ord_less_rat @ A2 @ B2 )
       => ( ( ord_less_eq_rat @ C2 @ A2 )
          & ( ord_less_eq_rat @ B2 @ D2 ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_4007_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( ord_less_eq_set_real @ ( set_or66887138388493659n_real @ A2 @ B2 ) @ ( set_or1222579329274155063t_real @ C2 @ D2 ) )
      = ( ( ord_less_real @ A2 @ B2 )
       => ( ( ord_less_eq_real @ C2 @ A2 )
          & ( ord_less_eq_real @ B2 @ D2 ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_4008_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ( set_or7139685690850216873Than_o
    = ( ^ [A7: $o,B7: $o] : ( minus_minus_set_o @ ( set_or8904488021354931149Most_o @ A7 @ B7 ) @ ( insert_o @ B7 @ bot_bot_set_o ) ) ) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_4009_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ( set_or66887138388493659n_real
    = ( ^ [A7: real,B7: real] : ( minus_minus_set_real @ ( set_or1222579329274155063t_real @ A7 @ B7 ) @ ( insert_real @ B7 @ bot_bot_set_real ) ) ) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_4010_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ( set_or4665077453230672383an_nat
    = ( ^ [A7: nat,B7: nat] : ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A7 @ B7 ) @ ( insert_nat @ B7 @ bot_bot_set_nat ) ) ) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_4011_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ( set_or4662586982721622107an_int
    = ( ^ [A7: int,B7: int] : ( minus_minus_set_int @ ( set_or1266510415728281911st_int @ A7 @ B7 ) @ ( insert_int @ B7 @ bot_bot_set_int ) ) ) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_4012_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ( set_or8404916559141939852nteger
    = ( ^ [A7: code_integer,B7: code_integer] : ( minus_2355218937544613996nteger @ ( set_or189985376899183464nteger @ A7 @ B7 ) @ ( insert_Code_integer @ B7 @ bot_bo3990330152332043303nteger ) ) ) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_4013_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc8306885398267862888on_nat] :
      ( ! [Uu2: nat > nat > nat,Uv: option_nat] :
          ( X
         != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv ) ) )
     => ( ! [Uw2: nat > nat > nat,V2: nat] :
            ( X
           != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F3: nat > nat > nat,A3: nat,B3: nat] :
              ( X
             != ( produc8929957630744042906on_nat @ F3 @ ( produc5098337634421038937on_nat @ ( some_nat @ A3 ) @ ( some_nat @ B3 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_4014_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc5542196010084753463at_nat] :
      ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
          ( X
         != ( produc2899441246263362727at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv ) ) )
     => ( ! [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 ) ) )
       => ~ ! [F3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( X
             != ( produc2899441246263362727at_nat @ F3 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A3 ) @ ( some_P7363390416028606310at_nat @ B3 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_4015_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc2233624965454879586on_nat] :
      ( ! [Uu2: nat > nat > $o,Uv: option_nat] :
          ( X
         != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv ) ) )
     => ( ! [Uw2: nat > nat > $o,V2: nat] :
            ( X
           != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F3: nat > nat > $o,X4: nat,Y3: nat] :
              ( X
             != ( produc4035269172776083154on_nat @ F3 @ ( produc5098337634421038937on_nat @ ( some_nat @ X4 ) @ ( some_nat @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_4016_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc5491161045314408544at_nat] :
      ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv: option4927543243414619207at_nat] :
          ( X
         != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv ) ) )
     => ( ! [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 ) ) )
       => ~ ! [F3: product_prod_nat_nat > product_prod_nat_nat > $o,X4: product_prod_nat_nat,Y3: product_prod_nat_nat] :
              ( X
             != ( produc3994169339658061776at_nat @ F3 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X4 ) @ ( some_P7363390416028606310at_nat @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_4017_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ F2 @ ( some_P7363390416028606310at_nat @ A2 ) @ ( some_P7363390416028606310at_nat @ B2 ) )
      = ( some_P7363390416028606310at_nat @ ( F2 @ A2 @ B2 ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_4018_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F2: nat > nat > nat,A2: nat,B2: nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ F2 @ ( some_nat @ A2 ) @ ( some_nat @ B2 ) )
      = ( some_nat @ ( F2 @ A2 @ B2 ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_4019_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ Uu @ none_P5556105721700978146at_nat @ Uv2 )
      = none_P5556105721700978146at_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_4020_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: nat > nat > nat,Uv2: option_nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ Uu @ none_nat @ Uv2 )
      = none_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_4021_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( ( vEBT_V1502963449132264192at_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = none_P5556105721700978146at_nat )
         => ( Y != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] :
                ( Xa
                = ( some_P7363390416028606310at_nat @ V2 ) )
           => ( ( Xb = none_P5556105721700978146at_nat )
             => ( Y != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A3: product_prod_nat_nat] :
                ( ( Xa
                  = ( some_P7363390416028606310at_nat @ A3 ) )
               => ! [B3: product_prod_nat_nat] :
                    ( ( Xb
                      = ( some_P7363390416028606310at_nat @ B3 ) )
                   => ( Y
                     != ( some_P7363390416028606310at_nat @ ( X @ A3 @ B3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_4022_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: nat > nat > nat,Xa: option_nat,Xb: option_nat,Y: option_nat] :
      ( ( ( vEBT_V4262088993061758097ft_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = none_nat )
         => ( Y != none_nat ) )
       => ( ( ? [V2: nat] :
                ( Xa
                = ( some_nat @ V2 ) )
           => ( ( Xb = none_nat )
             => ( Y != none_nat ) ) )
         => ~ ! [A3: nat] :
                ( ( Xa
                  = ( some_nat @ A3 ) )
               => ! [B3: nat] :
                    ( ( Xb
                      = ( some_nat @ B3 ) )
                   => ( Y
                     != ( some_nat @ ( X @ A3 @ B3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_4023_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_4024_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_4025_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv2 @ Uw ) )
      = none_nat ) ).

% vebt_mint.simps(2)
thf(fact_4026_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv2 @ Uw ) )
      = none_nat ) ).

% vebt_maxt.simps(2)
thf(fact_4027_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) )
      = ( some_nat @ Mi ) ) ).

% vebt_mint.simps(3)
thf(fact_4028_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) )
      = ( some_nat @ Ma ) ) ).

% vebt_maxt.simps(3)
thf(fact_4029_vebt__mint_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      ( ( A2
       => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B2 ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A2
       => ( ( B2
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B2 ) )
              = ( some_nat @ one_one_nat ) ) )
          & ( ~ B2
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B2 ) )
              = none_nat ) ) ) ) ) ).

% vebt_mint.simps(1)
thf(fact_4030_VEBT__internal_OTb_Osimps_I2_J,axiom,
    ( ( vEBT_VEBT_Tb @ ( suc @ zero_zero_nat ) )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% VEBT_internal.Tb.simps(2)
thf(fact_4031_VEBT__internal_OTb_H_Osimps_I2_J,axiom,
    ( ( vEBT_VEBT_Tb2 @ ( suc @ zero_zero_nat ) )
    = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% VEBT_internal.Tb'.simps(2)
thf(fact_4032_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I2_J,axiom,
    ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ zero_zero_nat ) )
    = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(2)
thf(fact_4033_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) ) ).

% vebt_delete.simps(6)
thf(fact_4034_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I2_J,axiom,
    ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ zero_zero_nat ) )
    = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(2)
thf(fact_4035_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_4036_heaphelp,axiom,
    ! [Xa: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb: vEBT_VEBTi,N3: nat,Xc: vEBT_VEBTi,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn
        @ ( times_times_assn
          @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Xa @ Tree_is ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ Summary @ Xb ) )
            @ ( pure_assn
              @ ( ( none_nat = none_nat )
                & ( N3 = N3 ) ) ) )
          @ ( pure_assn
            @ ( Xc
              = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ N3 @ Xa @ Xb ) ) ) )
        @ H2 )
     => ( rep_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ none_P5556105721700978146at_nat @ N3 @ TreeList @ Summary ) @ Xc ) @ H2 ) ) ).

% heaphelp
thf(fact_4037_heaphelp,axiom,
    ! [Xa: array_VEBT_VEBTi,Tree_is: list_VEBT_VEBTi,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Xb: vEBT_VEBTi,N3: nat,Xc: vEBT_VEBTi,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn
        @ ( times_times_assn
          @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Xa @ Tree_is ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList @ Tree_is ) ) @ ( vEBT_vebt_assn_raw @ Summary @ Xb ) )
            @ ( pure_assn
              @ ( ( none_P5556105721700978146at_nat = none_P5556105721700978146at_nat )
                & ( N3 = N3 ) ) ) )
          @ ( pure_assn
            @ ( Xc
              = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ N3 @ Xa @ Xb ) ) ) )
        @ H2 )
     => ( rep_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ none_P5556105721700978146at_nat @ N3 @ TreeList @ Summary ) @ Xc ) @ H2 ) ) ).

% heaphelp
thf(fact_4038_nat__induct2,axiom,
    ! [P: nat > $o,N3: nat] :
      ( ( P @ zero_zero_nat )
     => ( ( P @ one_one_nat )
       => ( ! [N: nat] :
              ( ( P @ N )
             => ( P @ ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( P @ N3 ) ) ) ) ).

% nat_induct2
thf(fact_4039_VEBTi_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi,Y11: option4927543243414619207at_nat,Y12: nat,Y13: array_VEBT_VEBTi,Y14: vEBT_VEBTi] :
      ( ( ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 )
        = ( vEBT_Nodei @ Y11 @ Y12 @ Y13 @ Y14 ) )
      = ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBTi.inject(1)
thf(fact_4040_Rep__assn__inject,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( rep_assn @ X )
        = ( rep_assn @ Y ) )
      = ( X = Y ) ) ).

% Rep_assn_inject
thf(fact_4041_mod__h__bot__iff_I5_J,axiom,
    ! [P: assn,Q: assn,H2: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( times_times_assn @ P @ Q ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
        & ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(5)
thf(fact_4042_mod__pure__star__dist,axiom,
    ! [P: assn,B2: $o,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P @ ( pure_assn @ B2 ) ) @ H2 )
      = ( ( rep_assn @ P @ H2 )
        & B2 ) ) ).

% mod_pure_star_dist
thf(fact_4043_mod__h__bot__iff_I1_J,axiom,
    ! [B2: $o,H2: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( pure_assn @ B2 ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = B2 ) ).

% mod_h_bot_iff(1)
thf(fact_4044_mod__h__bot__iff_I4_J,axiom,
    ! [Q3: array_VEBT_VEBTi,Y: list_VEBT_VEBTi,H2: heap_e7401611519738050253t_unit] :
      ~ ( rep_assn @ ( snga_assn_VEBT_VEBTi @ Q3 @ Y ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ).

% mod_h_bot_iff(4)
thf(fact_4045_ent__pure__post__iff,axiom,
    ! [P: assn,Q: assn,B2: $o] :
      ( ( entails @ P @ ( times_times_assn @ Q @ ( pure_assn @ B2 ) ) )
      = ( ! [H: produc3658429121746597890et_nat] :
            ( ( rep_assn @ P @ H )
           => B2 )
        & ( entails @ P @ Q ) ) ) ).

% ent_pure_post_iff
thf(fact_4046_ent__pure__post__iff__sng,axiom,
    ! [P: assn,B2: $o] :
      ( ( entails @ P @ ( pure_assn @ B2 ) )
      = ( ! [H: produc3658429121746597890et_nat] :
            ( ( rep_assn @ P @ H )
           => B2 )
        & ( entails @ P @ one_one_assn ) ) ) ).

% ent_pure_post_iff_sng
thf(fact_4047_mod__starE,axiom,
    ! [A2: assn,B2: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A2 @ B2 ) @ H2 )
     => ~ ( ? [X_1: produc3658429121746597890et_nat] : ( rep_assn @ A2 @ X_1 )
         => ! [H_2: produc3658429121746597890et_nat] :
              ~ ( rep_assn @ B2 @ H_2 ) ) ) ).

% mod_starE
thf(fact_4048_mod__starD,axiom,
    ! [A: assn,B: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A @ B ) @ H2 )
     => ? [H1: produc3658429121746597890et_nat,H22: produc3658429121746597890et_nat] :
          ( ( rep_assn @ A @ H1 )
          & ( rep_assn @ B @ H22 ) ) ) ).

% mod_starD
thf(fact_4049_entails__def,axiom,
    ( entails
    = ( ^ [P3: assn,Q6: assn] :
        ! [H: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P3 @ H )
         => ( rep_assn @ Q6 @ H ) ) ) ) ).

% entails_def
thf(fact_4050_entailsI,axiom,
    ! [P: assn,Q: assn] :
      ( ! [H4: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P @ H4 )
         => ( rep_assn @ Q @ H4 ) )
     => ( entails @ P @ Q ) ) ).

% entailsI
thf(fact_4051_entailsD,axiom,
    ! [P: assn,Q: assn,H2: produc3658429121746597890et_nat] :
      ( ( entails @ P @ Q )
     => ( ( rep_assn @ P @ H2 )
       => ( rep_assn @ Q @ H2 ) ) ) ).

% entailsD
thf(fact_4052_ent__fwd,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat,Q: assn] :
      ( ( rep_assn @ P @ H2 )
     => ( ( entails @ P @ Q )
       => ( rep_assn @ Q @ H2 ) ) ) ).

% ent_fwd
thf(fact_4053_mod__h__bot__indep,axiom,
    ! [P: assn,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ bot_bot_set_nat ) ) ) ).

% mod_h_bot_indep
thf(fact_4054_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L2: int,U: int] :
      ( ( set_or4662586982721622107an_int @ L2 @ ( plus_plus_int @ U @ one_one_int ) )
      = ( set_or1266510415728281911st_int @ L2 @ U ) ) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_4055_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I2: int,J: int] : ( if_set_int @ ( ord_less_int @ J @ I2 ) @ bot_bot_set_int @ ( insert_int @ I2 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I2 @ one_one_int ) @ J ) ) ) ) ) ).

% simp_from_to
thf(fact_4056_aset_I2_J,axiom,
    ! [D: int,A: set_int,P: int > $o,Q: int > $o] :
      ( ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A )
                 => ( X4
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X4 )
           => ( P @ ( plus_plus_int @ X4 @ D ) ) ) )
     => ( ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X4 )
             => ( Q @ ( plus_plus_int @ X4 @ D ) ) ) )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A )
                   => ( X5
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
             => ( ( P @ ( plus_plus_int @ X5 @ D ) )
                | ( Q @ ( plus_plus_int @ X5 @ D ) ) ) ) ) ) ) ).

% aset(2)
thf(fact_4057_aset_I1_J,axiom,
    ! [D: int,A: set_int,P: int > $o,Q: int > $o] :
      ( ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A )
                 => ( X4
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X4 )
           => ( P @ ( plus_plus_int @ X4 @ D ) ) ) )
     => ( ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X4 )
             => ( Q @ ( plus_plus_int @ X4 @ D ) ) ) )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A )
                   => ( X5
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
             => ( ( P @ ( plus_plus_int @ X5 @ D ) )
                & ( Q @ ( plus_plus_int @ X5 @ D ) ) ) ) ) ) ) ).

% aset(1)
thf(fact_4058_bset_I2_J,axiom,
    ! [D: int,B: set_int,P: int > $o,Q: int > $o] :
      ( ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B )
                 => ( X4
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X4 )
           => ( P @ ( minus_minus_int @ X4 @ D ) ) ) )
     => ( ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X4 )
             => ( Q @ ( minus_minus_int @ X4 @ D ) ) ) )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B )
                   => ( X5
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
             => ( ( P @ ( minus_minus_int @ X5 @ D ) )
                | ( Q @ ( minus_minus_int @ X5 @ D ) ) ) ) ) ) ) ).

% bset(2)
thf(fact_4059_bset_I1_J,axiom,
    ! [D: int,B: set_int,P: int > $o,Q: int > $o] :
      ( ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B )
                 => ( X4
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( P @ X4 )
           => ( P @ ( minus_minus_int @ X4 @ D ) ) ) )
     => ( ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( Q @ X4 )
             => ( Q @ ( minus_minus_int @ X4 @ D ) ) ) )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B )
                   => ( X5
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
             => ( ( P @ ( minus_minus_int @ X5 @ D ) )
                & ( Q @ ( minus_minus_int @ X5 @ D ) ) ) ) ) ) ) ).

% bset(1)
thf(fact_4060_extract__pre__list__assn__lengthD,axiom,
    ! [A: vEBT_VEBT > vEBT_VEBT > assn,Xs2: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L1279224858307276611T_VEBT @ A @ Xs2 @ Xsi ) @ H2 )
     => ( ( size_s6755466524823107622T_VEBT @ Xsi )
        = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_4061_extract__pre__list__assn__lengthD,axiom,
    ! [A: real > vEBT_VEBT > assn,Xs2: list_real,Xsi: list_VEBT_VEBT,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L4595930785310033027T_VEBT @ A @ Xs2 @ Xsi ) @ H2 )
     => ( ( size_s6755466524823107622T_VEBT @ Xsi )
        = ( size_size_list_real @ Xs2 ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_4062_extract__pre__list__assn__lengthD,axiom,
    ! [A: $o > vEBT_VEBT > assn,Xs2: list_o,Xsi: list_VEBT_VEBT,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L1750719106661372127T_VEBT @ A @ Xs2 @ Xsi ) @ H2 )
     => ( ( size_s6755466524823107622T_VEBT @ Xsi )
        = ( size_size_list_o @ Xs2 ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_4063_extract__pre__list__assn__lengthD,axiom,
    ! [A: nat > vEBT_VEBT > assn,Xs2: list_nat,Xsi: list_VEBT_VEBT,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L8158188754432654943T_VEBT @ A @ Xs2 @ Xsi ) @ H2 )
     => ( ( size_s6755466524823107622T_VEBT @ Xsi )
        = ( size_size_list_nat @ Xs2 ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_4064_extract__pre__list__assn__lengthD,axiom,
    ! [A: vEBT_VEBTi > vEBT_VEBT > assn,Xs2: list_VEBT_VEBTi,Xsi: list_VEBT_VEBT,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L7265847600308530106T_VEBT @ A @ Xs2 @ Xsi ) @ H2 )
     => ( ( size_s6755466524823107622T_VEBT @ Xsi )
        = ( size_s7982070591426661849_VEBTi @ Xs2 ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_4065_extract__pre__list__assn__lengthD,axiom,
    ! [A: vEBT_VEBT > real > assn,Xs2: list_VEBT_VEBT,Xsi: list_real,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L5781919052683127133T_real @ A @ Xs2 @ Xsi ) @ H2 )
     => ( ( size_size_list_real @ Xsi )
        = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_4066_extract__pre__list__assn__lengthD,axiom,
    ! [A: real > real > assn,Xs2: list_real,Xsi: list_real,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L1930518968523514909l_real @ A @ Xs2 @ Xsi ) @ H2 )
     => ( ( size_size_list_real @ Xsi )
        = ( size_size_list_real @ Xs2 ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_4067_extract__pre__list__assn__lengthD,axiom,
    ! [A: $o > real > assn,Xs2: list_o,Xsi: list_real,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L4725278957065240257o_real @ A @ Xs2 @ Xsi ) @ H2 )
     => ( ( size_size_list_real @ Xsi )
        = ( size_size_list_o @ Xs2 ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_4068_extract__pre__list__assn__lengthD,axiom,
    ! [A: nat > real > assn,Xs2: list_nat,Xsi: list_real,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L6102073776069194049t_real @ A @ Xs2 @ Xsi ) @ H2 )
     => ( ( size_size_list_real @ Xsi )
        = ( size_size_list_nat @ Xs2 ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_4069_extract__pre__list__assn__lengthD,axiom,
    ! [A: vEBT_VEBTi > real > assn,Xs2: list_VEBT_VEBTi,Xsi: list_real,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( vEBT_L8937798142398754470i_real @ A @ Xs2 @ Xsi ) @ H2 )
     => ( ( size_size_list_real @ Xsi )
        = ( size_s7982070591426661849_VEBTi @ Xs2 ) ) ) ).

% extract_pre_list_assn_lengthD
thf(fact_4070_mod__emp__simp,axiom,
    ! [H2: heap_e7401611519738050253t_unit] : ( rep_assn @ one_one_assn @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ).

% mod_emp_simp
thf(fact_4071_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N3: int] :
      ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N3 ) )
     => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N3 ) )
        = ( insert_int @ ( plus_plus_int @ one_one_int @ N3 ) @ ( set_or1266510415728281911st_int @ M @ N3 ) ) ) ) ).

% atLeastAtMostPlus1_int_conv
thf(fact_4072_pinf_I1_J,axiom,
    ! [P: real > $o,P2: real > $o,Q: real > $o,Q2: real > $o] :
      ( ? [Z4: real] :
        ! [X4: real] :
          ( ( ord_less_real @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P2 @ X4 ) ) )
     => ( ? [Z4: real] :
          ! [X4: real] :
            ( ( ord_less_real @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z2: real] :
          ! [X5: real] :
            ( ( ord_less_real @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P2 @ X5 )
                & ( Q2 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_4073_pinf_I1_J,axiom,
    ! [P: rat > $o,P2: rat > $o,Q: rat > $o,Q2: rat > $o] :
      ( ? [Z4: rat] :
        ! [X4: rat] :
          ( ( ord_less_rat @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P2 @ X4 ) ) )
     => ( ? [Z4: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P2 @ X5 )
                & ( Q2 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_4074_pinf_I1_J,axiom,
    ! [P: num > $o,P2: num > $o,Q: num > $o,Q2: num > $o] :
      ( ? [Z4: num] :
        ! [X4: num] :
          ( ( ord_less_num @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P2 @ X4 ) ) )
     => ( ? [Z4: num] :
          ! [X4: num] :
            ( ( ord_less_num @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( ( ord_less_num @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P2 @ X5 )
                & ( Q2 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_4075_pinf_I1_J,axiom,
    ! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z4: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P2 @ X4 ) ) )
     => ( ? [Z4: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P2 @ X5 )
                & ( Q2 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_4076_pinf_I1_J,axiom,
    ! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z4: int] :
        ! [X4: int] :
          ( ( ord_less_int @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P2 @ X4 ) ) )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P2 @ X5 )
                & ( Q2 @ X5 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_4077_pinf_I2_J,axiom,
    ! [P: real > $o,P2: real > $o,Q: real > $o,Q2: real > $o] :
      ( ? [Z4: real] :
        ! [X4: real] :
          ( ( ord_less_real @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P2 @ X4 ) ) )
     => ( ? [Z4: real] :
          ! [X4: real] :
            ( ( ord_less_real @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z2: real] :
          ! [X5: real] :
            ( ( ord_less_real @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P2 @ X5 )
                | ( Q2 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_4078_pinf_I2_J,axiom,
    ! [P: rat > $o,P2: rat > $o,Q: rat > $o,Q2: rat > $o] :
      ( ? [Z4: rat] :
        ! [X4: rat] :
          ( ( ord_less_rat @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P2 @ X4 ) ) )
     => ( ? [Z4: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P2 @ X5 )
                | ( Q2 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_4079_pinf_I2_J,axiom,
    ! [P: num > $o,P2: num > $o,Q: num > $o,Q2: num > $o] :
      ( ? [Z4: num] :
        ! [X4: num] :
          ( ( ord_less_num @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P2 @ X4 ) ) )
     => ( ? [Z4: num] :
          ! [X4: num] :
            ( ( ord_less_num @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( ( ord_less_num @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P2 @ X5 )
                | ( Q2 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_4080_pinf_I2_J,axiom,
    ! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z4: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P2 @ X4 ) ) )
     => ( ? [Z4: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P2 @ X5 )
                | ( Q2 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_4081_pinf_I2_J,axiom,
    ! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z4: int] :
        ! [X4: int] :
          ( ( ord_less_int @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P2 @ X4 ) ) )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z2 @ X5 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P2 @ X5 )
                | ( Q2 @ X5 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_4082_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_4083_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_4084_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_4085_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_4086_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(3)
thf(fact_4087_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_4088_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_4089_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_4090_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_4091_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ( X5 != T ) ) ).

% pinf(4)
thf(fact_4092_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ~ ( ord_less_real @ X5 @ T ) ) ).

% pinf(5)
thf(fact_4093_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ~ ( ord_less_rat @ X5 @ T ) ) ).

% pinf(5)
thf(fact_4094_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z2 @ X5 )
     => ~ ( ord_less_num @ X5 @ T ) ) ).

% pinf(5)
thf(fact_4095_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ~ ( ord_less_nat @ X5 @ T ) ) ).

% pinf(5)
thf(fact_4096_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ~ ( ord_less_int @ X5 @ T ) ) ).

% pinf(5)
thf(fact_4097_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ( ord_less_real @ T @ X5 ) ) ).

% pinf(7)
thf(fact_4098_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ( ord_less_rat @ T @ X5 ) ) ).

% pinf(7)
thf(fact_4099_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z2 @ X5 )
     => ( ord_less_num @ T @ X5 ) ) ).

% pinf(7)
thf(fact_4100_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( ord_less_nat @ T @ X5 ) ) ).

% pinf(7)
thf(fact_4101_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ( ord_less_int @ T @ X5 ) ) ).

% pinf(7)
thf(fact_4102_minf_I1_J,axiom,
    ! [P: real > $o,P2: real > $o,Q: real > $o,Q2: real > $o] :
      ( ? [Z4: real] :
        ! [X4: real] :
          ( ( ord_less_real @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P2 @ X4 ) ) )
     => ( ? [Z4: real] :
          ! [X4: real] :
            ( ( ord_less_real @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z2: real] :
          ! [X5: real] :
            ( ( ord_less_real @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P2 @ X5 )
                & ( Q2 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_4103_minf_I1_J,axiom,
    ! [P: rat > $o,P2: rat > $o,Q: rat > $o,Q2: rat > $o] :
      ( ? [Z4: rat] :
        ! [X4: rat] :
          ( ( ord_less_rat @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P2 @ X4 ) ) )
     => ( ? [Z4: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P2 @ X5 )
                & ( Q2 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_4104_minf_I1_J,axiom,
    ! [P: num > $o,P2: num > $o,Q: num > $o,Q2: num > $o] :
      ( ? [Z4: num] :
        ! [X4: num] :
          ( ( ord_less_num @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P2 @ X4 ) ) )
     => ( ? [Z4: num] :
          ! [X4: num] :
            ( ( ord_less_num @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( ( ord_less_num @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P2 @ X5 )
                & ( Q2 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_4105_minf_I1_J,axiom,
    ! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z4: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P2 @ X4 ) ) )
     => ( ? [Z4: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P2 @ X5 )
                & ( Q2 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_4106_minf_I1_J,axiom,
    ! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z4: int] :
        ! [X4: int] :
          ( ( ord_less_int @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P2 @ X4 ) ) )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                & ( Q @ X5 ) )
              = ( ( P2 @ X5 )
                & ( Q2 @ X5 ) ) ) ) ) ) ).

% minf(1)
thf(fact_4107_minf_I2_J,axiom,
    ! [P: real > $o,P2: real > $o,Q: real > $o,Q2: real > $o] :
      ( ? [Z4: real] :
        ! [X4: real] :
          ( ( ord_less_real @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P2 @ X4 ) ) )
     => ( ? [Z4: real] :
          ! [X4: real] :
            ( ( ord_less_real @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z2: real] :
          ! [X5: real] :
            ( ( ord_less_real @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P2 @ X5 )
                | ( Q2 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_4108_minf_I2_J,axiom,
    ! [P: rat > $o,P2: rat > $o,Q: rat > $o,Q2: rat > $o] :
      ( ? [Z4: rat] :
        ! [X4: rat] :
          ( ( ord_less_rat @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P2 @ X4 ) ) )
     => ( ? [Z4: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z2: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P2 @ X5 )
                | ( Q2 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_4109_minf_I2_J,axiom,
    ! [P: num > $o,P2: num > $o,Q: num > $o,Q2: num > $o] :
      ( ? [Z4: num] :
        ! [X4: num] :
          ( ( ord_less_num @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P2 @ X4 ) ) )
     => ( ? [Z4: num] :
          ! [X4: num] :
            ( ( ord_less_num @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z2: num] :
          ! [X5: num] :
            ( ( ord_less_num @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P2 @ X5 )
                | ( Q2 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_4110_minf_I2_J,axiom,
    ! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z4: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P2 @ X4 ) ) )
     => ( ? [Z4: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z2: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P2 @ X5 )
                | ( Q2 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_4111_minf_I2_J,axiom,
    ! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
      ( ? [Z4: int] :
        ! [X4: int] :
          ( ( ord_less_int @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P2 @ X4 ) ) )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q2 @ X4 ) ) )
       => ? [Z2: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z2 )
           => ( ( ( P @ X5 )
                | ( Q @ X5 ) )
              = ( ( P2 @ X5 )
                | ( Q2 @ X5 ) ) ) ) ) ) ).

% minf(2)
thf(fact_4112_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_4113_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_4114_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_4115_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_4116_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(3)
thf(fact_4117_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_4118_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_4119_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_4120_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_4121_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ( X5 != T ) ) ).

% minf(4)
thf(fact_4122_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ( ord_less_real @ X5 @ T ) ) ).

% minf(5)
thf(fact_4123_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ( ord_less_rat @ X5 @ T ) ) ).

% minf(5)
thf(fact_4124_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z2 )
     => ( ord_less_num @ X5 @ T ) ) ).

% minf(5)
thf(fact_4125_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( ord_less_nat @ X5 @ T ) ) ).

% minf(5)
thf(fact_4126_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ( ord_less_int @ X5 @ T ) ) ).

% minf(5)
thf(fact_4127_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ~ ( ord_less_real @ T @ X5 ) ) ).

% minf(7)
thf(fact_4128_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ~ ( ord_less_rat @ T @ X5 ) ) ).

% minf(7)
thf(fact_4129_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z2 )
     => ~ ( ord_less_num @ T @ X5 ) ) ).

% minf(7)
thf(fact_4130_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ~ ( ord_less_nat @ T @ X5 ) ) ).

% minf(7)
thf(fact_4131_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ~ ( ord_less_int @ T @ X5 ) ) ).

% minf(7)
thf(fact_4132_bset_I3_J,axiom,
    ! [D: int,T: int,B: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B )
                   => ( X5
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X5 = T )
             => ( ( minus_minus_int @ X5 @ D )
                = T ) ) ) ) ) ).

% bset(3)
thf(fact_4133_bset_I4_J,axiom,
    ! [D: int,T: int,B: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ( member_int @ T @ B )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B )
                   => ( X5
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X5 != T )
             => ( ( minus_minus_int @ X5 @ D )
               != T ) ) ) ) ) ).

% bset(4)
thf(fact_4134_bset_I5_J,axiom,
    ! [D: int,B: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B )
                 => ( X5
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ X5 @ T )
           => ( ord_less_int @ ( minus_minus_int @ X5 @ D ) @ T ) ) ) ) ).

% bset(5)
thf(fact_4135_bset_I7_J,axiom,
    ! [D: int,T: int,B: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ( member_int @ T @ B )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B )
                   => ( X5
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ T @ X5 )
             => ( ord_less_int @ T @ ( minus_minus_int @ X5 @ D ) ) ) ) ) ) ).

% bset(7)
thf(fact_4136_aset_I3_J,axiom,
    ! [D: int,T: int,A: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A )
                   => ( X5
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X5 = T )
             => ( ( plus_plus_int @ X5 @ D )
                = T ) ) ) ) ) ).

% aset(3)
thf(fact_4137_aset_I4_J,axiom,
    ! [D: int,T: int,A: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ( member_int @ T @ A )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A )
                   => ( X5
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( X5 != T )
             => ( ( plus_plus_int @ X5 @ D )
               != T ) ) ) ) ) ).

% aset(4)
thf(fact_4138_aset_I5_J,axiom,
    ! [D: int,T: int,A: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ( member_int @ T @ A )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A )
                   => ( X5
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ X5 @ T )
             => ( ord_less_int @ ( plus_plus_int @ X5 @ D ) @ T ) ) ) ) ) ).

% aset(5)
thf(fact_4139_aset_I7_J,axiom,
    ! [D: int,A: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A )
                 => ( X5
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ T @ X5 )
           => ( ord_less_int @ T @ ( plus_plus_int @ X5 @ D ) ) ) ) ) ).

% aset(7)
thf(fact_4140_periodic__finite__ex,axiom,
    ! [D2: int,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X4: int,K2: int] :
            ( ( P @ X4 )
            = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D2 ) ) ) )
       => ( ( ? [X8: int] : ( P @ X8 ) )
          = ( ? [X3: int] :
                ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D2 ) )
                & ( P @ X3 ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_4141_bset_I6_J,axiom,
    ! [D: int,B: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B )
                 => ( X5
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ X5 @ T )
           => ( ord_less_eq_int @ ( minus_minus_int @ X5 @ D ) @ T ) ) ) ) ).

% bset(6)
thf(fact_4142_bset_I8_J,axiom,
    ! [D: int,T: int,B: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B )
                   => ( X5
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ T @ X5 )
             => ( ord_less_eq_int @ T @ ( minus_minus_int @ X5 @ D ) ) ) ) ) ) ).

% bset(8)
thf(fact_4143_aset_I6_J,axiom,
    ! [D: int,T: int,A: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A )
       => ! [X5: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A )
                   => ( X5
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ X5 @ T )
             => ( ord_less_eq_int @ ( plus_plus_int @ X5 @ D ) @ T ) ) ) ) ) ).

% aset(6)
thf(fact_4144_aset_I8_J,axiom,
    ! [D: int,A: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A )
                 => ( X5
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ T @ X5 )
           => ( ord_less_eq_int @ T @ ( plus_plus_int @ X5 @ D ) ) ) ) ) ).

% aset(8)
thf(fact_4145_cppi,axiom,
    ! [D: int,P: int > $o,P2: int > $o,A: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z4 @ X4 )
           => ( ( P @ X4 )
              = ( P2 @ X4 ) ) )
       => ( ! [X4: int] :
              ( ! [Xa2: int] :
                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                 => ! [Xb2: int] :
                      ( ( member_int @ Xb2 @ A )
                     => ( X4
                       != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
             => ( ( P @ X4 )
               => ( P @ ( plus_plus_int @ X4 @ D ) ) ) )
         => ( ! [X4: int,K2: int] :
                ( ( P2 @ X4 )
                = ( P2 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
           => ( ( ? [X8: int] : ( P @ X8 ) )
              = ( ? [X3: int] :
                    ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                    & ( P2 @ X3 ) )
                | ? [X3: int] :
                    ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                    & ? [Y2: int] :
                        ( ( member_int @ Y2 @ A )
                        & ( P @ ( minus_minus_int @ Y2 @ X3 ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_4146_cpmi,axiom,
    ! [D: int,P: int > $o,P2: int > $o,B: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z4 )
           => ( ( P @ X4 )
              = ( P2 @ X4 ) ) )
       => ( ! [X4: int] :
              ( ! [Xa2: int] :
                  ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                 => ! [Xb2: int] :
                      ( ( member_int @ Xb2 @ B )
                     => ( X4
                       != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
             => ( ( P @ X4 )
               => ( P @ ( minus_minus_int @ X4 @ D ) ) ) )
         => ( ! [X4: int,K2: int] :
                ( ( P2 @ X4 )
                = ( P2 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
           => ( ( ? [X8: int] : ( P @ X8 ) )
              = ( ? [X3: int] :
                    ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                    & ( P2 @ X3 ) )
                | ? [X3: int] :
                    ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                    & ? [Y2: int] :
                        ( ( member_int @ Y2 @ B )
                        & ( P @ ( plus_plus_int @ Y2 @ X3 ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_4147_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ~ ( ord_less_eq_real @ X5 @ T ) ) ).

% pinf(6)
thf(fact_4148_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ~ ( ord_less_eq_rat @ X5 @ T ) ) ).

% pinf(6)
thf(fact_4149_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z2 @ X5 )
     => ~ ( ord_less_eq_num @ X5 @ T ) ) ).

% pinf(6)
thf(fact_4150_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ~ ( ord_less_eq_nat @ X5 @ T ) ) ).

% pinf(6)
thf(fact_4151_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ~ ( ord_less_eq_int @ X5 @ T ) ) ).

% pinf(6)
thf(fact_4152_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ( ord_less_eq_real @ T @ X5 ) ) ).

% pinf(8)
thf(fact_4153_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ( ord_less_eq_rat @ T @ X5 ) ) ).

% pinf(8)
thf(fact_4154_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z2 @ X5 )
     => ( ord_less_eq_num @ T @ X5 ) ) ).

% pinf(8)
thf(fact_4155_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( ord_less_eq_nat @ T @ X5 ) ) ).

% pinf(8)
thf(fact_4156_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ( ord_less_eq_int @ T @ X5 ) ) ).

% pinf(8)
thf(fact_4157_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ( ord_less_eq_real @ X5 @ T ) ) ).

% minf(6)
thf(fact_4158_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ( ord_less_eq_rat @ X5 @ T ) ) ).

% minf(6)
thf(fact_4159_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z2 )
     => ( ord_less_eq_num @ X5 @ T ) ) ).

% minf(6)
thf(fact_4160_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( ord_less_eq_nat @ X5 @ T ) ) ).

% minf(6)
thf(fact_4161_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ( ord_less_eq_int @ X5 @ T ) ) ).

% minf(6)
thf(fact_4162_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ~ ( ord_less_eq_real @ T @ X5 ) ) ).

% minf(8)
thf(fact_4163_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ~ ( ord_less_eq_rat @ T @ X5 ) ) ).

% minf(8)
thf(fact_4164_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z2: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z2 )
     => ~ ( ord_less_eq_num @ T @ X5 ) ) ).

% minf(8)
thf(fact_4165_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ~ ( ord_less_eq_nat @ T @ X5 ) ) ).

% minf(8)
thf(fact_4166_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ~ ( ord_less_eq_int @ T @ X5 ) ) ).

% minf(8)
thf(fact_4167_inf__period_I1_J,axiom,
    ! [P: real > $o,D: real,Q: real > $o] :
      ( ! [X4: real,K2: real] :
          ( ( P @ X4 )
          = ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D ) ) ) )
     => ( ! [X4: real,K2: real] :
            ( ( Q @ X4 )
            = ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D ) ) ) )
       => ! [X5: real,K5: real] :
            ( ( ( P @ X5 )
              & ( Q @ X5 ) )
            = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K5 @ D ) ) )
              & ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K5 @ D ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_4168_inf__period_I1_J,axiom,
    ! [P: rat > $o,D: rat,Q: rat > $o] :
      ( ! [X4: rat,K2: rat] :
          ( ( P @ X4 )
          = ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K2 @ D ) ) ) )
     => ( ! [X4: rat,K2: rat] :
            ( ( Q @ X4 )
            = ( Q @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K2 @ D ) ) ) )
       => ! [X5: rat,K5: rat] :
            ( ( ( P @ X5 )
              & ( Q @ X5 ) )
            = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K5 @ D ) ) )
              & ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K5 @ D ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_4169_inf__period_I1_J,axiom,
    ! [P: int > $o,D: int,Q: int > $o] :
      ( ! [X4: int,K2: int] :
          ( ( P @ X4 )
          = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
     => ( ! [X4: int,K2: int] :
            ( ( Q @ X4 )
            = ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
       => ! [X5: int,K5: int] :
            ( ( ( P @ X5 )
              & ( Q @ X5 ) )
            = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K5 @ D ) ) )
              & ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K5 @ D ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_4170_inf__period_I2_J,axiom,
    ! [P: real > $o,D: real,Q: real > $o] :
      ( ! [X4: real,K2: real] :
          ( ( P @ X4 )
          = ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D ) ) ) )
     => ( ! [X4: real,K2: real] :
            ( ( Q @ X4 )
            = ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D ) ) ) )
       => ! [X5: real,K5: real] :
            ( ( ( P @ X5 )
              | ( Q @ X5 ) )
            = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K5 @ D ) ) )
              | ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K5 @ D ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_4171_inf__period_I2_J,axiom,
    ! [P: rat > $o,D: rat,Q: rat > $o] :
      ( ! [X4: rat,K2: rat] :
          ( ( P @ X4 )
          = ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K2 @ D ) ) ) )
     => ( ! [X4: rat,K2: rat] :
            ( ( Q @ X4 )
            = ( Q @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K2 @ D ) ) ) )
       => ! [X5: rat,K5: rat] :
            ( ( ( P @ X5 )
              | ( Q @ X5 ) )
            = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K5 @ D ) ) )
              | ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K5 @ D ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_4172_inf__period_I2_J,axiom,
    ! [P: int > $o,D: int,Q: int > $o] :
      ( ! [X4: int,K2: int] :
          ( ( P @ X4 )
          = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
     => ( ! [X4: int,K2: int] :
            ( ( Q @ X4 )
            = ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
       => ! [X5: int,K5: int] :
            ( ( ( P @ X5 )
              | ( Q @ X5 ) )
            = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K5 @ D ) ) )
              | ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K5 @ D ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_4173_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ( ( X
         != ( suc @ zero_zero_nat ) )
       => ~ ! [Va3: nat] :
              ( X
             != ( suc @ ( suc @ Va3 ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
thf(fact_4174_vebt__delete_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,N3: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ N3 ) ) )
      = ( vEBT_Leaf @ A2 @ B2 ) ) ).

% vebt_delete.simps(3)
thf(fact_4175_vebt__delete_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B2 ) @ zero_zero_nat )
      = ( vEBT_Leaf @ $false @ B2 ) ) ).

% vebt_delete.simps(1)
thf(fact_4176_VEBT__internal_OT__vebt__buildupi_Osimps_I1_J,axiom,
    ( ( vEBT_V441764108873111860ildupi @ zero_zero_nat )
    = ( suc @ zero_zero_nat ) ) ).

% VEBT_internal.T_vebt_buildupi.simps(1)
thf(fact_4177_VEBT__internal_OT__vebt__buildupi_Osimps_I2_J,axiom,
    ( ( vEBT_V441764108873111860ildupi @ ( suc @ zero_zero_nat ) )
    = ( suc @ zero_zero_nat ) ) ).

% VEBT_internal.T_vebt_buildupi.simps(2)
thf(fact_4178_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_4179_VEBT__internal_OT__vebt__buildupi_H_Osimps_I1_J,axiom,
    ( ( vEBT_V9176841429113362141ildupi @ zero_zero_nat )
    = one_one_int ) ).

% VEBT_internal.T_vebt_buildupi'.simps(1)
thf(fact_4180_VEBT__internal_Ocnt_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      ( ( vEBT_VEBT_cnt @ ( vEBT_Leaf @ A2 @ B2 ) )
      = one_one_real ) ).

% VEBT_internal.cnt.simps(1)
thf(fact_4181_vebt__delete_Osimps_I2_J,axiom,
    ! [A2: $o,B2: $o] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ zero_zero_nat ) )
      = ( vEBT_Leaf @ A2 @ $false ) ) ).

% vebt_delete.simps(2)
thf(fact_4182_minusinfinity,axiom,
    ! [D2: int,P1: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X4: int,K2: int] :
            ( ( P1 @ X4 )
            = ( P1 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D2 ) ) ) )
       => ( ? [Z4: int] :
            ! [X4: int] :
              ( ( ord_less_int @ X4 @ Z4 )
             => ( ( P @ X4 )
                = ( P1 @ X4 ) ) )
         => ( ? [X_12: int] : ( P1 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% minusinfinity
thf(fact_4183_plusinfinity,axiom,
    ! [D2: int,P2: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X4: int,K2: int] :
            ( ( P2 @ X4 )
            = ( P2 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D2 ) ) ) )
       => ( ? [Z4: int] :
            ! [X4: int] :
              ( ( ord_less_int @ Z4 @ X4 )
             => ( ( P @ X4 )
                = ( P2 @ X4 ) ) )
         => ( ? [X_12: int] : ( P2 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% plusinfinity
thf(fact_4184_VEBT__internal_OT__vebt__buildupi_H_Osimps_I2_J,axiom,
    ( ( vEBT_V9176841429113362141ildupi @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% VEBT_internal.T_vebt_buildupi'.simps(2)
thf(fact_4185_double__not__eq__Suc__double,axiom,
    ! [M: nat,N3: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
     != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% double_not_eq_Suc_double
thf(fact_4186_Suc__double__not__eq__double,axiom,
    ! [M: nat,N3: nat] :
      ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
     != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% Suc_double_not_eq_double
thf(fact_4187_incr__mult__lemma,axiom,
    ! [D2: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X4: int] :
            ( ( P @ X4 )
           => ( P @ ( plus_plus_int @ X4 @ D2 ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X5: int] :
              ( ( P @ X5 )
             => ( P @ ( plus_plus_int @ X5 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_4188_decr__mult__lemma,axiom,
    ! [D2: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ! [X4: int] :
            ( ( P @ X4 )
           => ( P @ ( minus_minus_int @ X4 @ D2 ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X5: int] :
              ( ( P @ X5 )
             => ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D2 ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_4189_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I1_J,axiom,
    ( ( vEBT_V8646137997579335489_i_l_d @ zero_zero_nat )
    = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(1)
thf(fact_4190_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) ) ).

% vebt_delete.simps(5)
thf(fact_4191_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I1_J,axiom,
    ( ( vEBT_V8346862874174094_d_u_p @ zero_zero_nat )
    = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(1)
thf(fact_4192_VEBT__internal_OTb_H_Osimps_I1_J,axiom,
    ( ( vEBT_VEBT_Tb2 @ zero_zero_nat )
    = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% VEBT_internal.Tb'.simps(1)
thf(fact_4193_VEBT__internal_OTb_Osimps_I1_J,axiom,
    ( ( vEBT_VEBT_Tb @ zero_zero_nat )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% VEBT_internal.Tb.simps(1)
thf(fact_4194_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      ( ( vEBT_VEBT_space2 @ ( vEBT_Leaf @ A2 @ B2 ) )
      = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ).

% VEBT_internal.space'.simps(1)
thf(fact_4195_VEBT__internal_Ospace_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      ( ( vEBT_VEBT_space @ ( vEBT_Leaf @ A2 @ B2 ) )
      = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% VEBT_internal.space.simps(1)
thf(fact_4196_nat__approx__posE,axiom,
    ! [E: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ E )
     => ~ ! [N: nat] :
            ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_4197_nat__approx__posE,axiom,
    ! [E: real] :
      ( ( ord_less_real @ zero_zero_real @ E )
     => ~ ! [N: nat] :
            ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_4198_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: $o] :
      ( ( ( vEBT_VEBT_minNull @ X )
        = Y )
     => ( ( ( X
            = ( vEBT_Leaf @ $false @ $false ) )
         => ~ Y )
       => ( ( ? [Uv: $o] :
                ( X
                = ( vEBT_Leaf @ $true @ Uv ) )
           => Y )
         => ( ( ? [Uu2: $o] :
                  ( X
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
             => Y )
           => ( ( ? [Uw2: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux @ Uy ) )
               => ~ Y )
             => ~ ( ? [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va2 @ Vb @ Vc ) )
                 => Y ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
thf(fact_4199_member__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N3: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_m_e_m_b_e_r @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).

% member_bound_size_univ
thf(fact_4200_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X ) ).

% vebt_member.simps(4)
thf(fact_4201_int__ops_I6_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B2 ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B2 ) )
          = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% int_ops(6)
thf(fact_4202_max__bot2,axiom,
    ! [X: set_real] :
      ( ( ord_max_set_real @ X @ bot_bot_set_real )
      = X ) ).

% max_bot2
thf(fact_4203_max__bot2,axiom,
    ! [X: set_o] :
      ( ( ord_max_set_o @ X @ bot_bot_set_o )
      = X ) ).

% max_bot2
thf(fact_4204_max__bot2,axiom,
    ! [X: set_nat] :
      ( ( ord_max_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% max_bot2
thf(fact_4205_max__bot2,axiom,
    ! [X: set_int] :
      ( ( ord_max_set_int @ X @ bot_bot_set_int )
      = X ) ).

% max_bot2
thf(fact_4206_max__bot2,axiom,
    ! [X: nat] :
      ( ( ord_max_nat @ X @ bot_bot_nat )
      = X ) ).

% max_bot2
thf(fact_4207_max__bot,axiom,
    ! [X: set_real] :
      ( ( ord_max_set_real @ bot_bot_set_real @ X )
      = X ) ).

% max_bot
thf(fact_4208_max__bot,axiom,
    ! [X: set_o] :
      ( ( ord_max_set_o @ bot_bot_set_o @ X )
      = X ) ).

% max_bot
thf(fact_4209_max__bot,axiom,
    ! [X: set_nat] :
      ( ( ord_max_set_nat @ bot_bot_set_nat @ X )
      = X ) ).

% max_bot
thf(fact_4210_max__bot,axiom,
    ! [X: set_int] :
      ( ( ord_max_set_int @ bot_bot_set_int @ X )
      = X ) ).

% max_bot
thf(fact_4211_max__bot,axiom,
    ! [X: nat] :
      ( ( ord_max_nat @ bot_bot_nat @ X )
      = X ) ).

% max_bot
thf(fact_4212_pos__mult__pos__ge,axiom,
    ! [X: int,N3: int] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ N3 )
       => ( ord_less_eq_int @ ( times_times_int @ N3 @ one_one_int ) @ ( times_times_int @ N3 @ X ) ) ) ) ).

% pos_mult_pos_ge
thf(fact_4213_dual__order_Orefl,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_4214_dual__order_Orefl,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_4215_dual__order_Orefl,axiom,
    ! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_4216_dual__order_Orefl,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_4217_dual__order_Orefl,axiom,
    ! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_4218_order__refl,axiom,
    ! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).

% order_refl
thf(fact_4219_order__refl,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).

% order_refl
thf(fact_4220_order__refl,axiom,
    ! [X: num] : ( ord_less_eq_num @ X @ X ) ).

% order_refl
thf(fact_4221_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_4222_order__refl,axiom,
    ! [X: int] : ( ord_less_eq_int @ X @ X ) ).

% order_refl
thf(fact_4223_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y22: num] :
      ( ( ( bit0 @ X22 )
        = ( bit0 @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% verit_eq_simplify(8)
thf(fact_4224_star__false__left,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ bot_bot_assn @ P )
      = bot_bot_assn ) ).

% star_false_left
thf(fact_4225_star__false__right,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ P @ bot_bot_assn )
      = bot_bot_assn ) ).

% star_false_right
thf(fact_4226_pure__assn__eq__false__iff,axiom,
    ! [P: $o] :
      ( ( ( pure_assn @ P )
        = bot_bot_assn )
      = ~ P ) ).

% pure_assn_eq_false_iff
thf(fact_4227_pure__false,axiom,
    ( ( pure_assn @ $false )
    = bot_bot_assn ) ).

% pure_false
thf(fact_4228_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
thf(fact_4229_ent__false__iff,axiom,
    ! [P: assn] :
      ( ( entails @ P @ bot_bot_assn )
      = ( ! [H: produc3658429121746597890et_nat] :
            ~ ( rep_assn @ P @ H ) ) ) ).

% ent_false_iff
thf(fact_4230_snga__same__false,axiom,
    ! [P6: array_VEBT_VEBTi,X: list_VEBT_VEBTi,Y: list_VEBT_VEBTi] :
      ( ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ P6 @ X ) @ ( snga_assn_VEBT_VEBTi @ P6 @ Y ) )
      = bot_bot_assn ) ).

% snga_same_false
thf(fact_4231_bot__set__def,axiom,
    ( bot_bot_set_complex
    = ( collect_complex @ bot_bot_complex_o ) ) ).

% bot_set_def
thf(fact_4232_bot__set__def,axiom,
    ( bot_bo1796632182523588997nt_int
    = ( collec213857154873943460nt_int @ bot_bo8147686125503663512_int_o ) ) ).

% bot_set_def
thf(fact_4233_bot__set__def,axiom,
    ( bot_bot_set_real
    = ( collect_real @ bot_bot_real_o ) ) ).

% bot_set_def
thf(fact_4234_bot__set__def,axiom,
    ( bot_bot_set_o
    = ( collect_o @ bot_bot_o_o ) ) ).

% bot_set_def
thf(fact_4235_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_4236_bot__set__def,axiom,
    ( bot_bot_set_int
    = ( collect_int @ bot_bot_int_o ) ) ).

% bot_set_def
thf(fact_4237_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_4238_bot__option__def,axiom,
    bot_bot_option_nat = none_nat ).

% bot_option_def
thf(fact_4239_ent__false,axiom,
    ! [P: assn] : ( entails @ bot_bot_assn @ P ) ).

% ent_false
thf(fact_4240_mod__false,axiom,
    ! [H2: produc3658429121746597890et_nat] :
      ~ ( rep_assn @ bot_bot_assn @ H2 ) ).

% mod_false
thf(fact_4241_list__assn__aux__ineq__len,axiom,
    ! [L2: list_VEBT_VEBT,Li2: list_VEBT_VEBT,A: vEBT_VEBT > vEBT_VEBT > assn] :
      ( ( ( size_s6755466524823107622T_VEBT @ L2 )
       != ( size_s6755466524823107622T_VEBT @ Li2 ) )
     => ( ( vEBT_L1279224858307276611T_VEBT @ A @ L2 @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_4242_list__assn__aux__ineq__len,axiom,
    ! [L2: list_VEBT_VEBT,Li2: list_real,A: vEBT_VEBT > real > assn] :
      ( ( ( size_s6755466524823107622T_VEBT @ L2 )
       != ( size_size_list_real @ Li2 ) )
     => ( ( vEBT_L5781919052683127133T_real @ A @ L2 @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_4243_list__assn__aux__ineq__len,axiom,
    ! [L2: list_VEBT_VEBT,Li2: list_o,A: vEBT_VEBT > $o > assn] :
      ( ( ( size_s6755466524823107622T_VEBT @ L2 )
       != ( size_size_list_o @ Li2 ) )
     => ( ( vEBT_L7489408758114837031VEBT_o @ A @ L2 @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_4244_list__assn__aux__ineq__len,axiom,
    ! [L2: list_VEBT_VEBT,Li2: list_nat,A: vEBT_VEBT > nat > assn] :
      ( ( ( size_s6755466524823107622T_VEBT @ L2 )
       != ( size_size_list_nat @ Li2 ) )
     => ( ( vEBT_L8296926524756676353BT_nat @ A @ L2 @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_4245_list__assn__aux__ineq__len,axiom,
    ! [L2: list_real,Li2: list_VEBT_VEBT,A: real > vEBT_VEBT > assn] :
      ( ( ( size_size_list_real @ L2 )
       != ( size_s6755466524823107622T_VEBT @ Li2 ) )
     => ( ( vEBT_L4595930785310033027T_VEBT @ A @ L2 @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_4246_list__assn__aux__ineq__len,axiom,
    ! [L2: list_real,Li2: list_real,A: real > real > assn] :
      ( ( ( size_size_list_real @ L2 )
       != ( size_size_list_real @ Li2 ) )
     => ( ( vEBT_L1930518968523514909l_real @ A @ L2 @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_4247_list__assn__aux__ineq__len,axiom,
    ! [L2: list_real,Li2: list_o,A: real > $o > assn] :
      ( ( ( size_size_list_real @ L2 )
       != ( size_size_list_o @ Li2 ) )
     => ( ( vEBT_L6234343332106409831real_o @ A @ L2 @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_4248_list__assn__aux__ineq__len,axiom,
    ! [L2: list_real,Li2: list_nat,A: real > nat > assn] :
      ( ( ( size_size_list_real @ L2 )
       != ( size_size_list_nat @ Li2 ) )
     => ( ( vEBT_L1446010312343316929al_nat @ A @ L2 @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_4249_list__assn__aux__ineq__len,axiom,
    ! [L2: list_real,Li2: list_VEBT_VEBTi,A: real > vEBT_VEBTi > assn] :
      ( ( ( size_size_list_real @ L2 )
       != ( size_s7982070591426661849_VEBTi @ Li2 ) )
     => ( ( vEBT_L9060850011106065574_VEBTi @ A @ L2 @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_4250_list__assn__aux__ineq__len,axiom,
    ! [L2: list_o,Li2: list_VEBT_VEBT,A: $o > vEBT_VEBT > assn] :
      ( ( ( size_size_list_o @ L2 )
       != ( size_s6755466524823107622T_VEBT @ Li2 ) )
     => ( ( vEBT_L1750719106661372127T_VEBT @ A @ L2 @ Li2 )
        = bot_bot_assn ) ) ).

% list_assn_aux_ineq_len
thf(fact_4251_order__antisym__conv,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_less_eq_set_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_4252_order__antisym__conv,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( ord_less_eq_rat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_4253_order__antisym__conv,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ( ( ord_less_eq_num @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_4254_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_4255_order__antisym__conv,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_4256_linorder__le__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_eq_rat @ X @ Y )
     => ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_4257_linorder__le__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_eq_num @ X @ Y )
     => ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_4258_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_4259_linorder__le__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_eq_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_4260_ord__le__eq__subst,axiom,
    ! [A2: rat,B2: rat,F2: rat > rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_4261_ord__le__eq__subst,axiom,
    ! [A2: rat,B2: rat,F2: rat > num,C2: num] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_4262_ord__le__eq__subst,axiom,
    ! [A2: rat,B2: rat,F2: rat > nat,C2: nat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_4263_ord__le__eq__subst,axiom,
    ! [A2: rat,B2: rat,F2: rat > int,C2: int] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_4264_ord__le__eq__subst,axiom,
    ! [A2: num,B2: num,F2: num > rat,C2: rat] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_4265_ord__le__eq__subst,axiom,
    ! [A2: num,B2: num,F2: num > num,C2: num] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_4266_ord__le__eq__subst,axiom,
    ! [A2: num,B2: num,F2: num > nat,C2: nat] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_4267_ord__le__eq__subst,axiom,
    ! [A2: num,B2: num,F2: num > int,C2: int] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_4268_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F2: nat > rat,C2: rat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_4269_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F2: nat > num,C2: num] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_4270_ord__eq__le__subst,axiom,
    ! [A2: rat,F2: rat > rat,B2: rat,C2: rat] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_4271_ord__eq__le__subst,axiom,
    ! [A2: num,F2: rat > num,B2: rat,C2: rat] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_4272_ord__eq__le__subst,axiom,
    ! [A2: nat,F2: rat > nat,B2: rat,C2: rat] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_4273_ord__eq__le__subst,axiom,
    ! [A2: int,F2: rat > int,B2: rat,C2: rat] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_4274_ord__eq__le__subst,axiom,
    ! [A2: rat,F2: num > rat,B2: num,C2: num] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_4275_ord__eq__le__subst,axiom,
    ! [A2: num,F2: num > num,B2: num,C2: num] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_4276_ord__eq__le__subst,axiom,
    ! [A2: nat,F2: num > nat,B2: num,C2: num] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_4277_ord__eq__le__subst,axiom,
    ! [A2: int,F2: num > int,B2: num,C2: num] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_4278_ord__eq__le__subst,axiom,
    ! [A2: rat,F2: nat > rat,B2: nat,C2: nat] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_4279_ord__eq__le__subst,axiom,
    ! [A2: num,F2: nat > num,B2: nat,C2: nat] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_4280_linorder__linear,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
      | ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_linear
thf(fact_4281_linorder__linear,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
      | ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_linear
thf(fact_4282_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_4283_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_linear
thf(fact_4284_verit__la__disequality,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 = B2 )
      | ~ ( ord_less_eq_rat @ A2 @ B2 )
      | ~ ( ord_less_eq_rat @ B2 @ A2 ) ) ).

% verit_la_disequality
thf(fact_4285_verit__la__disequality,axiom,
    ! [A2: num,B2: num] :
      ( ( A2 = B2 )
      | ~ ( ord_less_eq_num @ A2 @ B2 )
      | ~ ( ord_less_eq_num @ B2 @ A2 ) ) ).

% verit_la_disequality
thf(fact_4286_verit__la__disequality,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 = B2 )
      | ~ ( ord_less_eq_nat @ A2 @ B2 )
      | ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ).

% verit_la_disequality
thf(fact_4287_verit__la__disequality,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 = B2 )
      | ~ ( ord_less_eq_int @ A2 @ B2 )
      | ~ ( ord_less_eq_int @ B2 @ A2 ) ) ).

% verit_la_disequality
thf(fact_4288_order__eq__refl,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
     => ( ord_less_eq_set_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_4289_order__eq__refl,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_4290_order__eq__refl,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
     => ( ord_less_eq_num @ X @ Y ) ) ).

% order_eq_refl
thf(fact_4291_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_4292_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_eq_refl
thf(fact_4293_order__subst2,axiom,
    ! [A2: rat,B2: rat,F2: rat > rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_4294_order__subst2,axiom,
    ! [A2: rat,B2: rat,F2: rat > num,C2: num] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_num @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_4295_order__subst2,axiom,
    ! [A2: rat,B2: rat,F2: rat > nat,C2: nat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_4296_order__subst2,axiom,
    ! [A2: rat,B2: rat,F2: rat > int,C2: int] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_int @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_4297_order__subst2,axiom,
    ! [A2: num,B2: num,F2: num > rat,C2: rat] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_4298_order__subst2,axiom,
    ! [A2: num,B2: num,F2: num > num,C2: num] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_eq_num @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_4299_order__subst2,axiom,
    ! [A2: num,B2: num,F2: num > nat,C2: nat] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_4300_order__subst2,axiom,
    ! [A2: num,B2: num,F2: num > int,C2: int] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_eq_int @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_4301_order__subst2,axiom,
    ! [A2: nat,B2: nat,F2: nat > rat,C2: rat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_4302_order__subst2,axiom,
    ! [A2: nat,B2: nat,F2: nat > num,C2: num] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_num @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_4303_order__subst1,axiom,
    ! [A2: rat,F2: rat > rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_4304_order__subst1,axiom,
    ! [A2: rat,F2: num > rat,B2: num,C2: num] :
      ( ( ord_less_eq_rat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_4305_order__subst1,axiom,
    ! [A2: rat,F2: nat > rat,B2: nat,C2: nat] :
      ( ( ord_less_eq_rat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_4306_order__subst1,axiom,
    ! [A2: rat,F2: int > rat,B2: int,C2: int] :
      ( ( ord_less_eq_rat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_int @ B2 @ C2 )
       => ( ! [X4: int,Y3: int] :
              ( ( ord_less_eq_int @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_4307_order__subst1,axiom,
    ! [A2: num,F2: rat > num,B2: rat,C2: rat] :
      ( ( ord_less_eq_num @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_4308_order__subst1,axiom,
    ! [A2: num,F2: num > num,B2: num,C2: num] :
      ( ( ord_less_eq_num @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_4309_order__subst1,axiom,
    ! [A2: num,F2: nat > num,B2: nat,C2: nat] :
      ( ( ord_less_eq_num @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_4310_order__subst1,axiom,
    ! [A2: num,F2: int > num,B2: int,C2: int] :
      ( ( ord_less_eq_num @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_int @ B2 @ C2 )
       => ( ! [X4: int,Y3: int] :
              ( ( ord_less_eq_int @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_4311_order__subst1,axiom,
    ! [A2: nat,F2: rat > nat,B2: rat,C2: rat] :
      ( ( ord_less_eq_nat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_4312_order__subst1,axiom,
    ! [A2: nat,F2: num > nat,B2: num,C2: num] :
      ( ( ord_less_eq_nat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_4313_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_nat,Z3: set_nat] : Y5 = Z3 )
    = ( ^ [A7: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ A7 @ B7 )
          & ( ord_less_eq_set_nat @ B7 @ A7 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_4314_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: rat,Z3: rat] : Y5 = Z3 )
    = ( ^ [A7: rat,B7: rat] :
          ( ( ord_less_eq_rat @ A7 @ B7 )
          & ( ord_less_eq_rat @ B7 @ A7 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_4315_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: num,Z3: num] : Y5 = Z3 )
    = ( ^ [A7: num,B7: num] :
          ( ( ord_less_eq_num @ A7 @ B7 )
          & ( ord_less_eq_num @ B7 @ A7 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_4316_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z3: nat] : Y5 = Z3 )
    = ( ^ [A7: nat,B7: nat] :
          ( ( ord_less_eq_nat @ A7 @ B7 )
          & ( ord_less_eq_nat @ B7 @ A7 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_4317_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: int,Z3: int] : Y5 = Z3 )
    = ( ^ [A7: int,B7: int] :
          ( ( ord_less_eq_int @ A7 @ B7 )
          & ( ord_less_eq_int @ B7 @ A7 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_4318_antisym,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_4319_antisym,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_4320_antisym,axiom,
    ! [A2: num,B2: num] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_eq_num @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_4321_antisym,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_4322_antisym,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_4323_dual__order_Otrans,axiom,
    ! [B2: set_nat,A2: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( ord_less_eq_set_nat @ C2 @ B2 )
       => ( ord_less_eq_set_nat @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_4324_dual__order_Otrans,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( ord_less_eq_rat @ C2 @ B2 )
       => ( ord_less_eq_rat @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_4325_dual__order_Otrans,axiom,
    ! [B2: num,A2: num,C2: num] :
      ( ( ord_less_eq_num @ B2 @ A2 )
     => ( ( ord_less_eq_num @ C2 @ B2 )
       => ( ord_less_eq_num @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_4326_dual__order_Otrans,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ C2 @ B2 )
       => ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_4327_dual__order_Otrans,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_less_eq_int @ C2 @ B2 )
       => ( ord_less_eq_int @ C2 @ A2 ) ) ) ).

% dual_order.trans
thf(fact_4328_dual__order_Oantisym,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_4329_dual__order_Oantisym,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( ord_less_eq_rat @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_4330_dual__order_Oantisym,axiom,
    ! [B2: num,A2: num] :
      ( ( ord_less_eq_num @ B2 @ A2 )
     => ( ( ord_less_eq_num @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_4331_dual__order_Oantisym,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_4332_dual__order_Oantisym,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_less_eq_int @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_4333_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: set_nat,Z3: set_nat] : Y5 = Z3 )
    = ( ^ [A7: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ B7 @ A7 )
          & ( ord_less_eq_set_nat @ A7 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_4334_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: rat,Z3: rat] : Y5 = Z3 )
    = ( ^ [A7: rat,B7: rat] :
          ( ( ord_less_eq_rat @ B7 @ A7 )
          & ( ord_less_eq_rat @ A7 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_4335_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: num,Z3: num] : Y5 = Z3 )
    = ( ^ [A7: num,B7: num] :
          ( ( ord_less_eq_num @ B7 @ A7 )
          & ( ord_less_eq_num @ A7 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_4336_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: nat,Z3: nat] : Y5 = Z3 )
    = ( ^ [A7: nat,B7: nat] :
          ( ( ord_less_eq_nat @ B7 @ A7 )
          & ( ord_less_eq_nat @ A7 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_4337_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: int,Z3: int] : Y5 = Z3 )
    = ( ^ [A7: int,B7: int] :
          ( ( ord_less_eq_int @ B7 @ A7 )
          & ( ord_less_eq_int @ A7 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_4338_linorder__wlog,axiom,
    ! [P: rat > rat > $o,A2: rat,B2: rat] :
      ( ! [A3: rat,B3: rat] :
          ( ( ord_less_eq_rat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: rat,B3: rat] :
            ( ( P @ B3 @ A3 )
           => ( P @ A3 @ B3 ) )
       => ( P @ A2 @ B2 ) ) ) ).

% linorder_wlog
thf(fact_4339_linorder__wlog,axiom,
    ! [P: num > num > $o,A2: num,B2: num] :
      ( ! [A3: num,B3: num] :
          ( ( ord_less_eq_num @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: num,B3: num] :
            ( ( P @ B3 @ A3 )
           => ( P @ A3 @ B3 ) )
       => ( P @ A2 @ B2 ) ) ) ).

% linorder_wlog
thf(fact_4340_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A2: nat,B2: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: nat,B3: nat] :
            ( ( P @ B3 @ A3 )
           => ( P @ A3 @ B3 ) )
       => ( P @ A2 @ B2 ) ) ) ).

% linorder_wlog
thf(fact_4341_linorder__wlog,axiom,
    ! [P: int > int > $o,A2: int,B2: int] :
      ( ! [A3: int,B3: int] :
          ( ( ord_less_eq_int @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: int,B3: int] :
            ( ( P @ B3 @ A3 )
           => ( P @ A3 @ B3 ) )
       => ( P @ A2 @ B2 ) ) ) ).

% linorder_wlog
thf(fact_4342_order__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ Z )
       => ( ord_less_eq_set_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_4343_order__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ Z )
       => ( ord_less_eq_rat @ X @ Z ) ) ) ).

% order_trans
thf(fact_4344_order__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ Z )
       => ( ord_less_eq_num @ X @ Z ) ) ) ).

% order_trans
thf(fact_4345_order__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_eq_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_4346_order__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ X @ Z ) ) ) ).

% order_trans
thf(fact_4347_order_Otrans,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C2 )
       => ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_4348_order_Otrans,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ B2 @ C2 )
       => ( ord_less_eq_rat @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_4349_order_Otrans,axiom,
    ! [A2: num,B2: num,C2: num] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_eq_num @ B2 @ C2 )
       => ( ord_less_eq_num @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_4350_order_Otrans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_4351_order_Otrans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ B2 @ C2 )
       => ( ord_less_eq_int @ A2 @ C2 ) ) ) ).

% order.trans
thf(fact_4352_order__antisym,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_4353_order__antisym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_4354_order__antisym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_4355_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_4356_order__antisym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_4357_ord__le__eq__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_4358_ord__le__eq__trans,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_eq_rat @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_4359_ord__le__eq__trans,axiom,
    ! [A2: num,B2: num,C2: num] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_eq_num @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_4360_ord__le__eq__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_4361_ord__le__eq__trans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_eq_int @ A2 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_4362_ord__eq__le__trans,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C2 )
       => ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_4363_ord__eq__le__trans,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_rat @ B2 @ C2 )
       => ( ord_less_eq_rat @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_4364_ord__eq__le__trans,axiom,
    ! [A2: num,B2: num,C2: num] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_num @ B2 @ C2 )
       => ( ord_less_eq_num @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_4365_ord__eq__le__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_4366_ord__eq__le__trans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_int @ B2 @ C2 )
       => ( ord_less_eq_int @ A2 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_4367_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_nat,Z3: set_nat] : Y5 = Z3 )
    = ( ^ [X3: set_nat,Y2: set_nat] :
          ( ( ord_less_eq_set_nat @ X3 @ Y2 )
          & ( ord_less_eq_set_nat @ Y2 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_4368_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: rat,Z3: rat] : Y5 = Z3 )
    = ( ^ [X3: rat,Y2: rat] :
          ( ( ord_less_eq_rat @ X3 @ Y2 )
          & ( ord_less_eq_rat @ Y2 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_4369_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: num,Z3: num] : Y5 = Z3 )
    = ( ^ [X3: num,Y2: num] :
          ( ( ord_less_eq_num @ X3 @ Y2 )
          & ( ord_less_eq_num @ Y2 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_4370_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z3: nat] : Y5 = Z3 )
    = ( ^ [X3: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y2 )
          & ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_4371_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: int,Z3: int] : Y5 = Z3 )
    = ( ^ [X3: int,Y2: int] :
          ( ( ord_less_eq_int @ X3 @ Y2 )
          & ( ord_less_eq_int @ Y2 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_4372_le__cases3,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ( ord_less_eq_rat @ X @ Y )
       => ~ ( ord_less_eq_rat @ Y @ Z ) )
     => ( ( ( ord_less_eq_rat @ Y @ X )
         => ~ ( ord_less_eq_rat @ X @ Z ) )
       => ( ( ( ord_less_eq_rat @ X @ Z )
           => ~ ( ord_less_eq_rat @ Z @ Y ) )
         => ( ( ( ord_less_eq_rat @ Z @ Y )
             => ~ ( ord_less_eq_rat @ Y @ X ) )
           => ( ( ( ord_less_eq_rat @ Y @ Z )
               => ~ ( ord_less_eq_rat @ Z @ X ) )
             => ~ ( ( ord_less_eq_rat @ Z @ X )
                 => ~ ( ord_less_eq_rat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_4373_le__cases3,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ( ord_less_eq_num @ X @ Y )
       => ~ ( ord_less_eq_num @ Y @ Z ) )
     => ( ( ( ord_less_eq_num @ Y @ X )
         => ~ ( ord_less_eq_num @ X @ Z ) )
       => ( ( ( ord_less_eq_num @ X @ Z )
           => ~ ( ord_less_eq_num @ Z @ Y ) )
         => ( ( ( ord_less_eq_num @ Z @ Y )
             => ~ ( ord_less_eq_num @ Y @ X ) )
           => ( ( ( ord_less_eq_num @ Y @ Z )
               => ~ ( ord_less_eq_num @ Z @ X ) )
             => ~ ( ( ord_less_eq_num @ Z @ X )
                 => ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_4374_le__cases3,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z ) )
       => ( ( ( ord_less_eq_nat @ X @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_4375_le__cases3,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ( ord_less_eq_int @ X @ Y )
       => ~ ( ord_less_eq_int @ Y @ Z ) )
     => ( ( ( ord_less_eq_int @ Y @ X )
         => ~ ( ord_less_eq_int @ X @ Z ) )
       => ( ( ( ord_less_eq_int @ X @ Z )
           => ~ ( ord_less_eq_int @ Z @ Y ) )
         => ( ( ( ord_less_eq_int @ Z @ Y )
             => ~ ( ord_less_eq_int @ Y @ X ) )
           => ( ( ( ord_less_eq_int @ Y @ Z )
               => ~ ( ord_less_eq_int @ Z @ X ) )
             => ~ ( ( ord_less_eq_int @ Z @ X )
                 => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_4376_nle__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ~ ( ord_less_eq_rat @ A2 @ B2 ) )
      = ( ( ord_less_eq_rat @ B2 @ A2 )
        & ( B2 != A2 ) ) ) ).

% nle_le
thf(fact_4377_nle__le,axiom,
    ! [A2: num,B2: num] :
      ( ( ~ ( ord_less_eq_num @ A2 @ B2 ) )
      = ( ( ord_less_eq_num @ B2 @ A2 )
        & ( B2 != A2 ) ) ) ).

% nle_le
thf(fact_4378_nle__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
      = ( ( ord_less_eq_nat @ B2 @ A2 )
        & ( B2 != A2 ) ) ) ).

% nle_le
thf(fact_4379_nle__le,axiom,
    ! [A2: int,B2: int] :
      ( ( ~ ( ord_less_eq_int @ A2 @ B2 ) )
      = ( ( ord_less_eq_int @ B2 @ A2 )
        & ( B2 != A2 ) ) ) ).

% nle_le
thf(fact_4380_verit__comp__simplify1_I2_J,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).

% verit_comp_simplify1(2)
thf(fact_4381_verit__comp__simplify1_I2_J,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ A2 @ A2 ) ).

% verit_comp_simplify1(2)
thf(fact_4382_verit__comp__simplify1_I2_J,axiom,
    ! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).

% verit_comp_simplify1(2)
thf(fact_4383_verit__comp__simplify1_I2_J,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).

% verit_comp_simplify1(2)
thf(fact_4384_verit__comp__simplify1_I2_J,axiom,
    ! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).

% verit_comp_simplify1(2)
thf(fact_4385_verit__comp__simplify1_I1_J,axiom,
    ! [A2: real] :
      ~ ( ord_less_real @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_4386_verit__comp__simplify1_I1_J,axiom,
    ! [A2: rat] :
      ~ ( ord_less_rat @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_4387_verit__comp__simplify1_I1_J,axiom,
    ! [A2: num] :
      ~ ( ord_less_num @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_4388_verit__comp__simplify1_I1_J,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_4389_verit__comp__simplify1_I1_J,axiom,
    ! [A2: int] :
      ~ ( ord_less_int @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_4390_lt__ex,axiom,
    ! [X: real] :
    ? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).

% lt_ex
thf(fact_4391_lt__ex,axiom,
    ! [X: rat] :
    ? [Y3: rat] : ( ord_less_rat @ Y3 @ X ) ).

% lt_ex
thf(fact_4392_lt__ex,axiom,
    ! [X: int] :
    ? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).

% lt_ex
thf(fact_4393_gt__ex,axiom,
    ! [X: real] :
    ? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).

% gt_ex
thf(fact_4394_gt__ex,axiom,
    ! [X: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).

% gt_ex
thf(fact_4395_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).

% gt_ex
thf(fact_4396_gt__ex,axiom,
    ! [X: int] :
    ? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).

% gt_ex
thf(fact_4397_dense,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ? [Z2: real] :
          ( ( ord_less_real @ X @ Z2 )
          & ( ord_less_real @ Z2 @ Y ) ) ) ).

% dense
thf(fact_4398_dense,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ? [Z2: rat] :
          ( ( ord_less_rat @ X @ Z2 )
          & ( ord_less_rat @ Z2 @ Y ) ) ) ).

% dense
thf(fact_4399_less__imp__neq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_4400_less__imp__neq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_4401_less__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_4402_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_4403_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_4404_order_Oasym,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ~ ( ord_less_real @ B2 @ A2 ) ) ).

% order.asym
thf(fact_4405_order_Oasym,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ~ ( ord_less_rat @ B2 @ A2 ) ) ).

% order.asym
thf(fact_4406_order_Oasym,axiom,
    ! [A2: num,B2: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ~ ( ord_less_num @ B2 @ A2 ) ) ).

% order.asym
thf(fact_4407_order_Oasym,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ~ ( ord_less_nat @ B2 @ A2 ) ) ).

% order.asym
thf(fact_4408_order_Oasym,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ~ ( ord_less_int @ B2 @ A2 ) ) ).

% order.asym
thf(fact_4409_ord__eq__less__trans,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( A2 = B2 )
     => ( ( ord_less_real @ B2 @ C2 )
       => ( ord_less_real @ A2 @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_4410_ord__eq__less__trans,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( A2 = B2 )
     => ( ( ord_less_rat @ B2 @ C2 )
       => ( ord_less_rat @ A2 @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_4411_ord__eq__less__trans,axiom,
    ! [A2: num,B2: num,C2: num] :
      ( ( A2 = B2 )
     => ( ( ord_less_num @ B2 @ C2 )
       => ( ord_less_num @ A2 @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_4412_ord__eq__less__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ord_less_nat @ A2 @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_4413_ord__eq__less__trans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( A2 = B2 )
     => ( ( ord_less_int @ B2 @ C2 )
       => ( ord_less_int @ A2 @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_4414_ord__less__eq__trans,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_real @ A2 @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_4415_ord__less__eq__trans,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_rat @ A2 @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_4416_ord__less__eq__trans,axiom,
    ! [A2: num,B2: num,C2: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_num @ A2 @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_4417_ord__less__eq__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_nat @ A2 @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_4418_ord__less__eq__trans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_int @ A2 @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_4419_less__induct,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ! [X4: nat] :
          ( ! [Y4: nat] :
              ( ( ord_less_nat @ Y4 @ X4 )
             => ( P @ Y4 ) )
         => ( P @ X4 ) )
     => ( P @ A2 ) ) ).

% less_induct
thf(fact_4420_antisym__conv3,axiom,
    ! [Y: real,X: real] :
      ( ~ ( ord_less_real @ Y @ X )
     => ( ( ~ ( ord_less_real @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_4421_antisym__conv3,axiom,
    ! [Y: rat,X: rat] :
      ( ~ ( ord_less_rat @ Y @ X )
     => ( ( ~ ( ord_less_rat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_4422_antisym__conv3,axiom,
    ! [Y: num,X: num] :
      ( ~ ( ord_less_num @ Y @ X )
     => ( ( ~ ( ord_less_num @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_4423_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_4424_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_int @ Y @ X )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_4425_linorder__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_4426_linorder__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_4427_linorder__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_num @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_4428_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_4429_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_4430_dual__order_Oasym,axiom,
    ! [B2: real,A2: real] :
      ( ( ord_less_real @ B2 @ A2 )
     => ~ ( ord_less_real @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_4431_dual__order_Oasym,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ~ ( ord_less_rat @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_4432_dual__order_Oasym,axiom,
    ! [B2: num,A2: num] :
      ( ( ord_less_num @ B2 @ A2 )
     => ~ ( ord_less_num @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_4433_dual__order_Oasym,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ~ ( ord_less_nat @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_4434_dual__order_Oasym,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ~ ( ord_less_int @ A2 @ B2 ) ) ).

% dual_order.asym
thf(fact_4435_dual__order_Oirrefl,axiom,
    ! [A2: real] :
      ~ ( ord_less_real @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_4436_dual__order_Oirrefl,axiom,
    ! [A2: rat] :
      ~ ( ord_less_rat @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_4437_dual__order_Oirrefl,axiom,
    ! [A2: num] :
      ~ ( ord_less_num @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_4438_dual__order_Oirrefl,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_4439_dual__order_Oirrefl,axiom,
    ! [A2: int] :
      ~ ( ord_less_int @ A2 @ A2 ) ).

% dual_order.irrefl
thf(fact_4440_exists__least__iff,axiom,
    ( ( ^ [P5: nat > $o] :
        ? [X9: nat] : ( P5 @ X9 ) )
    = ( ^ [P3: nat > $o] :
        ? [N2: nat] :
          ( ( P3 @ N2 )
          & ! [M5: nat] :
              ( ( ord_less_nat @ M5 @ N2 )
             => ~ ( P3 @ M5 ) ) ) ) ) ).

% exists_least_iff
thf(fact_4441_linorder__less__wlog,axiom,
    ! [P: real > real > $o,A2: real,B2: real] :
      ( ! [A3: real,B3: real] :
          ( ( ord_less_real @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: real] : ( P @ A3 @ A3 )
       => ( ! [A3: real,B3: real] :
              ( ( P @ B3 @ A3 )
             => ( P @ A3 @ B3 ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_4442_linorder__less__wlog,axiom,
    ! [P: rat > rat > $o,A2: rat,B2: rat] :
      ( ! [A3: rat,B3: rat] :
          ( ( ord_less_rat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: rat] : ( P @ A3 @ A3 )
       => ( ! [A3: rat,B3: rat] :
              ( ( P @ B3 @ A3 )
             => ( P @ A3 @ B3 ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_4443_linorder__less__wlog,axiom,
    ! [P: num > num > $o,A2: num,B2: num] :
      ( ! [A3: num,B3: num] :
          ( ( ord_less_num @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: num] : ( P @ A3 @ A3 )
       => ( ! [A3: num,B3: num] :
              ( ( P @ B3 @ A3 )
             => ( P @ A3 @ B3 ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_4444_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A2: nat,B2: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_nat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: nat] : ( P @ A3 @ A3 )
       => ( ! [A3: nat,B3: nat] :
              ( ( P @ B3 @ A3 )
             => ( P @ A3 @ B3 ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_4445_linorder__less__wlog,axiom,
    ! [P: int > int > $o,A2: int,B2: int] :
      ( ! [A3: int,B3: int] :
          ( ( ord_less_int @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: int] : ( P @ A3 @ A3 )
       => ( ! [A3: int,B3: int] :
              ( ( P @ B3 @ A3 )
             => ( P @ A3 @ B3 ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_4446_order_Ostrict__trans,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_real @ B2 @ C2 )
       => ( ord_less_real @ A2 @ C2 ) ) ) ).

% order.strict_trans
thf(fact_4447_order_Ostrict__trans,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ B2 @ C2 )
       => ( ord_less_rat @ A2 @ C2 ) ) ) ).

% order.strict_trans
thf(fact_4448_order_Ostrict__trans,axiom,
    ! [A2: num,B2: num,C2: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ( ( ord_less_num @ B2 @ C2 )
       => ( ord_less_num @ A2 @ C2 ) ) ) ).

% order.strict_trans
thf(fact_4449_order_Ostrict__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ord_less_nat @ A2 @ C2 ) ) ) ).

% order.strict_trans
thf(fact_4450_order_Ostrict__trans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ B2 @ C2 )
       => ( ord_less_int @ A2 @ C2 ) ) ) ).

% order.strict_trans
thf(fact_4451_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_4452_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_4453_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_4454_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_4455_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_4456_dual__order_Ostrict__trans,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( ord_less_real @ B2 @ A2 )
     => ( ( ord_less_real @ C2 @ B2 )
       => ( ord_less_real @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_4457_dual__order_Ostrict__trans,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ( ord_less_rat @ C2 @ B2 )
       => ( ord_less_rat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_4458_dual__order_Ostrict__trans,axiom,
    ! [B2: num,A2: num,C2: num] :
      ( ( ord_less_num @ B2 @ A2 )
     => ( ( ord_less_num @ C2 @ B2 )
       => ( ord_less_num @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_4459_dual__order_Ostrict__trans,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( ord_less_nat @ C2 @ B2 )
       => ( ord_less_nat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_4460_dual__order_Ostrict__trans,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( ord_less_int @ C2 @ B2 )
       => ( ord_less_int @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans
thf(fact_4461_order_Ostrict__implies__not__eq,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_4462_order_Ostrict__implies__not__eq,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_4463_order_Ostrict__implies__not__eq,axiom,
    ! [A2: num,B2: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_4464_order_Ostrict__implies__not__eq,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_4465_order_Ostrict__implies__not__eq,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( A2 != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_4466_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: real,A2: real] :
      ( ( ord_less_real @ B2 @ A2 )
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_4467_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_4468_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: num,A2: num] :
      ( ( ord_less_num @ B2 @ A2 )
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_4469_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_4470_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( A2 != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_4471_linorder__neqE,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ ( ord_less_real @ X @ Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_4472_linorder__neqE,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ ( ord_less_rat @ X @ Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_4473_linorder__neqE,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
     => ( ~ ( ord_less_num @ X @ Y )
       => ( ord_less_num @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_4474_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_4475_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_4476_order__less__asym,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_asym
thf(fact_4477_order__less__asym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_asym
thf(fact_4478_order__less__asym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_asym
thf(fact_4479_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_4480_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_asym
thf(fact_4481_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_4482_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_4483_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_4484_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_4485_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_4486_order__less__asym_H,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ~ ( ord_less_real @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_4487_order__less__asym_H,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ~ ( ord_less_rat @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_4488_order__less__asym_H,axiom,
    ! [A2: num,B2: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ~ ( ord_less_num @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_4489_order__less__asym_H,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ~ ( ord_less_nat @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_4490_order__less__asym_H,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ~ ( ord_less_int @ B2 @ A2 ) ) ).

% order_less_asym'
thf(fact_4491_order__less__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_real @ Y @ Z )
       => ( ord_less_real @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_4492_order__less__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ Z )
       => ( ord_less_rat @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_4493_order__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_num @ Y @ Z )
       => ( ord_less_num @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_4494_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_4495_order__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_4496_ord__eq__less__subst,axiom,
    ! [A2: real,F2: real > real,B2: real,C2: real] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_real @ B2 @ C2 )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_4497_ord__eq__less__subst,axiom,
    ! [A2: rat,F2: real > rat,B2: real,C2: real] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_real @ B2 @ C2 )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_4498_ord__eq__less__subst,axiom,
    ! [A2: num,F2: real > num,B2: real,C2: real] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_real @ B2 @ C2 )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_num @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_num @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_4499_ord__eq__less__subst,axiom,
    ! [A2: nat,F2: real > nat,B2: real,C2: real] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_real @ B2 @ C2 )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_4500_ord__eq__less__subst,axiom,
    ! [A2: int,F2: real > int,B2: real,C2: real] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_real @ B2 @ C2 )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_4501_ord__eq__less__subst,axiom,
    ! [A2: real,F2: rat > real,B2: rat,C2: rat] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_4502_ord__eq__less__subst,axiom,
    ! [A2: rat,F2: rat > rat,B2: rat,C2: rat] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_4503_ord__eq__less__subst,axiom,
    ! [A2: num,F2: rat > num,B2: rat,C2: rat] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_num @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_num @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_4504_ord__eq__less__subst,axiom,
    ! [A2: nat,F2: rat > nat,B2: rat,C2: rat] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_4505_ord__eq__less__subst,axiom,
    ! [A2: int,F2: rat > int,B2: rat,C2: rat] :
      ( ( A2
        = ( F2 @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_4506_ord__less__eq__subst,axiom,
    ! [A2: real,B2: real,F2: real > real,C2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C2 )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_4507_ord__less__eq__subst,axiom,
    ! [A2: real,B2: real,F2: real > rat,C2: rat] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C2 )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_4508_ord__less__eq__subst,axiom,
    ! [A2: real,B2: real,F2: real > num,C2: num] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C2 )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_num @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_num @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_4509_ord__less__eq__subst,axiom,
    ! [A2: real,B2: real,F2: real > nat,C2: nat] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C2 )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_4510_ord__less__eq__subst,axiom,
    ! [A2: real,B2: real,F2: real > int,C2: int] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C2 )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_4511_ord__less__eq__subst,axiom,
    ! [A2: rat,B2: rat,F2: rat > real,C2: real] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_4512_ord__less__eq__subst,axiom,
    ! [A2: rat,B2: rat,F2: rat > rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_4513_ord__less__eq__subst,axiom,
    ! [A2: rat,B2: rat,F2: rat > num,C2: num] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_num @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_num @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_4514_ord__less__eq__subst,axiom,
    ! [A2: rat,B2: rat,F2: rat > nat,C2: nat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_4515_ord__less__eq__subst,axiom,
    ! [A2: rat,B2: rat,F2: rat > int,C2: int] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ( F2 @ B2 )
          = C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_4516_order__less__irrefl,axiom,
    ! [X: real] :
      ~ ( ord_less_real @ X @ X ) ).

% order_less_irrefl
thf(fact_4517_order__less__irrefl,axiom,
    ! [X: rat] :
      ~ ( ord_less_rat @ X @ X ) ).

% order_less_irrefl
thf(fact_4518_order__less__irrefl,axiom,
    ! [X: num] :
      ~ ( ord_less_num @ X @ X ) ).

% order_less_irrefl
thf(fact_4519_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_4520_order__less__irrefl,axiom,
    ! [X: int] :
      ~ ( ord_less_int @ X @ X ) ).

% order_less_irrefl
thf(fact_4521_order__less__subst1,axiom,
    ! [A2: real,F2: real > real,B2: real,C2: real] :
      ( ( ord_less_real @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_real @ B2 @ C2 )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_4522_order__less__subst1,axiom,
    ! [A2: real,F2: rat > real,B2: rat,C2: rat] :
      ( ( ord_less_real @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_4523_order__less__subst1,axiom,
    ! [A2: real,F2: num > real,B2: num,C2: num] :
      ( ( ord_less_real @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_num @ B2 @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_num @ X4 @ Y3 )
             => ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_4524_order__less__subst1,axiom,
    ! [A2: real,F2: nat > real,B2: nat,C2: nat] :
      ( ( ord_less_real @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_4525_order__less__subst1,axiom,
    ! [A2: real,F2: int > real,B2: int,C2: int] :
      ( ( ord_less_real @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_int @ B2 @ C2 )
       => ( ! [X4: int,Y3: int] :
              ( ( ord_less_int @ X4 @ Y3 )
             => ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_4526_order__less__subst1,axiom,
    ! [A2: rat,F2: real > rat,B2: real,C2: real] :
      ( ( ord_less_rat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_real @ B2 @ C2 )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_4527_order__less__subst1,axiom,
    ! [A2: rat,F2: rat > rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_4528_order__less__subst1,axiom,
    ! [A2: rat,F2: num > rat,B2: num,C2: num] :
      ( ( ord_less_rat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_num @ B2 @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_num @ X4 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_4529_order__less__subst1,axiom,
    ! [A2: rat,F2: nat > rat,B2: nat,C2: nat] :
      ( ( ord_less_rat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_4530_order__less__subst1,axiom,
    ! [A2: rat,F2: int > rat,B2: int,C2: int] :
      ( ( ord_less_rat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_int @ B2 @ C2 )
       => ( ! [X4: int,Y3: int] :
              ( ( ord_less_int @ X4 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_4531_order__less__subst2,axiom,
    ! [A2: real,B2: real,F2: real > real,C2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_real @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_4532_order__less__subst2,axiom,
    ! [A2: real,B2: real,F2: real > rat,C2: rat] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_rat @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_4533_order__less__subst2,axiom,
    ! [A2: real,B2: real,F2: real > num,C2: num] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_num @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_num @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_num @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_4534_order__less__subst2,axiom,
    ! [A2: real,B2: real,F2: real > nat,C2: nat] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_4535_order__less__subst2,axiom,
    ! [A2: real,B2: real,F2: real > int,C2: int] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_int @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_4536_order__less__subst2,axiom,
    ! [A2: rat,B2: rat,F2: rat > real,C2: real] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_real @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_4537_order__less__subst2,axiom,
    ! [A2: rat,B2: rat,F2: rat > rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_4538_order__less__subst2,axiom,
    ! [A2: rat,B2: rat,F2: rat > num,C2: num] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_num @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_num @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_num @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_4539_order__less__subst2,axiom,
    ! [A2: rat,B2: rat,F2: rat > nat,C2: nat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_4540_order__less__subst2,axiom,
    ! [A2: rat,B2: rat,F2: rat > int,C2: int] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_int @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_4541_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_4542_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_4543_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_4544_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_4545_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_4546_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_4547_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_4548_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_4549_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_4550_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_4551_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_4552_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_4553_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_4554_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_4555_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_4556_order__less__imp__not__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_4557_order__less__imp__not__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_4558_order__less__imp__not__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_4559_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_4560_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_4561_order__less__imp__not__eq2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_4562_order__less__imp__not__eq2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_4563_order__less__imp__not__eq2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_4564_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_4565_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_4566_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_4567_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_4568_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_4569_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_4570_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_4571_vebt__assn__raw_Osimps_I4_J,axiom,
    ! [Vd2: $o,Ve2: $o,V: option4927543243414619207at_nat,Va: nat,Vb2: array_VEBT_VEBTi,Vc2: vEBT_VEBTi] :
      ( ( vEBT_vebt_assn_raw @ ( vEBT_Leaf @ Vd2 @ Ve2 ) @ ( vEBT_Nodei @ V @ Va @ Vb2 @ Vc2 ) )
      = bot_bot_assn ) ).

% vebt_assn_raw.simps(4)
thf(fact_4572_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_4573_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_4574_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_4575_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_4576_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_4577_leD,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ Y @ X )
     => ~ ( ord_less_real @ X @ Y ) ) ).

% leD
thf(fact_4578_leD,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ~ ( ord_less_set_nat @ X @ Y ) ) ).

% leD
thf(fact_4579_leD,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ~ ( ord_less_rat @ X @ Y ) ) ).

% leD
thf(fact_4580_leD,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ~ ( ord_less_num @ X @ Y ) ) ).

% leD
thf(fact_4581_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_4582_leD,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ~ ( ord_less_int @ X @ Y ) ) ).

% leD
thf(fact_4583_leI,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ord_less_eq_real @ Y @ X ) ) ).

% leI
thf(fact_4584_leI,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ord_less_eq_rat @ Y @ X ) ) ).

% leI
thf(fact_4585_leI,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ord_less_eq_num @ Y @ X ) ) ).

% leI
thf(fact_4586_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_4587_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% leI
thf(fact_4588_nless__le,axiom,
    ! [A2: real,B2: real] :
      ( ( ~ ( ord_less_real @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_real @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_4589_nless__le,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ~ ( ord_less_set_nat @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_set_nat @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_4590_nless__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ~ ( ord_less_rat @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_rat @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_4591_nless__le,axiom,
    ! [A2: num,B2: num] :
      ( ( ~ ( ord_less_num @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_num @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_4592_nless__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ~ ( ord_less_nat @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_nat @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_4593_nless__le,axiom,
    ! [A2: int,B2: int] :
      ( ( ~ ( ord_less_int @ A2 @ B2 ) )
      = ( ~ ( ord_less_eq_int @ A2 @ B2 )
        | ( A2 = B2 ) ) ) ).

% nless_le
thf(fact_4594_antisym__conv1,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ( ord_less_eq_real @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_4595_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_4596_antisym__conv1,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_4597_antisym__conv1,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ( ord_less_eq_num @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_4598_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_4599_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_4600_antisym__conv2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ~ ( ord_less_real @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_4601_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_4602_antisym__conv2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ~ ( ord_less_rat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_4603_antisym__conv2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ~ ( ord_less_num @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_4604_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_4605_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_4606_dense__ge,axiom,
    ! [Z: real,Y: real] :
      ( ! [X4: real] :
          ( ( ord_less_real @ Z @ X4 )
         => ( ord_less_eq_real @ Y @ X4 ) )
     => ( ord_less_eq_real @ Y @ Z ) ) ).

% dense_ge
thf(fact_4607_dense__ge,axiom,
    ! [Z: rat,Y: rat] :
      ( ! [X4: rat] :
          ( ( ord_less_rat @ Z @ X4 )
         => ( ord_less_eq_rat @ Y @ X4 ) )
     => ( ord_less_eq_rat @ Y @ Z ) ) ).

% dense_ge
thf(fact_4608_dense__le,axiom,
    ! [Y: real,Z: real] :
      ( ! [X4: real] :
          ( ( ord_less_real @ X4 @ Y )
         => ( ord_less_eq_real @ X4 @ Z ) )
     => ( ord_less_eq_real @ Y @ Z ) ) ).

% dense_le
thf(fact_4609_dense__le,axiom,
    ! [Y: rat,Z: rat] :
      ( ! [X4: rat] :
          ( ( ord_less_rat @ X4 @ Y )
         => ( ord_less_eq_rat @ X4 @ Z ) )
     => ( ord_less_eq_rat @ Y @ Z ) ) ).

% dense_le
thf(fact_4610_less__le__not__le,axiom,
    ( ord_less_real
    = ( ^ [X3: real,Y2: real] :
          ( ( ord_less_eq_real @ X3 @ Y2 )
          & ~ ( ord_less_eq_real @ Y2 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_4611_less__le__not__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X3: set_nat,Y2: set_nat] :
          ( ( ord_less_eq_set_nat @ X3 @ Y2 )
          & ~ ( ord_less_eq_set_nat @ Y2 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_4612_less__le__not__le,axiom,
    ( ord_less_rat
    = ( ^ [X3: rat,Y2: rat] :
          ( ( ord_less_eq_rat @ X3 @ Y2 )
          & ~ ( ord_less_eq_rat @ Y2 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_4613_less__le__not__le,axiom,
    ( ord_less_num
    = ( ^ [X3: num,Y2: num] :
          ( ( ord_less_eq_num @ X3 @ Y2 )
          & ~ ( ord_less_eq_num @ Y2 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_4614_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y2 )
          & ~ ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_4615_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X3: int,Y2: int] :
          ( ( ord_less_eq_int @ X3 @ Y2 )
          & ~ ( ord_less_eq_int @ Y2 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_4616_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_4617_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_4618_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_4619_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_4620_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_4621_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [A7: real,B7: real] :
          ( ( ord_less_real @ A7 @ B7 )
          | ( A7 = B7 ) ) ) ) ).

% order.order_iff_strict
thf(fact_4622_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A7: set_nat,B7: set_nat] :
          ( ( ord_less_set_nat @ A7 @ B7 )
          | ( A7 = B7 ) ) ) ) ).

% order.order_iff_strict
thf(fact_4623_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [A7: rat,B7: rat] :
          ( ( ord_less_rat @ A7 @ B7 )
          | ( A7 = B7 ) ) ) ) ).

% order.order_iff_strict
thf(fact_4624_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [A7: num,B7: num] :
          ( ( ord_less_num @ A7 @ B7 )
          | ( A7 = B7 ) ) ) ) ).

% order.order_iff_strict
thf(fact_4625_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A7: nat,B7: nat] :
          ( ( ord_less_nat @ A7 @ B7 )
          | ( A7 = B7 ) ) ) ) ).

% order.order_iff_strict
thf(fact_4626_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A7: int,B7: int] :
          ( ( ord_less_int @ A7 @ B7 )
          | ( A7 = B7 ) ) ) ) ).

% order.order_iff_strict
thf(fact_4627_order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [A7: real,B7: real] :
          ( ( ord_less_eq_real @ A7 @ B7 )
          & ( A7 != B7 ) ) ) ) ).

% order.strict_iff_order
thf(fact_4628_order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [A7: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ A7 @ B7 )
          & ( A7 != B7 ) ) ) ) ).

% order.strict_iff_order
thf(fact_4629_order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [A7: rat,B7: rat] :
          ( ( ord_less_eq_rat @ A7 @ B7 )
          & ( A7 != B7 ) ) ) ) ).

% order.strict_iff_order
thf(fact_4630_order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [A7: num,B7: num] :
          ( ( ord_less_eq_num @ A7 @ B7 )
          & ( A7 != B7 ) ) ) ) ).

% order.strict_iff_order
thf(fact_4631_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A7: nat,B7: nat] :
          ( ( ord_less_eq_nat @ A7 @ B7 )
          & ( A7 != B7 ) ) ) ) ).

% order.strict_iff_order
thf(fact_4632_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A7: int,B7: int] :
          ( ( ord_less_eq_int @ A7 @ B7 )
          & ( A7 != B7 ) ) ) ) ).

% order.strict_iff_order
thf(fact_4633_order_Ostrict__trans1,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_real @ B2 @ C2 )
       => ( ord_less_real @ A2 @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_4634_order_Ostrict__trans1,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_set_nat @ B2 @ C2 )
       => ( ord_less_set_nat @ A2 @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_4635_order_Ostrict__trans1,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ B2 @ C2 )
       => ( ord_less_rat @ A2 @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_4636_order_Ostrict__trans1,axiom,
    ! [A2: num,B2: num,C2: num] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_num @ B2 @ C2 )
       => ( ord_less_num @ A2 @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_4637_order_Ostrict__trans1,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ord_less_nat @ A2 @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_4638_order_Ostrict__trans1,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_int @ B2 @ C2 )
       => ( ord_less_int @ A2 @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_4639_order_Ostrict__trans2,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ B2 @ C2 )
       => ( ord_less_real @ A2 @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_4640_order_Ostrict__trans2,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C2 )
       => ( ord_less_set_nat @ A2 @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_4641_order_Ostrict__trans2,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ B2 @ C2 )
       => ( ord_less_rat @ A2 @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_4642_order_Ostrict__trans2,axiom,
    ! [A2: num,B2: num,C2: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ( ( ord_less_eq_num @ B2 @ C2 )
       => ( ord_less_num @ A2 @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_4643_order_Ostrict__trans2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ord_less_nat @ A2 @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_4644_order_Ostrict__trans2,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ B2 @ C2 )
       => ( ord_less_int @ A2 @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_4645_order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [A7: real,B7: real] :
          ( ( ord_less_eq_real @ A7 @ B7 )
          & ~ ( ord_less_eq_real @ B7 @ A7 ) ) ) ) ).

% order.strict_iff_not
thf(fact_4646_order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [A7: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ A7 @ B7 )
          & ~ ( ord_less_eq_set_nat @ B7 @ A7 ) ) ) ) ).

% order.strict_iff_not
thf(fact_4647_order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [A7: rat,B7: rat] :
          ( ( ord_less_eq_rat @ A7 @ B7 )
          & ~ ( ord_less_eq_rat @ B7 @ A7 ) ) ) ) ).

% order.strict_iff_not
thf(fact_4648_order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [A7: num,B7: num] :
          ( ( ord_less_eq_num @ A7 @ B7 )
          & ~ ( ord_less_eq_num @ B7 @ A7 ) ) ) ) ).

% order.strict_iff_not
thf(fact_4649_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A7: nat,B7: nat] :
          ( ( ord_less_eq_nat @ A7 @ B7 )
          & ~ ( ord_less_eq_nat @ B7 @ A7 ) ) ) ) ).

% order.strict_iff_not
thf(fact_4650_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A7: int,B7: int] :
          ( ( ord_less_eq_int @ A7 @ B7 )
          & ~ ( ord_less_eq_int @ B7 @ A7 ) ) ) ) ).

% order.strict_iff_not
thf(fact_4651_dense__ge__bounded,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ Z @ X )
     => ( ! [W3: real] :
            ( ( ord_less_real @ Z @ W3 )
           => ( ( ord_less_real @ W3 @ X )
             => ( ord_less_eq_real @ Y @ W3 ) ) )
       => ( ord_less_eq_real @ Y @ Z ) ) ) ).

% dense_ge_bounded
thf(fact_4652_dense__ge__bounded,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ Z @ X )
     => ( ! [W3: rat] :
            ( ( ord_less_rat @ Z @ W3 )
           => ( ( ord_less_rat @ W3 @ X )
             => ( ord_less_eq_rat @ Y @ W3 ) ) )
       => ( ord_less_eq_rat @ Y @ Z ) ) ) ).

% dense_ge_bounded
thf(fact_4653_dense__le__bounded,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ! [W3: real] :
            ( ( ord_less_real @ X @ W3 )
           => ( ( ord_less_real @ W3 @ Y )
             => ( ord_less_eq_real @ W3 @ Z ) ) )
       => ( ord_less_eq_real @ Y @ Z ) ) ) ).

% dense_le_bounded
thf(fact_4654_dense__le__bounded,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ! [W3: rat] :
            ( ( ord_less_rat @ X @ W3 )
           => ( ( ord_less_rat @ W3 @ Y )
             => ( ord_less_eq_rat @ W3 @ Z ) ) )
       => ( ord_less_eq_rat @ Y @ Z ) ) ) ).

% dense_le_bounded
thf(fact_4655_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [B7: real,A7: real] :
          ( ( ord_less_real @ B7 @ A7 )
          | ( A7 = B7 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_4656_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B7: set_nat,A7: set_nat] :
          ( ( ord_less_set_nat @ B7 @ A7 )
          | ( A7 = B7 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_4657_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [B7: rat,A7: rat] :
          ( ( ord_less_rat @ B7 @ A7 )
          | ( A7 = B7 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_4658_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [B7: num,A7: num] :
          ( ( ord_less_num @ B7 @ A7 )
          | ( A7 = B7 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_4659_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B7: nat,A7: nat] :
          ( ( ord_less_nat @ B7 @ A7 )
          | ( A7 = B7 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_4660_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B7: int,A7: int] :
          ( ( ord_less_int @ B7 @ A7 )
          | ( A7 = B7 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_4661_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [B7: real,A7: real] :
          ( ( ord_less_eq_real @ B7 @ A7 )
          & ( A7 != B7 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_4662_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [B7: set_nat,A7: set_nat] :
          ( ( ord_less_eq_set_nat @ B7 @ A7 )
          & ( A7 != B7 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_4663_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [B7: rat,A7: rat] :
          ( ( ord_less_eq_rat @ B7 @ A7 )
          & ( A7 != B7 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_4664_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [B7: num,A7: num] :
          ( ( ord_less_eq_num @ B7 @ A7 )
          & ( A7 != B7 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_4665_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B7: nat,A7: nat] :
          ( ( ord_less_eq_nat @ B7 @ A7 )
          & ( A7 != B7 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_4666_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B7: int,A7: int] :
          ( ( ord_less_eq_int @ B7 @ A7 )
          & ( A7 != B7 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_4667_dual__order_Ostrict__trans1,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( ord_less_eq_real @ B2 @ A2 )
     => ( ( ord_less_real @ C2 @ B2 )
       => ( ord_less_real @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_4668_dual__order_Ostrict__trans1,axiom,
    ! [B2: set_nat,A2: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( ord_less_set_nat @ C2 @ B2 )
       => ( ord_less_set_nat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_4669_dual__order_Ostrict__trans1,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( ord_less_rat @ C2 @ B2 )
       => ( ord_less_rat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_4670_dual__order_Ostrict__trans1,axiom,
    ! [B2: num,A2: num,C2: num] :
      ( ( ord_less_eq_num @ B2 @ A2 )
     => ( ( ord_less_num @ C2 @ B2 )
       => ( ord_less_num @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_4671_dual__order_Ostrict__trans1,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_nat @ C2 @ B2 )
       => ( ord_less_nat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_4672_dual__order_Ostrict__trans1,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_less_int @ C2 @ B2 )
       => ( ord_less_int @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans1
thf(fact_4673_dual__order_Ostrict__trans2,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( ord_less_real @ B2 @ A2 )
     => ( ( ord_less_eq_real @ C2 @ B2 )
       => ( ord_less_real @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_4674_dual__order_Ostrict__trans2,axiom,
    ! [B2: set_nat,A2: set_nat,C2: set_nat] :
      ( ( ord_less_set_nat @ B2 @ A2 )
     => ( ( ord_less_eq_set_nat @ C2 @ B2 )
       => ( ord_less_set_nat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_4675_dual__order_Ostrict__trans2,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ( ord_less_eq_rat @ C2 @ B2 )
       => ( ord_less_rat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_4676_dual__order_Ostrict__trans2,axiom,
    ! [B2: num,A2: num,C2: num] :
      ( ( ord_less_num @ B2 @ A2 )
     => ( ( ord_less_eq_num @ C2 @ B2 )
       => ( ord_less_num @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_4677_dual__order_Ostrict__trans2,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ C2 @ B2 )
       => ( ord_less_nat @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_4678_dual__order_Ostrict__trans2,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( ord_less_eq_int @ C2 @ B2 )
       => ( ord_less_int @ C2 @ A2 ) ) ) ).

% dual_order.strict_trans2
thf(fact_4679_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [B7: real,A7: real] :
          ( ( ord_less_eq_real @ B7 @ A7 )
          & ~ ( ord_less_eq_real @ A7 @ B7 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_4680_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [B7: set_nat,A7: set_nat] :
          ( ( ord_less_eq_set_nat @ B7 @ A7 )
          & ~ ( ord_less_eq_set_nat @ A7 @ B7 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_4681_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [B7: rat,A7: rat] :
          ( ( ord_less_eq_rat @ B7 @ A7 )
          & ~ ( ord_less_eq_rat @ A7 @ B7 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_4682_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [B7: num,A7: num] :
          ( ( ord_less_eq_num @ B7 @ A7 )
          & ~ ( ord_less_eq_num @ A7 @ B7 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_4683_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B7: nat,A7: nat] :
          ( ( ord_less_eq_nat @ B7 @ A7 )
          & ~ ( ord_less_eq_nat @ A7 @ B7 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_4684_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B7: int,A7: int] :
          ( ( ord_less_eq_int @ B7 @ A7 )
          & ~ ( ord_less_eq_int @ A7 @ B7 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_4685_order_Ostrict__implies__order,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ord_less_eq_real @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_4686_order_Ostrict__implies__order,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_4687_order_Ostrict__implies__order,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ord_less_eq_rat @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_4688_order_Ostrict__implies__order,axiom,
    ! [A2: num,B2: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ( ord_less_eq_num @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_4689_order_Ostrict__implies__order,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ord_less_eq_nat @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_4690_order_Ostrict__implies__order,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ord_less_eq_int @ A2 @ B2 ) ) ).

% order.strict_implies_order
thf(fact_4691_dual__order_Ostrict__implies__order,axiom,
    ! [B2: real,A2: real] :
      ( ( ord_less_real @ B2 @ A2 )
     => ( ord_less_eq_real @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_4692_dual__order_Ostrict__implies__order,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ B2 @ A2 )
     => ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_4693_dual__order_Ostrict__implies__order,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ord_less_eq_rat @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_4694_dual__order_Ostrict__implies__order,axiom,
    ! [B2: num,A2: num] :
      ( ( ord_less_num @ B2 @ A2 )
     => ( ord_less_eq_num @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_4695_dual__order_Ostrict__implies__order,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ord_less_eq_nat @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_4696_dual__order_Ostrict__implies__order,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ord_less_eq_int @ B2 @ A2 ) ) ).

% dual_order.strict_implies_order
thf(fact_4697_order__le__less,axiom,
    ( ord_less_eq_real
    = ( ^ [X3: real,Y2: real] :
          ( ( ord_less_real @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_4698_order__le__less,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X3: set_nat,Y2: set_nat] :
          ( ( ord_less_set_nat @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_4699_order__le__less,axiom,
    ( ord_less_eq_rat
    = ( ^ [X3: rat,Y2: rat] :
          ( ( ord_less_rat @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_4700_order__le__less,axiom,
    ( ord_less_eq_num
    = ( ^ [X3: num,Y2: num] :
          ( ( ord_less_num @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_4701_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X3: nat,Y2: nat] :
          ( ( ord_less_nat @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_4702_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X3: int,Y2: int] :
          ( ( ord_less_int @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% order_le_less
thf(fact_4703_order__less__le,axiom,
    ( ord_less_real
    = ( ^ [X3: real,Y2: real] :
          ( ( ord_less_eq_real @ X3 @ Y2 )
          & ( X3 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_4704_order__less__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X3: set_nat,Y2: set_nat] :
          ( ( ord_less_eq_set_nat @ X3 @ Y2 )
          & ( X3 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_4705_order__less__le,axiom,
    ( ord_less_rat
    = ( ^ [X3: rat,Y2: rat] :
          ( ( ord_less_eq_rat @ X3 @ Y2 )
          & ( X3 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_4706_order__less__le,axiom,
    ( ord_less_num
    = ( ^ [X3: num,Y2: num] :
          ( ( ord_less_eq_num @ X3 @ Y2 )
          & ( X3 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_4707_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y2: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y2 )
          & ( X3 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_4708_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X3: int,Y2: int] :
          ( ( ord_less_eq_int @ X3 @ Y2 )
          & ( X3 != Y2 ) ) ) ) ).

% order_less_le
thf(fact_4709_linorder__not__le,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_eq_real @ X @ Y ) )
      = ( ord_less_real @ Y @ X ) ) ).

% linorder_not_le
thf(fact_4710_linorder__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_eq_rat @ X @ Y ) )
      = ( ord_less_rat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_4711_linorder__not__le,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_eq_num @ X @ Y ) )
      = ( ord_less_num @ Y @ X ) ) ).

% linorder_not_le
thf(fact_4712_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_4713_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_eq_int @ X @ Y ) )
      = ( ord_less_int @ Y @ X ) ) ).

% linorder_not_le
thf(fact_4714_linorder__not__less,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_real @ X @ Y ) )
      = ( ord_less_eq_real @ Y @ X ) ) ).

% linorder_not_less
thf(fact_4715_linorder__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_rat @ X @ Y ) )
      = ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_4716_linorder__not__less,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_num @ X @ Y ) )
      = ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_not_less
thf(fact_4717_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_4718_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_not_less
thf(fact_4719_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_4720_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_4721_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_4722_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_4723_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_4724_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_4725_order__le__neq__trans,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_real @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_4726_order__le__neq__trans,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_set_nat @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_4727_order__le__neq__trans,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_rat @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_4728_order__le__neq__trans,axiom,
    ! [A2: num,B2: num] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_num @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_4729_order__le__neq__trans,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_nat @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_4730_order__le__neq__trans,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_int @ A2 @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_4731_order__neq__le__trans,axiom,
    ! [A2: real,B2: real] :
      ( ( A2 != B2 )
     => ( ( ord_less_eq_real @ A2 @ B2 )
       => ( ord_less_real @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_4732_order__neq__le__trans,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( A2 != B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( ord_less_set_nat @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_4733_order__neq__le__trans,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2 != B2 )
     => ( ( ord_less_eq_rat @ A2 @ B2 )
       => ( ord_less_rat @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_4734_order__neq__le__trans,axiom,
    ! [A2: num,B2: num] :
      ( ( A2 != B2 )
     => ( ( ord_less_eq_num @ A2 @ B2 )
       => ( ord_less_num @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_4735_order__neq__le__trans,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != B2 )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( ord_less_nat @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_4736_order__neq__le__trans,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != B2 )
     => ( ( ord_less_eq_int @ A2 @ B2 )
       => ( ord_less_int @ A2 @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_4737_order__le__less__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_real @ Y @ Z )
       => ( ord_less_real @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_4738_order__le__less__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_set_nat @ Y @ Z )
       => ( ord_less_set_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_4739_order__le__less__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ Z )
       => ( ord_less_rat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_4740_order__le__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_num @ Y @ Z )
       => ( ord_less_num @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_4741_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_4742_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_4743_order__less__le__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_eq_real @ Y @ Z )
       => ( ord_less_real @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_4744_order__less__le__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ Z )
       => ( ord_less_set_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_4745_order__less__le__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ Z )
       => ( ord_less_rat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_4746_order__less__le__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ Z )
       => ( ord_less_num @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_4747_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_4748_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_4749_order__le__less__subst1,axiom,
    ! [A2: real,F2: real > real,B2: real,C2: real] :
      ( ( ord_less_eq_real @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_real @ B2 @ C2 )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4750_order__le__less__subst1,axiom,
    ! [A2: real,F2: rat > real,B2: rat,C2: rat] :
      ( ( ord_less_eq_real @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4751_order__le__less__subst1,axiom,
    ! [A2: real,F2: num > real,B2: num,C2: num] :
      ( ( ord_less_eq_real @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_num @ B2 @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_num @ X4 @ Y3 )
             => ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4752_order__le__less__subst1,axiom,
    ! [A2: real,F2: nat > real,B2: nat,C2: nat] :
      ( ( ord_less_eq_real @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4753_order__le__less__subst1,axiom,
    ! [A2: real,F2: int > real,B2: int,C2: int] :
      ( ( ord_less_eq_real @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_int @ B2 @ C2 )
       => ( ! [X4: int,Y3: int] :
              ( ( ord_less_int @ X4 @ Y3 )
             => ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4754_order__le__less__subst1,axiom,
    ! [A2: rat,F2: real > rat,B2: real,C2: real] :
      ( ( ord_less_eq_rat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_real @ B2 @ C2 )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4755_order__le__less__subst1,axiom,
    ! [A2: rat,F2: rat > rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_rat @ B2 @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4756_order__le__less__subst1,axiom,
    ! [A2: rat,F2: num > rat,B2: num,C2: num] :
      ( ( ord_less_eq_rat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_num @ B2 @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_num @ X4 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4757_order__le__less__subst1,axiom,
    ! [A2: rat,F2: nat > rat,B2: nat,C2: nat] :
      ( ( ord_less_eq_rat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4758_order__le__less__subst1,axiom,
    ! [A2: rat,F2: int > rat,B2: int,C2: int] :
      ( ( ord_less_eq_rat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_int @ B2 @ C2 )
       => ( ! [X4: int,Y3: int] :
              ( ( ord_less_int @ X4 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4759_order__le__less__subst2,axiom,
    ! [A2: rat,B2: rat,F2: rat > real,C2: real] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_real @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_4760_order__le__less__subst2,axiom,
    ! [A2: rat,B2: rat,F2: rat > rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_rat @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_4761_order__le__less__subst2,axiom,
    ! [A2: rat,B2: rat,F2: rat > num,C2: num] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_num @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_num @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_4762_order__le__less__subst2,axiom,
    ! [A2: rat,B2: rat,F2: rat > nat,C2: nat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_4763_order__le__less__subst2,axiom,
    ! [A2: rat,B2: rat,F2: rat > int,C2: int] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_int @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_4764_order__le__less__subst2,axiom,
    ! [A2: num,B2: num,F2: num > real,C2: real] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_real @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_4765_order__le__less__subst2,axiom,
    ! [A2: num,B2: num,F2: num > rat,C2: rat] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_rat @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_4766_order__le__less__subst2,axiom,
    ! [A2: num,B2: num,F2: num > num,C2: num] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_num @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_num @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_4767_order__le__less__subst2,axiom,
    ! [A2: num,B2: num,F2: num > nat,C2: nat] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_nat @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_4768_order__le__less__subst2,axiom,
    ! [A2: num,B2: num,F2: num > int,C2: int] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_less_int @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_4769_order__less__le__subst1,axiom,
    ! [A2: real,F2: rat > real,B2: rat,C2: rat] :
      ( ( ord_less_real @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4770_order__less__le__subst1,axiom,
    ! [A2: rat,F2: rat > rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4771_order__less__le__subst1,axiom,
    ! [A2: num,F2: rat > num,B2: rat,C2: rat] :
      ( ( ord_less_num @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_num @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4772_order__less__le__subst1,axiom,
    ! [A2: nat,F2: rat > nat,B2: rat,C2: rat] :
      ( ( ord_less_nat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4773_order__less__le__subst1,axiom,
    ! [A2: int,F2: rat > int,B2: rat,C2: rat] :
      ( ( ord_less_int @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_rat @ B2 @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4774_order__less__le__subst1,axiom,
    ! [A2: real,F2: num > real,B2: num,C2: num] :
      ( ( ord_less_real @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4775_order__less__le__subst1,axiom,
    ! [A2: rat,F2: num > rat,B2: num,C2: num] :
      ( ( ord_less_rat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4776_order__less__le__subst1,axiom,
    ! [A2: num,F2: num > num,B2: num,C2: num] :
      ( ( ord_less_num @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_num @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4777_order__less__le__subst1,axiom,
    ! [A2: nat,F2: num > nat,B2: num,C2: num] :
      ( ( ord_less_nat @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4778_order__less__le__subst1,axiom,
    ! [A2: int,F2: num > int,B2: num,C2: num] :
      ( ( ord_less_int @ A2 @ ( F2 @ B2 ) )
     => ( ( ord_less_eq_num @ B2 @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_eq_num @ X4 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ A2 @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4779_order__less__le__subst2,axiom,
    ! [A2: real,B2: real,F2: real > real,C2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_4780_order__less__le__subst2,axiom,
    ! [A2: rat,B2: rat,F2: rat > real,C2: real] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_eq_real @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_4781_order__less__le__subst2,axiom,
    ! [A2: num,B2: num,F2: num > real,C2: real] :
      ( ( ord_less_num @ A2 @ B2 )
     => ( ( ord_less_eq_real @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_num @ X4 @ Y3 )
             => ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_4782_order__less__le__subst2,axiom,
    ! [A2: nat,B2: nat,F2: nat > real,C2: real] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_real @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_4783_order__less__le__subst2,axiom,
    ! [A2: int,B2: int,F2: int > real,C2: real] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_eq_real @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: int,Y3: int] :
              ( ( ord_less_int @ X4 @ Y3 )
             => ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_4784_order__less__le__subst2,axiom,
    ! [A2: real,B2: real,F2: real > rat,C2: rat] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: real,Y3: real] :
              ( ( ord_less_real @ X4 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_4785_order__less__le__subst2,axiom,
    ! [A2: rat,B2: rat,F2: rat > rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: rat,Y3: rat] :
              ( ( ord_less_rat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_4786_order__less__le__subst2,axiom,
    ! [A2: num,B2: num,F2: num > rat,C2: rat] :
      ( ( ord_less_num @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: num,Y3: num] :
              ( ( ord_less_num @ X4 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_4787_order__less__le__subst2,axiom,
    ! [A2: nat,B2: nat,F2: nat > rat,C2: rat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_4788_order__less__le__subst2,axiom,
    ! [A2: int,B2: int,F2: int > rat,C2: rat] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ ( F2 @ B2 ) @ C2 )
       => ( ! [X4: int,Y3: int] :
              ( ( ord_less_int @ X4 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X4 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ ( F2 @ A2 ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_4789_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_4790_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_4791_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_4792_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_4793_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_4794_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_4795_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_4796_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_4797_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_4798_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_4799_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_4800_verit__sum__simplify,axiom,
    ! [A2: complex] :
      ( ( plus_plus_complex @ A2 @ zero_zero_complex )
      = A2 ) ).

% verit_sum_simplify
thf(fact_4801_verit__sum__simplify,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ A2 @ zero_zero_real )
      = A2 ) ).

% verit_sum_simplify
thf(fact_4802_verit__sum__simplify,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ A2 @ zero_zero_rat )
      = A2 ) ).

% verit_sum_simplify
thf(fact_4803_verit__sum__simplify,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% verit_sum_simplify
thf(fact_4804_verit__sum__simplify,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ A2 @ zero_zero_int )
      = A2 ) ).

% verit_sum_simplify
thf(fact_4805_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] :
      ( one
     != ( bit0 @ X22 ) ) ).

% verit_eq_simplify(10)
thf(fact_4806_bot_Oextremum,axiom,
    ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).

% bot.extremum
thf(fact_4807_bot_Oextremum,axiom,
    ! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).

% bot.extremum
thf(fact_4808_bot_Oextremum,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).

% bot.extremum
thf(fact_4809_bot_Oextremum,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).

% bot.extremum
thf(fact_4810_bot_Oextremum,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).

% bot.extremum
thf(fact_4811_bot_Oextremum__unique,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
      = ( A2 = bot_bot_set_real ) ) ).

% bot.extremum_unique
thf(fact_4812_bot_Oextremum__unique,axiom,
    ! [A2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
      = ( A2 = bot_bot_set_o ) ) ).

% bot.extremum_unique
thf(fact_4813_bot_Oextremum__unique,axiom,
    ! [A2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
      = ( A2 = bot_bot_set_int ) ) ).

% bot.extremum_unique
thf(fact_4814_bot_Oextremum__unique,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
      = ( A2 = bot_bot_set_nat ) ) ).

% bot.extremum_unique
thf(fact_4815_bot_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
      = ( A2 = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_4816_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
     => ( A2 = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
thf(fact_4817_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
     => ( A2 = bot_bot_set_o ) ) ).

% bot.extremum_uniqueI
thf(fact_4818_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
     => ( A2 = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
thf(fact_4819_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
     => ( A2 = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_4820_bot_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
     => ( A2 = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_4821_bot_Oextremum__strict,axiom,
    ! [A2: set_real] :
      ~ ( ord_less_set_real @ A2 @ bot_bot_set_real ) ).

% bot.extremum_strict
thf(fact_4822_bot_Oextremum__strict,axiom,
    ! [A2: set_o] :
      ~ ( ord_less_set_o @ A2 @ bot_bot_set_o ) ).

% bot.extremum_strict
thf(fact_4823_bot_Oextremum__strict,axiom,
    ! [A2: set_nat] :
      ~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_4824_bot_Oextremum__strict,axiom,
    ! [A2: set_int] :
      ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).

% bot.extremum_strict
thf(fact_4825_bot_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_4826_bot_Onot__eq__extremum,axiom,
    ! [A2: set_real] :
      ( ( A2 != bot_bot_set_real )
      = ( ord_less_set_real @ bot_bot_set_real @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_4827_bot_Onot__eq__extremum,axiom,
    ! [A2: set_o] :
      ( ( A2 != bot_bot_set_o )
      = ( ord_less_set_o @ bot_bot_set_o @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_4828_bot_Onot__eq__extremum,axiom,
    ! [A2: set_nat] :
      ( ( A2 != bot_bot_set_nat )
      = ( ord_less_set_nat @ bot_bot_set_nat @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_4829_bot_Onot__eq__extremum,axiom,
    ! [A2: set_int] :
      ( ( A2 != bot_bot_set_int )
      = ( ord_less_set_int @ bot_bot_set_int @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_4830_bot_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).

% bot.not_eq_extremum
thf(fact_4831_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X33: num] :
      ( ( bit0 @ X22 )
     != ( bit1 @ X33 ) ) ).

% verit_eq_simplify(14)
thf(fact_4832_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_12: nat] : ( P @ X_12 )
       => ? [N: nat] :
            ( ~ ( P @ N )
            & ( P @ ( suc @ N ) ) ) ) ) ).

% exists_least_lemma
thf(fact_4833_real__arch__simple,axiom,
    ! [X: real] :
    ? [N: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ).

% real_arch_simple
thf(fact_4834_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N ) ) ).

% real_arch_simple
thf(fact_4835_verit__eq__simplify_I12_J,axiom,
    ! [X33: num] :
      ( one
     != ( bit1 @ X33 ) ) ).

% verit_eq_simplify(12)
thf(fact_4836_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N ) ) ).

% reals_Archimedean2
thf(fact_4837_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ).

% reals_Archimedean2
thf(fact_4838_max__absorb2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_max_set_nat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4839_max__absorb2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_max_rat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4840_max__absorb2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_max_num @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4841_max__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_max_nat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4842_max__absorb2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_max_int @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4843_max__absorb1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_max_set_nat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4844_max__absorb1,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( ord_max_rat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4845_max__absorb1,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ( ( ord_max_num @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4846_max__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_max_nat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4847_max__absorb1,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_max_int @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4848_max__def,axiom,
    ( ord_max_set_nat
    = ( ^ [A7: set_nat,B7: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A7 @ B7 ) @ B7 @ A7 ) ) ) ).

% max_def
thf(fact_4849_max__def,axiom,
    ( ord_max_rat
    = ( ^ [A7: rat,B7: rat] : ( if_rat @ ( ord_less_eq_rat @ A7 @ B7 ) @ B7 @ A7 ) ) ) ).

% max_def
thf(fact_4850_max__def,axiom,
    ( ord_max_num
    = ( ^ [A7: num,B7: num] : ( if_num @ ( ord_less_eq_num @ A7 @ B7 ) @ B7 @ A7 ) ) ) ).

% max_def
thf(fact_4851_max__def,axiom,
    ( ord_max_nat
    = ( ^ [A7: nat,B7: nat] : ( if_nat @ ( ord_less_eq_nat @ A7 @ B7 ) @ B7 @ A7 ) ) ) ).

% max_def
thf(fact_4852_max__def,axiom,
    ( ord_max_int
    = ( ^ [A7: int,B7: int] : ( if_int @ ( ord_less_eq_int @ A7 @ B7 ) @ B7 @ A7 ) ) ) ).

% max_def
thf(fact_4853_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).

% VEBT_internal.minNull.simps(1)
thf(fact_4854_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv2: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv2 ) ) ).

% VEBT_internal.minNull.simps(2)
thf(fact_4855_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).

% VEBT_internal.minNull.simps(3)
thf(fact_4856_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv2 @ Uw ) @ X )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
thf(fact_4857_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
thf(fact_4858_int__ops_I3_J,axiom,
    ! [N3: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N3 ) )
      = ( numeral_numeral_int @ N3 ) ) ).

% int_ops(3)
thf(fact_4859_int__ops_I1_J,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% int_ops(1)
thf(fact_4860_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A7: nat,B7: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A7 ) @ ( semiri1314217659103216013at_int @ B7 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_4861_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A7: nat,B7: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A7 ) @ ( semiri1314217659103216013at_int @ B7 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_4862_int__ops_I5_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A2 @ B2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).

% int_ops(5)
thf(fact_4863_int__plus,axiom,
    ! [N3: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N3 @ M ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N3 ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% int_plus
thf(fact_4864_int__ops_I7_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A2 @ B2 ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).

% int_ops(7)
thf(fact_4865_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz2: product_prod_nat_nat,Va: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va @ Vb2 @ Vc2 ) ) ).

% VEBT_internal.minNull.simps(5)
thf(fact_4866_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) ) ).

% VEBT_internal.minNull.simps(4)
thf(fact_4867_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
thf(fact_4868_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv2 @ Uw ) @ X ) ).

% vebt_member.simps(2)
thf(fact_4869_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( X = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(1)
thf(fact_4870_ex__less__of__nat__mult,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ? [N: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_4871_ex__less__of__nat__mult,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [N: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_4872_int__ops_I4_J,axiom,
    ! [A2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ A2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ one_one_int ) ) ).

% int_ops(4)
thf(fact_4873_int__Suc,axiom,
    ! [N3: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N3 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N3 ) @ one_one_int ) ) ).

% int_Suc
thf(fact_4874_less__1__helper,axiom,
    ! [N3: int,M: int] :
      ( ( ord_less_eq_int @ N3 @ M )
     => ( ord_less_int @ ( minus_minus_int @ N3 @ one_one_int ) @ M ) ) ).

% less_1_helper
thf(fact_4875_vebt__member_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
      = ( ( ( X = zero_zero_nat )
         => A2 )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => B2 )
            & ( X = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
thf(fact_4876_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X ) ).

% vebt_member.simps(3)
thf(fact_4877_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X )
     => ( ! [Uv: $o] :
            ( X
           != ( vEBT_Leaf @ $true @ Uv ) )
       => ( ! [Uu2: $o] :
              ( X
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ~ ! [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va2 @ Vb @ Vc ) ) ) ) ) ).

% VEBT_internal.minNull.elims(3)
thf(fact_4878_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X )
     => ( ( X
         != ( vEBT_Leaf @ $false @ $false ) )
       => ~ ! [Uw2: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux @ Uy ) ) ) ) ).

% VEBT_internal.minNull.elims(2)
thf(fact_4879_member__bound__height,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_nat @ ( vEBT_T_m_e_m_b_e_r @ T @ X ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ).

% member_bound_height
thf(fact_4880_max_Oabsorb3,axiom,
    ! [B2: real,A2: real] :
      ( ( ord_less_real @ B2 @ A2 )
     => ( ( ord_max_real @ A2 @ B2 )
        = A2 ) ) ).

% max.absorb3
thf(fact_4881_max_Oabsorb3,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_rat @ B2 @ A2 )
     => ( ( ord_max_rat @ A2 @ B2 )
        = A2 ) ) ).

% max.absorb3
thf(fact_4882_max_Oabsorb3,axiom,
    ! [B2: num,A2: num] :
      ( ( ord_less_num @ B2 @ A2 )
     => ( ( ord_max_num @ A2 @ B2 )
        = A2 ) ) ).

% max.absorb3
thf(fact_4883_max_Oabsorb3,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( ord_max_nat @ A2 @ B2 )
        = A2 ) ) ).

% max.absorb3
thf(fact_4884_max_Oabsorb3,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ B2 @ A2 )
     => ( ( ord_max_int @ A2 @ B2 )
        = A2 ) ) ).

% max.absorb3
thf(fact_4885_max_Oabsorb4,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_max_real @ A2 @ B2 )
        = B2 ) ) ).

% max.absorb4
thf(fact_4886_max_Oabsorb4,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ( ord_max_rat @ A2 @ B2 )
        = B2 ) ) ).

% max.absorb4
thf(fact_4887_max_Oabsorb4,axiom,
    ! [A2: num,B2: num] :
      ( ( ord_less_num @ A2 @ B2 )
     => ( ( ord_max_num @ A2 @ B2 )
        = B2 ) ) ).

% max.absorb4
thf(fact_4888_max_Oabsorb4,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_max_nat @ A2 @ B2 )
        = B2 ) ) ).

% max.absorb4
thf(fact_4889_max_Oabsorb4,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_max_int @ A2 @ B2 )
        = B2 ) ) ).

% max.absorb4
thf(fact_4890_max__less__iff__conj,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_real @ ( ord_max_real @ X @ Y ) @ Z )
      = ( ( ord_less_real @ X @ Z )
        & ( ord_less_real @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_4891_max__less__iff__conj,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ X @ Y ) @ Z )
      = ( ( ord_less_rat @ X @ Z )
        & ( ord_less_rat @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_4892_max__less__iff__conj,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_num @ ( ord_max_num @ X @ Y ) @ Z )
      = ( ( ord_less_num @ X @ Z )
        & ( ord_less_num @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_4893_max__less__iff__conj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z )
      = ( ( ord_less_nat @ X @ Z )
        & ( ord_less_nat @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_4894_max__less__iff__conj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ( ord_less_int @ X @ Z )
        & ( ord_less_int @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_4895_max_Obounded__iff,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B2 @ C2 ) @ A2 )
      = ( ( ord_less_eq_rat @ B2 @ A2 )
        & ( ord_less_eq_rat @ C2 @ A2 ) ) ) ).

% max.bounded_iff
thf(fact_4896_max_Obounded__iff,axiom,
    ! [B2: num,C2: num,A2: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B2 @ C2 ) @ A2 )
      = ( ( ord_less_eq_num @ B2 @ A2 )
        & ( ord_less_eq_num @ C2 @ A2 ) ) ) ).

% max.bounded_iff
thf(fact_4897_max_Obounded__iff,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B2 @ C2 ) @ A2 )
      = ( ( ord_less_eq_nat @ B2 @ A2 )
        & ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).

% max.bounded_iff
thf(fact_4898_max_Obounded__iff,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B2 @ C2 ) @ A2 )
      = ( ( ord_less_eq_int @ B2 @ A2 )
        & ( ord_less_eq_int @ C2 @ A2 ) ) ) ).

% max.bounded_iff
thf(fact_4899_max_Oabsorb2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_max_rat @ A2 @ B2 )
        = B2 ) ) ).

% max.absorb2
thf(fact_4900_max_Oabsorb2,axiom,
    ! [A2: num,B2: num] :
      ( ( ord_less_eq_num @ A2 @ B2 )
     => ( ( ord_max_num @ A2 @ B2 )
        = B2 ) ) ).

% max.absorb2
thf(fact_4901_max_Oabsorb2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_max_nat @ A2 @ B2 )
        = B2 ) ) ).

% max.absorb2
thf(fact_4902_max_Oabsorb2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_max_int @ A2 @ B2 )
        = B2 ) ) ).

% max.absorb2
thf(fact_4903_max_Oabsorb1,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( ord_max_rat @ A2 @ B2 )
        = A2 ) ) ).

% max.absorb1
thf(fact_4904_max_Oabsorb1,axiom,
    ! [B2: num,A2: num] :
      ( ( ord_less_eq_num @ B2 @ A2 )
     => ( ( ord_max_num @ A2 @ B2 )
        = A2 ) ) ).

% max.absorb1
thf(fact_4905_max_Oabsorb1,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_max_nat @ A2 @ B2 )
        = A2 ) ) ).

% max.absorb1
thf(fact_4906_max_Oabsorb1,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_max_int @ A2 @ B2 )
        = A2 ) ) ).

% max.absorb1
thf(fact_4907_enat__ord__number_I1_J,axiom,
    ! [M: num,N3: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N3 ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N3 ) ) ) ).

% enat_ord_number(1)
thf(fact_4908_enat__ord__number_I2_J,axiom,
    ! [M: num,N3: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N3 ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N3 ) ) ) ).

% enat_ord_number(2)
thf(fact_4909_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Z @ Y )
     => ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y @ Z ) )
        = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ Z ) ) ) ).

% add_diff_assoc_enat
thf(fact_4910_max_Omono,axiom,
    ! [C2: rat,A2: rat,D2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ C2 @ A2 )
     => ( ( ord_less_eq_rat @ D2 @ B2 )
       => ( ord_less_eq_rat @ ( ord_max_rat @ C2 @ D2 ) @ ( ord_max_rat @ A2 @ B2 ) ) ) ) ).

% max.mono
thf(fact_4911_max_Omono,axiom,
    ! [C2: num,A2: num,D2: num,B2: num] :
      ( ( ord_less_eq_num @ C2 @ A2 )
     => ( ( ord_less_eq_num @ D2 @ B2 )
       => ( ord_less_eq_num @ ( ord_max_num @ C2 @ D2 ) @ ( ord_max_num @ A2 @ B2 ) ) ) ) ).

% max.mono
thf(fact_4912_max_Omono,axiom,
    ! [C2: nat,A2: nat,D2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C2 @ A2 )
     => ( ( ord_less_eq_nat @ D2 @ B2 )
       => ( ord_less_eq_nat @ ( ord_max_nat @ C2 @ D2 ) @ ( ord_max_nat @ A2 @ B2 ) ) ) ) ).

% max.mono
thf(fact_4913_max_Omono,axiom,
    ! [C2: int,A2: int,D2: int,B2: int] :
      ( ( ord_less_eq_int @ C2 @ A2 )
     => ( ( ord_less_eq_int @ D2 @ B2 )
       => ( ord_less_eq_int @ ( ord_max_int @ C2 @ D2 ) @ ( ord_max_int @ A2 @ B2 ) ) ) ) ).

% max.mono
thf(fact_4914_max_OorderE,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( A2
        = ( ord_max_rat @ A2 @ B2 ) ) ) ).

% max.orderE
thf(fact_4915_max_OorderE,axiom,
    ! [B2: num,A2: num] :
      ( ( ord_less_eq_num @ B2 @ A2 )
     => ( A2
        = ( ord_max_num @ A2 @ B2 ) ) ) ).

% max.orderE
thf(fact_4916_max_OorderE,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( A2
        = ( ord_max_nat @ A2 @ B2 ) ) ) ).

% max.orderE
thf(fact_4917_max_OorderE,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( A2
        = ( ord_max_int @ A2 @ B2 ) ) ) ).

% max.orderE
thf(fact_4918_max_OorderI,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2
        = ( ord_max_rat @ A2 @ B2 ) )
     => ( ord_less_eq_rat @ B2 @ A2 ) ) ).

% max.orderI
thf(fact_4919_max_OorderI,axiom,
    ! [A2: num,B2: num] :
      ( ( A2
        = ( ord_max_num @ A2 @ B2 ) )
     => ( ord_less_eq_num @ B2 @ A2 ) ) ).

% max.orderI
thf(fact_4920_max_OorderI,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2
        = ( ord_max_nat @ A2 @ B2 ) )
     => ( ord_less_eq_nat @ B2 @ A2 ) ) ).

% max.orderI
thf(fact_4921_max_OorderI,axiom,
    ! [A2: int,B2: int] :
      ( ( A2
        = ( ord_max_int @ A2 @ B2 ) )
     => ( ord_less_eq_int @ B2 @ A2 ) ) ).

% max.orderI
thf(fact_4922_max_OboundedE,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_less_eq_rat @ B2 @ A2 )
         => ~ ( ord_less_eq_rat @ C2 @ A2 ) ) ) ).

% max.boundedE
thf(fact_4923_max_OboundedE,axiom,
    ! [B2: num,C2: num,A2: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_less_eq_num @ B2 @ A2 )
         => ~ ( ord_less_eq_num @ C2 @ A2 ) ) ) ).

% max.boundedE
thf(fact_4924_max_OboundedE,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_less_eq_nat @ B2 @ A2 )
         => ~ ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).

% max.boundedE
thf(fact_4925_max_OboundedE,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_less_eq_int @ B2 @ A2 )
         => ~ ( ord_less_eq_int @ C2 @ A2 ) ) ) ).

% max.boundedE
thf(fact_4926_max_OboundedI,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B2 @ A2 )
     => ( ( ord_less_eq_rat @ C2 @ A2 )
       => ( ord_less_eq_rat @ ( ord_max_rat @ B2 @ C2 ) @ A2 ) ) ) ).

% max.boundedI
thf(fact_4927_max_OboundedI,axiom,
    ! [B2: num,A2: num,C2: num] :
      ( ( ord_less_eq_num @ B2 @ A2 )
     => ( ( ord_less_eq_num @ C2 @ A2 )
       => ( ord_less_eq_num @ ( ord_max_num @ B2 @ C2 ) @ A2 ) ) ) ).

% max.boundedI
thf(fact_4928_max_OboundedI,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ C2 @ A2 )
       => ( ord_less_eq_nat @ ( ord_max_nat @ B2 @ C2 ) @ A2 ) ) ) ).

% max.boundedI
thf(fact_4929_max_OboundedI,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( ord_less_eq_int @ B2 @ A2 )
     => ( ( ord_less_eq_int @ C2 @ A2 )
       => ( ord_less_eq_int @ ( ord_max_int @ B2 @ C2 ) @ A2 ) ) ) ).

% max.boundedI
thf(fact_4930_max_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B7: rat,A7: rat] :
          ( A7
          = ( ord_max_rat @ A7 @ B7 ) ) ) ) ).

% max.order_iff
thf(fact_4931_max_Oorder__iff,axiom,
    ( ord_less_eq_num
    = ( ^ [B7: num,A7: num] :
          ( A7
          = ( ord_max_num @ A7 @ B7 ) ) ) ) ).

% max.order_iff
thf(fact_4932_max_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B7: nat,A7: nat] :
          ( A7
          = ( ord_max_nat @ A7 @ B7 ) ) ) ) ).

% max.order_iff
thf(fact_4933_max_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B7: int,A7: int] :
          ( A7
          = ( ord_max_int @ A7 @ B7 ) ) ) ) ).

% max.order_iff
thf(fact_4934_max_Ocobounded1,axiom,
    ! [A2: rat,B2: rat] : ( ord_less_eq_rat @ A2 @ ( ord_max_rat @ A2 @ B2 ) ) ).

% max.cobounded1
thf(fact_4935_max_Ocobounded1,axiom,
    ! [A2: num,B2: num] : ( ord_less_eq_num @ A2 @ ( ord_max_num @ A2 @ B2 ) ) ).

% max.cobounded1
thf(fact_4936_max_Ocobounded1,axiom,
    ! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( ord_max_nat @ A2 @ B2 ) ) ).

% max.cobounded1
thf(fact_4937_max_Ocobounded1,axiom,
    ! [A2: int,B2: int] : ( ord_less_eq_int @ A2 @ ( ord_max_int @ A2 @ B2 ) ) ).

% max.cobounded1
thf(fact_4938_max_Ocobounded2,axiom,
    ! [B2: rat,A2: rat] : ( ord_less_eq_rat @ B2 @ ( ord_max_rat @ A2 @ B2 ) ) ).

% max.cobounded2
thf(fact_4939_max_Ocobounded2,axiom,
    ! [B2: num,A2: num] : ( ord_less_eq_num @ B2 @ ( ord_max_num @ A2 @ B2 ) ) ).

% max.cobounded2
thf(fact_4940_max_Ocobounded2,axiom,
    ! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( ord_max_nat @ A2 @ B2 ) ) ).

% max.cobounded2
thf(fact_4941_max_Ocobounded2,axiom,
    ! [B2: int,A2: int] : ( ord_less_eq_int @ B2 @ ( ord_max_int @ A2 @ B2 ) ) ).

% max.cobounded2
thf(fact_4942_le__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X @ Y ) )
      = ( ( ord_less_eq_rat @ Z @ X )
        | ( ord_less_eq_rat @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_4943_le__max__iff__disj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( ( ord_less_eq_num @ Z @ ( ord_max_num @ X @ Y ) )
      = ( ( ord_less_eq_num @ Z @ X )
        | ( ord_less_eq_num @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_4944_le__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_eq_nat @ Z @ X )
        | ( ord_less_eq_nat @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_4945_le__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_eq_int @ Z @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_eq_int @ Z @ X )
        | ( ord_less_eq_int @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_4946_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B7: rat,A7: rat] :
          ( ( ord_max_rat @ A7 @ B7 )
          = A7 ) ) ) ).

% max.absorb_iff1
thf(fact_4947_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_num
    = ( ^ [B7: num,A7: num] :
          ( ( ord_max_num @ A7 @ B7 )
          = A7 ) ) ) ).

% max.absorb_iff1
thf(fact_4948_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B7: nat,A7: nat] :
          ( ( ord_max_nat @ A7 @ B7 )
          = A7 ) ) ) ).

% max.absorb_iff1
thf(fact_4949_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B7: int,A7: int] :
          ( ( ord_max_int @ A7 @ B7 )
          = A7 ) ) ) ).

% max.absorb_iff1
thf(fact_4950_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A7: rat,B7: rat] :
          ( ( ord_max_rat @ A7 @ B7 )
          = B7 ) ) ) ).

% max.absorb_iff2
thf(fact_4951_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_num
    = ( ^ [A7: num,B7: num] :
          ( ( ord_max_num @ A7 @ B7 )
          = B7 ) ) ) ).

% max.absorb_iff2
thf(fact_4952_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A7: nat,B7: nat] :
          ( ( ord_max_nat @ A7 @ B7 )
          = B7 ) ) ) ).

% max.absorb_iff2
thf(fact_4953_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A7: int,B7: int] :
          ( ( ord_max_int @ A7 @ B7 )
          = B7 ) ) ) ).

% max.absorb_iff2
thf(fact_4954_max_OcoboundedI1,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ C2 @ A2 )
     => ( ord_less_eq_rat @ C2 @ ( ord_max_rat @ A2 @ B2 ) ) ) ).

% max.coboundedI1
thf(fact_4955_max_OcoboundedI1,axiom,
    ! [C2: num,A2: num,B2: num] :
      ( ( ord_less_eq_num @ C2 @ A2 )
     => ( ord_less_eq_num @ C2 @ ( ord_max_num @ A2 @ B2 ) ) ) ).

% max.coboundedI1
thf(fact_4956_max_OcoboundedI1,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ C2 @ A2 )
     => ( ord_less_eq_nat @ C2 @ ( ord_max_nat @ A2 @ B2 ) ) ) ).

% max.coboundedI1
thf(fact_4957_max_OcoboundedI1,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ C2 @ A2 )
     => ( ord_less_eq_int @ C2 @ ( ord_max_int @ A2 @ B2 ) ) ) ).

% max.coboundedI1
thf(fact_4958_max_OcoboundedI2,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ C2 @ B2 )
     => ( ord_less_eq_rat @ C2 @ ( ord_max_rat @ A2 @ B2 ) ) ) ).

% max.coboundedI2
thf(fact_4959_max_OcoboundedI2,axiom,
    ! [C2: num,B2: num,A2: num] :
      ( ( ord_less_eq_num @ C2 @ B2 )
     => ( ord_less_eq_num @ C2 @ ( ord_max_num @ A2 @ B2 ) ) ) ).

% max.coboundedI2
thf(fact_4960_max_OcoboundedI2,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ C2 @ B2 )
     => ( ord_less_eq_nat @ C2 @ ( ord_max_nat @ A2 @ B2 ) ) ) ).

% max.coboundedI2
thf(fact_4961_max_OcoboundedI2,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( ord_less_eq_int @ C2 @ B2 )
     => ( ord_less_eq_int @ C2 @ ( ord_max_int @ A2 @ B2 ) ) ) ).

% max.coboundedI2
thf(fact_4962_less__max__iff__disj,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ Z @ ( ord_max_real @ X @ Y ) )
      = ( ( ord_less_real @ Z @ X )
        | ( ord_less_real @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_4963_less__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ Z @ ( ord_max_rat @ X @ Y ) )
      = ( ( ord_less_rat @ Z @ X )
        | ( ord_less_rat @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_4964_less__max__iff__disj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( ( ord_less_num @ Z @ ( ord_max_num @ X @ Y ) )
      = ( ( ord_less_num @ Z @ X )
        | ( ord_less_num @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_4965_less__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ Z @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Z @ X )
        | ( ord_less_nat @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_4966_less__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ Z @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_int @ Z @ X )
        | ( ord_less_int @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_4967_max_Ostrict__boundedE,axiom,
    ! [B2: real,C2: real,A2: real] :
      ( ( ord_less_real @ ( ord_max_real @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_less_real @ B2 @ A2 )
         => ~ ( ord_less_real @ C2 @ A2 ) ) ) ).

% max.strict_boundedE
thf(fact_4968_max_Ostrict__boundedE,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_less_rat @ B2 @ A2 )
         => ~ ( ord_less_rat @ C2 @ A2 ) ) ) ).

% max.strict_boundedE
thf(fact_4969_max_Ostrict__boundedE,axiom,
    ! [B2: num,C2: num,A2: num] :
      ( ( ord_less_num @ ( ord_max_num @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_less_num @ B2 @ A2 )
         => ~ ( ord_less_num @ C2 @ A2 ) ) ) ).

% max.strict_boundedE
thf(fact_4970_max_Ostrict__boundedE,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_less_nat @ B2 @ A2 )
         => ~ ( ord_less_nat @ C2 @ A2 ) ) ) ).

% max.strict_boundedE
thf(fact_4971_max_Ostrict__boundedE,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( ord_less_int @ ( ord_max_int @ B2 @ C2 ) @ A2 )
     => ~ ( ( ord_less_int @ B2 @ A2 )
         => ~ ( ord_less_int @ C2 @ A2 ) ) ) ).

% max.strict_boundedE
thf(fact_4972_max_Ostrict__order__iff,axiom,
    ( ord_less_real
    = ( ^ [B7: real,A7: real] :
          ( ( A7
            = ( ord_max_real @ A7 @ B7 ) )
          & ( A7 != B7 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4973_max_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B7: rat,A7: rat] :
          ( ( A7
            = ( ord_max_rat @ A7 @ B7 ) )
          & ( A7 != B7 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4974_max_Ostrict__order__iff,axiom,
    ( ord_less_num
    = ( ^ [B7: num,A7: num] :
          ( ( A7
            = ( ord_max_num @ A7 @ B7 ) )
          & ( A7 != B7 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4975_max_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B7: nat,A7: nat] :
          ( ( A7
            = ( ord_max_nat @ A7 @ B7 ) )
          & ( A7 != B7 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4976_max_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B7: int,A7: int] :
          ( ( A7
            = ( ord_max_int @ A7 @ B7 ) )
          & ( A7 != B7 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4977_max_Ostrict__coboundedI1,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ord_less_real @ C2 @ A2 )
     => ( ord_less_real @ C2 @ ( ord_max_real @ A2 @ B2 ) ) ) ).

% max.strict_coboundedI1
thf(fact_4978_max_Ostrict__coboundedI1,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ C2 @ A2 )
     => ( ord_less_rat @ C2 @ ( ord_max_rat @ A2 @ B2 ) ) ) ).

% max.strict_coboundedI1
thf(fact_4979_max_Ostrict__coboundedI1,axiom,
    ! [C2: num,A2: num,B2: num] :
      ( ( ord_less_num @ C2 @ A2 )
     => ( ord_less_num @ C2 @ ( ord_max_num @ A2 @ B2 ) ) ) ).

% max.strict_coboundedI1
thf(fact_4980_max_Ostrict__coboundedI1,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ C2 @ A2 )
     => ( ord_less_nat @ C2 @ ( ord_max_nat @ A2 @ B2 ) ) ) ).

% max.strict_coboundedI1
thf(fact_4981_max_Ostrict__coboundedI1,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_int @ C2 @ A2 )
     => ( ord_less_int @ C2 @ ( ord_max_int @ A2 @ B2 ) ) ) ).

% max.strict_coboundedI1
thf(fact_4982_max_Ostrict__coboundedI2,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( ( ord_less_real @ C2 @ B2 )
     => ( ord_less_real @ C2 @ ( ord_max_real @ A2 @ B2 ) ) ) ).

% max.strict_coboundedI2
thf(fact_4983_max_Ostrict__coboundedI2,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ C2 @ B2 )
     => ( ord_less_rat @ C2 @ ( ord_max_rat @ A2 @ B2 ) ) ) ).

% max.strict_coboundedI2
thf(fact_4984_max_Ostrict__coboundedI2,axiom,
    ! [C2: num,B2: num,A2: num] :
      ( ( ord_less_num @ C2 @ B2 )
     => ( ord_less_num @ C2 @ ( ord_max_num @ A2 @ B2 ) ) ) ).

% max.strict_coboundedI2
thf(fact_4985_max_Ostrict__coboundedI2,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( ord_less_nat @ C2 @ B2 )
     => ( ord_less_nat @ C2 @ ( ord_max_nat @ A2 @ B2 ) ) ) ).

% max.strict_coboundedI2
thf(fact_4986_max_Ostrict__coboundedI2,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( ord_less_int @ C2 @ B2 )
     => ( ord_less_int @ C2 @ ( ord_max_int @ A2 @ B2 ) ) ) ).

% max.strict_coboundedI2
thf(fact_4987_heigt__uplog__rel,axiom,
    ! [T: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ T ) )
        = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ).

% heigt_uplog_rel
thf(fact_4988_set__bit__0,axiom,
    ! [A2: int] :
      ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A2 )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_4989_set__bit__0,axiom,
    ! [A2: nat] :
      ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A2 )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_4990_i0__less,axiom,
    ! [N3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N3 )
      = ( N3 != zero_z5237406670263579293d_enat ) ) ).

% i0_less
thf(fact_4991_idiff__0__right,axiom,
    ! [N3: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ N3 @ zero_z5237406670263579293d_enat )
      = N3 ) ).

% idiff_0_right
thf(fact_4992_idiff__0,axiom,
    ! [N3: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N3 )
      = zero_z5237406670263579293d_enat ) ).

% idiff_0
thf(fact_4993_set__bit__nonnegative__int__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N3 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% set_bit_nonnegative_int_iff
thf(fact_4994_set__bit__negative__int__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N3 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% set_bit_negative_int_iff
thf(fact_4995_ceiling__zero,axiom,
    ( ( archim2889992004027027881ng_rat @ zero_zero_rat )
    = zero_zero_int ) ).

% ceiling_zero
thf(fact_4996_ceiling__zero,axiom,
    ( ( archim7802044766580827645g_real @ zero_zero_real )
    = zero_zero_int ) ).

% ceiling_zero
thf(fact_4997_ceiling__numeral,axiom,
    ! [V: num] :
      ( ( archim7802044766580827645g_real @ ( numeral_numeral_real @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% ceiling_numeral
thf(fact_4998_ceiling__numeral,axiom,
    ! [V: num] :
      ( ( archim2889992004027027881ng_rat @ ( numeral_numeral_rat @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% ceiling_numeral
thf(fact_4999_ceiling__le__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% ceiling_le_zero
thf(fact_5000_ceiling__le__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).

% ceiling_le_zero
thf(fact_5001_ceiling__le__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_real @ X @ ( numeral_numeral_real @ V ) ) ) ).

% ceiling_le_numeral
thf(fact_5002_ceiling__le__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_rat @ X @ ( numeral_numeral_rat @ V ) ) ) ).

% ceiling_le_numeral
thf(fact_5003_zero__less__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ zero_zero_rat @ X ) ) ).

% zero_less_ceiling
thf(fact_5004_zero__less__ceiling,axiom,
    ! [X: real] :
      ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% zero_less_ceiling
thf(fact_5005_numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X ) ) ).

% numeral_less_ceiling
thf(fact_5006_numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X ) ) ).

% numeral_less_ceiling
thf(fact_5007_ceiling__less__one,axiom,
    ! [X: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% ceiling_less_one
thf(fact_5008_ceiling__less__one,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
      = ( ord_less_eq_rat @ X @ zero_zero_rat ) ) ).

% ceiling_less_one
thf(fact_5009_one__le__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ zero_zero_rat @ X ) ) ).

% one_le_ceiling
thf(fact_5010_one__le__ceiling,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% one_le_ceiling
thf(fact_5011_ceiling__add__numeral,axiom,
    ! [X: real,V: num] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( numeral_numeral_real @ V ) ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_add_numeral
thf(fact_5012_ceiling__add__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_add_numeral
thf(fact_5013_ceiling__le__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int )
      = ( ord_less_eq_real @ X @ one_one_real ) ) ).

% ceiling_le_one
thf(fact_5014_ceiling__le__one,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int )
      = ( ord_less_eq_rat @ X @ one_one_rat ) ) ).

% ceiling_le_one
thf(fact_5015_one__less__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ one_one_rat @ X ) ) ).

% one_less_ceiling
thf(fact_5016_one__less__ceiling,axiom,
    ! [X: real] :
      ( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ one_one_real @ X ) ) ).

% one_less_ceiling
thf(fact_5017_ceiling__add__one,axiom,
    ! [X: rat] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ one_one_rat ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).

% ceiling_add_one
thf(fact_5018_ceiling__add__one,axiom,
    ! [X: real] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ one_one_real ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).

% ceiling_add_one
thf(fact_5019_ceiling__diff__numeral,axiom,
    ! [X: real,V: num] :
      ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ ( numeral_numeral_real @ V ) ) )
      = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_diff_numeral
thf(fact_5020_ceiling__diff__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ ( numeral_numeral_rat @ V ) ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_diff_numeral
thf(fact_5021_ceiling__numeral__power,axiom,
    ! [X: num,N3: nat] :
      ( ( archim7802044766580827645g_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N3 ) )
      = ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ).

% ceiling_numeral_power
thf(fact_5022_ceiling__numeral__power,axiom,
    ! [X: num,N3: nat] :
      ( ( archim2889992004027027881ng_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N3 ) )
      = ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ).

% ceiling_numeral_power
thf(fact_5023_ceiling__diff__one,axiom,
    ! [X: rat] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ one_one_rat ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ one_one_int ) ) ).

% ceiling_diff_one
thf(fact_5024_ceiling__diff__one,axiom,
    ! [X: real] :
      ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ one_one_real ) )
      = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ one_one_int ) ) ).

% ceiling_diff_one
thf(fact_5025_ceiling__less__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).

% ceiling_less_numeral
thf(fact_5026_ceiling__less__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).

% ceiling_less_numeral
thf(fact_5027_numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X ) ) ).

% numeral_le_ceiling
thf(fact_5028_numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X ) ) ).

% numeral_le_ceiling
thf(fact_5029_not__iless0,axiom,
    ! [N3: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ N3 @ zero_z5237406670263579293d_enat ) ).

% not_iless0
thf(fact_5030_enat__less__induct,axiom,
    ! [P: extended_enat > $o,N3: extended_enat] :
      ( ! [N: extended_enat] :
          ( ! [M2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ M2 @ N )
             => ( P @ M2 ) )
         => ( P @ N ) )
     => ( P @ N3 ) ) ).

% enat_less_induct
thf(fact_5031_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N3 ) )
      = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
        & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N3 ) ) ) ).

% enat_0_less_mult_iff
thf(fact_5032_set__bit__greater__eq,axiom,
    ! [K: int,N3: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N3 @ K ) ) ).

% set_bit_greater_eq
thf(fact_5033_ceiling__mono,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ Y @ X )
     => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y ) @ ( archim7802044766580827645g_real @ X ) ) ) ).

% ceiling_mono
thf(fact_5034_ceiling__mono,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y ) @ ( archim2889992004027027881ng_rat @ X ) ) ) ).

% ceiling_mono
thf(fact_5035_ceiling__less__cancel,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) )
     => ( ord_less_rat @ X @ Y ) ) ).

% ceiling_less_cancel
thf(fact_5036_ceiling__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) )
     => ( ord_less_real @ X @ Y ) ) ).

% ceiling_less_cancel
thf(fact_5037_ceiling__add__le,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ Y ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ ( archim2889992004027027881ng_rat @ Y ) ) ) ).

% ceiling_add_le
thf(fact_5038_ceiling__add__le,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ ( archim7802044766580827645g_real @ Y ) ) ) ).

% ceiling_add_le
thf(fact_5039_mult__ceiling__le,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ B2 )
       => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A2 @ B2 ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A2 ) @ ( archim7802044766580827645g_real @ B2 ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_5040_mult__ceiling__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A2 @ B2 ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A2 ) @ ( archim2889992004027027881ng_rat @ B2 ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_5041_Abs__fnat__hom__add,axiom,
    ! [A2: nat,B2: nat] :
      ( ( plus_plus_rat @ ( semiri681578069525770553at_rat @ A2 ) @ ( semiri681578069525770553at_rat @ B2 ) )
      = ( semiri681578069525770553at_rat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ).

% Abs_fnat_hom_add
thf(fact_5042_Abs__fnat__hom__add,axiom,
    ! [A2: nat,B2: nat] :
      ( ( plus_plus_real @ ( semiri5074537144036343181t_real @ A2 ) @ ( semiri5074537144036343181t_real @ B2 ) )
      = ( semiri5074537144036343181t_real @ ( plus_plus_nat @ A2 @ B2 ) ) ) ).

% Abs_fnat_hom_add
thf(fact_5043_Abs__fnat__hom__add,axiom,
    ! [A2: nat,B2: nat] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) )
      = ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A2 @ B2 ) ) ) ).

% Abs_fnat_hom_add
thf(fact_5044_Abs__fnat__hom__add,axiom,
    ! [A2: nat,B2: nat] :
      ( ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ A2 ) @ ( semiri1316708129612266289at_nat @ B2 ) )
      = ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ).

% Abs_fnat_hom_add
thf(fact_5045_Abs__fnat__hom__add,axiom,
    ! [A2: nat,B2: nat] :
      ( ( plus_plus_complex @ ( semiri8010041392384452111omplex @ A2 ) @ ( semiri8010041392384452111omplex @ B2 ) )
      = ( semiri8010041392384452111omplex @ ( plus_plus_nat @ A2 @ B2 ) ) ) ).

% Abs_fnat_hom_add
thf(fact_5046_Abs__fnat__hom__add,axiom,
    ! [A2: nat,B2: nat] :
      ( ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ A2 ) @ ( semiri4939895301339042750nteger @ B2 ) )
      = ( semiri4939895301339042750nteger @ ( plus_plus_nat @ A2 @ B2 ) ) ) ).

% Abs_fnat_hom_add
thf(fact_5047_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,N3: nat,K: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ B2 @ N3 ) @ K )
     => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N3 @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
         => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N3 ) @ one_one_int ) ) ) ) ) ).

% ceiling_log_nat_eq_if
thf(fact_5048_ceiling__log2__div2,axiom,
    ! [N3: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
        = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).

% ceiling_log2_div2
thf(fact_5049_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N3 ) @ one_one_int ) )
          = ( ( ord_less_nat @ ( power_power_nat @ B2 @ N3 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
thf(fact_5050_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri681578069525770553at_rat @ K )
       != zero_zero_rat )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_5051_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri5074537144036343181t_real @ K )
       != zero_zero_real )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_5052_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri1314217659103216013at_int @ K )
       != zero_zero_int )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_5053_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri1316708129612266289at_nat @ K )
       != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_5054_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri8010041392384452111omplex @ K )
       != zero_zero_complex )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_5055_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri4939895301339042750nteger @ K )
       != zero_z3403309356797280102nteger )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_5056_log__ceil__idem,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
        = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ) ) ) ).

% log_ceil_idem
thf(fact_5057_succ__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N3: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_s_u_c_c @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).

% succ_bound_size_univ
thf(fact_5058_pred__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N3: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_p_r_e_d @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).

% pred_bound_size_univ
thf(fact_5059_insert__bound__size__univ,axiom,
    ! [T: vEBT_VEBT,N3: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_i_n_s_e_r_t @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).

% insert_bound_size_univ
thf(fact_5060_lemma__termdiff3,axiom,
    ! [H2: real,Z: real,K6: real,N3: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K6 )
       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K6 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N3 ) @ ( power_power_real @ Z @ N3 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K6 @ ( minus_minus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_5061_lemma__termdiff3,axiom,
    ! [H2: complex,Z: complex,K6: real,N3: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K6 )
       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K6 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N3 ) @ ( power_power_complex @ Z @ N3 ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N3 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K6 @ ( minus_minus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_5062_succ__bound__height,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_nat @ ( vEBT_T_s_u_c_c @ T @ X ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ).

% succ_bound_height
thf(fact_5063_pred__bound__height,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_nat @ ( vEBT_T_p_r_e_d @ T @ X ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ).

% pred_bound_height
thf(fact_5064_of__int__ceiling__cancel,axiom,
    ! [X: rat] :
      ( ( ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) )
        = X )
      = ( ? [N2: int] :
            ( X
            = ( ring_1_of_int_rat @ N2 ) ) ) ) ).

% of_int_ceiling_cancel
thf(fact_5065_of__int__ceiling__cancel,axiom,
    ! [X: real] :
      ( ( ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) )
        = X )
      = ( ? [N2: int] :
            ( X
            = ( ring_1_of_int_real @ N2 ) ) ) ) ).

% of_int_ceiling_cancel
thf(fact_5066_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_int @ Z )
        = zero_zero_int )
      = ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_5067_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_real @ Z )
        = zero_zero_real )
      = ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_5068_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_rat @ Z )
        = zero_zero_rat )
      = ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_5069_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_17405671764205052669omplex @ Z )
        = zero_zero_complex )
      = ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_5070_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_int
        = ( ring_1_of_int_int @ Z ) )
      = ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_5071_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_real
        = ( ring_1_of_int_real @ Z ) )
      = ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_5072_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_rat
        = ( ring_1_of_int_rat @ Z ) )
      = ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_5073_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_complex
        = ( ring_17405671764205052669omplex @ Z ) )
      = ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_5074_of__int__0,axiom,
    ( ( ring_1_of_int_int @ zero_zero_int )
    = zero_zero_int ) ).

% of_int_0
thf(fact_5075_of__int__0,axiom,
    ( ( ring_1_of_int_real @ zero_zero_int )
    = zero_zero_real ) ).

% of_int_0
thf(fact_5076_of__int__0,axiom,
    ( ( ring_1_of_int_rat @ zero_zero_int )
    = zero_zero_rat ) ).

% of_int_0
thf(fact_5077_of__int__0,axiom,
    ( ( ring_17405671764205052669omplex @ zero_zero_int )
    = zero_zero_complex ) ).

% of_int_0
thf(fact_5078_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N3: num] :
      ( ( ( ring_17405671764205052669omplex @ Z )
        = ( numera6690914467698888265omplex @ N3 ) )
      = ( Z
        = ( numeral_numeral_int @ N3 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_5079_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N3: num] :
      ( ( ( ring_1_of_int_real @ Z )
        = ( numeral_numeral_real @ N3 ) )
      = ( Z
        = ( numeral_numeral_int @ N3 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_5080_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N3: num] :
      ( ( ( ring_1_of_int_rat @ Z )
        = ( numeral_numeral_rat @ N3 ) )
      = ( Z
        = ( numeral_numeral_int @ N3 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_5081_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N3: num] :
      ( ( ( ring_1_of_int_int @ Z )
        = ( numeral_numeral_int @ N3 ) )
      = ( Z
        = ( numeral_numeral_int @ N3 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_5082_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
      = ( numera6690914467698888265omplex @ K ) ) ).

% of_int_numeral
thf(fact_5083_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_real @ K ) ) ).

% of_int_numeral
thf(fact_5084_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_rat @ K ) ) ).

% of_int_numeral
thf(fact_5085_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% of_int_numeral
thf(fact_5086_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% of_int_le_iff
thf(fact_5087_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% of_int_le_iff
thf(fact_5088_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% of_int_le_iff
thf(fact_5089_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ W @ Z ) ) ).

% of_int_less_iff
thf(fact_5090_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ W @ Z ) ) ).

% of_int_less_iff
thf(fact_5091_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ W @ Z ) ) ).

% of_int_less_iff
thf(fact_5092_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_add
thf(fact_5093_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_add
thf(fact_5094_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_add
thf(fact_5095_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_17405671764205052669omplex @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_complex @ ( ring_17405671764205052669omplex @ W ) @ ( ring_17405671764205052669omplex @ Z ) ) ) ).

% of_int_add
thf(fact_5096_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_17405671764205052669omplex @ ( times_times_int @ W @ Z ) )
      = ( times_times_complex @ ( ring_17405671764205052669omplex @ W ) @ ( ring_17405671764205052669omplex @ Z ) ) ) ).

% of_int_mult
thf(fact_5097_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
      = ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_mult
thf(fact_5098_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z ) )
      = ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_mult
thf(fact_5099_of__int__mult,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
      = ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_mult
thf(fact_5100_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_17405671764205052669omplex @ ( minus_minus_int @ W @ Z ) )
      = ( minus_minus_complex @ ( ring_17405671764205052669omplex @ W ) @ ( ring_17405671764205052669omplex @ Z ) ) ) ).

% of_int_diff
thf(fact_5101_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
      = ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_diff
thf(fact_5102_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_rat @ ( minus_minus_int @ W @ Z ) )
      = ( minus_minus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_diff
thf(fact_5103_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
      = ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_diff
thf(fact_5104_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W: nat] :
      ( ( ( ring_1_of_int_rat @ X )
        = ( power_power_rat @ ( ring_1_of_int_rat @ B2 ) @ W ) )
      = ( X
        = ( power_power_int @ B2 @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_5105_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W: nat] :
      ( ( ( ring_1_of_int_real @ X )
        = ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W ) )
      = ( X
        = ( power_power_int @ B2 @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_5106_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W: nat] :
      ( ( ( ring_1_of_int_int @ X )
        = ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W ) )
      = ( X
        = ( power_power_int @ B2 @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_5107_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W: nat] :
      ( ( ( ring_17405671764205052669omplex @ X )
        = ( power_power_complex @ ( ring_17405671764205052669omplex @ B2 ) @ W ) )
      = ( X
        = ( power_power_int @ B2 @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_5108_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W: nat] :
      ( ( ( ring_18347121197199848620nteger @ X )
        = ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B2 ) @ W ) )
      = ( X
        = ( power_power_int @ B2 @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_5109_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X: int] :
      ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B2 ) @ W )
        = ( ring_1_of_int_rat @ X ) )
      = ( ( power_power_int @ B2 @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_5110_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X: int] :
      ( ( ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W )
        = ( ring_1_of_int_real @ X ) )
      = ( ( power_power_int @ B2 @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_5111_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X: int] :
      ( ( ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W )
        = ( ring_1_of_int_int @ X ) )
      = ( ( power_power_int @ B2 @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_5112_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X: int] :
      ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B2 ) @ W )
        = ( ring_17405671764205052669omplex @ X ) )
      = ( ( power_power_int @ B2 @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_5113_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X: int] :
      ( ( ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B2 ) @ W )
        = ( ring_18347121197199848620nteger @ X ) )
      = ( ( power_power_int @ B2 @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_5114_of__int__power,axiom,
    ! [Z: int,N3: nat] :
      ( ( ring_1_of_int_rat @ ( power_power_int @ Z @ N3 ) )
      = ( power_power_rat @ ( ring_1_of_int_rat @ Z ) @ N3 ) ) ).

% of_int_power
thf(fact_5115_of__int__power,axiom,
    ! [Z: int,N3: nat] :
      ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N3 ) )
      = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N3 ) ) ).

% of_int_power
thf(fact_5116_of__int__power,axiom,
    ! [Z: int,N3: nat] :
      ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N3 ) )
      = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N3 ) ) ).

% of_int_power
thf(fact_5117_of__int__power,axiom,
    ! [Z: int,N3: nat] :
      ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N3 ) )
      = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N3 ) ) ).

% of_int_power
thf(fact_5118_of__int__power,axiom,
    ! [Z: int,N3: nat] :
      ( ( ring_18347121197199848620nteger @ ( power_power_int @ Z @ N3 ) )
      = ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ Z ) @ N3 ) ) ).

% of_int_power
thf(fact_5119_ceiling__add__of__int,axiom,
    ! [X: rat,Z: int] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z ) ) ).

% ceiling_add_of_int
thf(fact_5120_ceiling__add__of__int,axiom,
    ! [X: real,Z: int] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X ) @ Z ) ) ).

% ceiling_add_of_int
thf(fact_5121_ceiling__diff__of__int,axiom,
    ! [X: rat,Z: int] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ Z ) ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X ) @ Z ) ) ).

% ceiling_diff_of_int
thf(fact_5122_ceiling__diff__of__int,axiom,
    ! [X: real,Z: int] :
      ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ Z ) ) )
      = ( minus_minus_int @ ( archim7802044766580827645g_real @ X ) @ Z ) ) ).

% ceiling_diff_of_int
thf(fact_5123_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_5124_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_5125_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_5126_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_5127_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_5128_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_5129_of__int__le__numeral__iff,axiom,
    ! [Z: int,N3: num] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N3 ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N3 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_5130_of__int__le__numeral__iff,axiom,
    ! [Z: int,N3: num] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N3 ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N3 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_5131_of__int__le__numeral__iff,axiom,
    ! [Z: int,N3: num] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N3 ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N3 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_5132_of__int__numeral__le__iff,axiom,
    ! [N3: num,Z: int] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N3 ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N3 ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_5133_of__int__numeral__le__iff,axiom,
    ! [N3: num,Z: int] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N3 ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N3 ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_5134_of__int__numeral__le__iff,axiom,
    ! [N3: num,Z: int] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N3 ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N3 ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_5135_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_5136_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_5137_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_5138_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_5139_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_5140_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_5141_of__int__numeral__less__iff,axiom,
    ! [N3: num,Z: int] :
      ( ( ord_less_real @ ( numeral_numeral_real @ N3 ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N3 ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_5142_of__int__numeral__less__iff,axiom,
    ! [N3: num,Z: int] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ N3 ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N3 ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_5143_of__int__numeral__less__iff,axiom,
    ! [N3: num,Z: int] :
      ( ( ord_less_int @ ( numeral_numeral_int @ N3 ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N3 ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_5144_of__int__less__numeral__iff,axiom,
    ! [Z: int,N3: num] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N3 ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N3 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_5145_of__int__less__numeral__iff,axiom,
    ! [Z: int,N3: num] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N3 ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N3 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_5146_of__int__less__numeral__iff,axiom,
    ! [Z: int,N3: num] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N3 ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N3 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_5147_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_5148_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_5149_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_5150_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_5151_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_5152_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_5153_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_5154_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_5155_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_5156_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_5157_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_5158_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_5159_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N3: nat] :
      ( ( ( ring_18347121197199848620nteger @ Y )
        = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N3 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_5160_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N3: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N3 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_5161_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N3: nat] :
      ( ( ( ring_1_of_int_real @ Y )
        = ( power_power_real @ ( numeral_numeral_real @ X ) @ N3 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_5162_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N3: nat] :
      ( ( ( ring_1_of_int_rat @ Y )
        = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N3 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_5163_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N3: nat] :
      ( ( ( ring_1_of_int_int @ Y )
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_5164_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: int] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N3 )
        = ( ring_18347121197199848620nteger @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_5165_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: int] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N3 )
        = ( ring_17405671764205052669omplex @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_5166_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: int] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N3 )
        = ( ring_1_of_int_real @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_5167_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: int] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N3 )
        = ( ring_1_of_int_rat @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_5168_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: int] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 )
        = ( ring_1_of_int_int @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_5169_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B2 @ W ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_5170_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B2 ) @ W ) @ ( ring_18347121197199848620nteger @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B2 @ W ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_5171_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B2 ) @ W ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B2 @ W ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_5172_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B2 @ W ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_5173_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B2 @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_5174_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ X ) @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B2 ) @ W ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B2 @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_5175_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B2 ) @ W ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B2 @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_5176_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B2 @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_5177_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ X ) @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B2 ) @ W ) )
      = ( ord_less_int @ X @ ( power_power_int @ B2 @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_5178_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W ) )
      = ( ord_less_int @ X @ ( power_power_int @ B2 @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_5179_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B2 ) @ W ) )
      = ( ord_less_int @ X @ ( power_power_int @ B2 @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_5180_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B2: int,W: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W ) )
      = ( ord_less_int @ X @ ( power_power_int @ B2 @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_5181_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B2 ) @ W ) @ ( ring_18347121197199848620nteger @ X ) )
      = ( ord_less_int @ ( power_power_int @ B2 @ W ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_5182_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X: int] :
      ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B2 ) @ W ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_int @ ( power_power_int @ B2 @ W ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_5183_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B2 ) @ W ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_int @ ( power_power_int @ B2 @ W ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_5184_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,X: int] :
      ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B2 ) @ W ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_int @ ( power_power_int @ B2 @ W ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_5185_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N3 ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_5186_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N3: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A2 ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N3 ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_5187_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N3: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N3 ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_5188_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N3: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A2 ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_5189_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A2: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N3 ) @ ( ring_18347121197199848620nteger @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) @ A2 ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_5190_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A2: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N3 ) @ ( ring_1_of_int_real @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) @ A2 ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_5191_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A2: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N3 ) @ ( ring_1_of_int_rat @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) @ A2 ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_5192_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) @ ( ring_1_of_int_int @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) @ A2 ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_5193_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N3 ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_5194_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N3: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A2 ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N3 ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_5195_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N3: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N3 ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_5196_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N3: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A2 ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_5197_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A2: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X ) @ N3 ) @ ( ring_18347121197199848620nteger @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) @ A2 ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_5198_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A2: int] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N3 ) @ ( ring_1_of_int_real @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) @ A2 ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_5199_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A2: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N3 ) @ ( ring_1_of_int_rat @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) @ A2 ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_5200_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A2: int] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) @ ( ring_1_of_int_int @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) @ A2 ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_5201_ex__le__of__int,axiom,
    ! [X: real] :
    ? [Z2: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ).

% ex_le_of_int
thf(fact_5202_ex__le__of__int,axiom,
    ! [X: rat] :
    ? [Z2: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ).

% ex_le_of_int
thf(fact_5203_ex__less__of__int,axiom,
    ! [X: real] :
    ? [Z2: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z2 ) ) ).

% ex_less_of_int
thf(fact_5204_ex__less__of__int,axiom,
    ! [X: rat] :
    ? [Z2: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z2 ) ) ).

% ex_less_of_int
thf(fact_5205_ex__of__int__less,axiom,
    ! [X: real] :
    ? [Z2: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ X ) ).

% ex_of_int_less
thf(fact_5206_ex__of__int__less,axiom,
    ! [X: rat] :
    ? [Z2: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ X ) ).

% ex_of_int_less
thf(fact_5207_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_5208_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_5209_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_5210_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_5211_le__of__int__ceiling,axiom,
    ! [X: real] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ).

% le_of_int_ceiling
thf(fact_5212_le__of__int__ceiling,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ).

% le_of_int_ceiling
thf(fact_5213_ceiling__le,axiom,
    ! [X: real,A2: int] :
      ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A2 ) )
     => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ A2 ) ) ).

% ceiling_le
thf(fact_5214_ceiling__le,axiom,
    ! [X: rat,A2: int] :
      ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A2 ) )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ A2 ) ) ).

% ceiling_le
thf(fact_5215_ceiling__le__iff,axiom,
    ! [X: real,Z: int] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ Z )
      = ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) ) ) ).

% ceiling_le_iff
thf(fact_5216_ceiling__le__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
      = ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) ) ) ).

% ceiling_le_iff
thf(fact_5217_less__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ X ) ) ).

% less_ceiling_iff
thf(fact_5218_less__ceiling__iff,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_int @ Z @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X ) ) ).

% less_ceiling_iff
thf(fact_5219_real__of__int__div4,axiom,
    ! [N3: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N3 @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N3 ) @ ( ring_1_of_int_real @ X ) ) ) ).

% real_of_int_div4
thf(fact_5220_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_nonneg
thf(fact_5221_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_nonneg
thf(fact_5222_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_nonneg
thf(fact_5223_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_pos
thf(fact_5224_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_pos
thf(fact_5225_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_pos
thf(fact_5226_floor__exists,axiom,
    ! [X: real] :
    ? [Z2: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_5227_floor__exists,axiom,
    ! [X: rat] :
    ? [Z2: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_5228_floor__exists1,axiom,
    ! [X: real] :
    ? [X4: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X4 ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X4 @ one_one_int ) ) )
      & ! [Y4: int] :
          ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y4 ) @ X )
            & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
         => ( Y4 = X4 ) ) ) ).

% floor_exists1
thf(fact_5229_floor__exists1,axiom,
    ! [X: rat] :
    ? [X4: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X4 ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X4 @ one_one_int ) ) )
      & ! [Y4: int] :
          ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y4 ) @ X )
            & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
         => ( Y4 = X4 ) ) ) ).

% floor_exists1
thf(fact_5230_of__int__ceiling__le__add__one,axiom,
    ! [R3: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R3 ) ) @ ( plus_plus_real @ R3 @ one_one_real ) ) ).

% of_int_ceiling_le_add_one
thf(fact_5231_of__int__ceiling__le__add__one,axiom,
    ! [R3: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R3 ) ) @ ( plus_plus_rat @ R3 @ one_one_rat ) ) ).

% of_int_ceiling_le_add_one
thf(fact_5232_of__int__ceiling__diff__one__le,axiom,
    ! [R3: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R3 ) ) @ one_one_real ) @ R3 ) ).

% of_int_ceiling_diff_one_le
thf(fact_5233_of__int__ceiling__diff__one__le,axiom,
    ! [R3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R3 ) ) @ one_one_rat ) @ R3 ) ).

% of_int_ceiling_diff_one_le
thf(fact_5234_of__nat__less__of__int__iff,axiom,
    ! [N3: nat,X: int] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N3 ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N3 ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_5235_of__nat__less__of__int__iff,axiom,
    ! [N3: nat,X: int] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N3 ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_5236_of__nat__less__of__int__iff,axiom,
    ! [N3: nat,X: int] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N3 ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N3 ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_5237_of__nat__less__of__int__iff,axiom,
    ! [N3: nat,X: int] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ N3 ) @ ( ring_18347121197199848620nteger @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N3 ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_5238_int__le__real__less,axiom,
    ( ord_less_eq_int
    = ( ^ [N2: int,M5: int] : ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M5 ) @ one_one_real ) ) ) ) ).

% int_le_real_less
thf(fact_5239_int__less__real__le,axiom,
    ( ord_less_int
    = ( ^ [N2: int,M5: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) @ ( ring_1_of_int_real @ M5 ) ) ) ) ).

% int_less_real_le
thf(fact_5240_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S2 ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
thf(fact_5241_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv2: $o] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ Uu @ Uv2 ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
thf(fact_5242_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw: $o,N3: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uv2 @ Uw ) @ ( suc @ N3 ) )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
thf(fact_5243_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I4_J,axiom,
    ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va: vEBT_VEBT,Vb2: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va ) @ Vb2 )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(4)
thf(fact_5244_ceiling__split,axiom,
    ! [P: int > $o,T: real] :
      ( ( P @ ( archim7802044766580827645g_real @ T ) )
      = ( ! [I2: int] :
            ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I2 ) @ one_one_real ) @ T )
              & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I2 ) ) )
           => ( P @ I2 ) ) ) ) ).

% ceiling_split
thf(fact_5245_ceiling__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
      = ( ! [I2: int] :
            ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I2 ) @ one_one_rat ) @ T )
              & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I2 ) ) )
           => ( P @ I2 ) ) ) ) ).

% ceiling_split
thf(fact_5246_ceiling__eq__iff,axiom,
    ! [X: real,A2: int] :
      ( ( ( archim7802044766580827645g_real @ X )
        = A2 )
      = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A2 ) @ one_one_real ) @ X )
        & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ A2 ) ) ) ) ).

% ceiling_eq_iff
thf(fact_5247_ceiling__eq__iff,axiom,
    ! [X: rat,A2: int] :
      ( ( ( archim2889992004027027881ng_rat @ X )
        = A2 )
      = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A2 ) @ one_one_rat ) @ X )
        & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ A2 ) ) ) ) ).

% ceiling_eq_iff
thf(fact_5248_ceiling__unique,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z ) )
       => ( ( archim7802044766580827645g_real @ X )
          = Z ) ) ) ).

% ceiling_unique
thf(fact_5249_ceiling__unique,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X )
     => ( ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z ) )
       => ( ( archim2889992004027027881ng_rat @ X )
          = Z ) ) ) ).

% ceiling_unique
thf(fact_5250_ceiling__correct,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) @ one_one_real ) @ X )
      & ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X ) ) ) ) ).

% ceiling_correct
thf(fact_5251_ceiling__correct,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) @ one_one_rat ) @ X )
      & ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X ) ) ) ) ).

% ceiling_correct
thf(fact_5252_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I3_J,axiom,
    ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
thf(fact_5253_ceiling__less__iff,axiom,
    ! [X: real,Z: int] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ Z )
      = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).

% ceiling_less_iff
thf(fact_5254_ceiling__less__iff,axiom,
    ! [X: rat,Z: int] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ Z )
      = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).

% ceiling_less_iff
thf(fact_5255_le__ceiling__iff,axiom,
    ! [Z: int,X: rat] :
      ( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X ) ) ).

% le_ceiling_iff
thf(fact_5256_le__ceiling__iff,axiom,
    ! [Z: int,X: real] :
      ( ( ord_less_eq_int @ Z @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X ) ) ).

% le_ceiling_iff
thf(fact_5257_real__of__int__div2,axiom,
    ! [N3: int,X: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N3 ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N3 @ X ) ) ) ) ).

% real_of_int_div2
thf(fact_5258_real__of__int__div3,axiom,
    ! [N3: int,X: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N3 ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N3 @ X ) ) ) @ one_one_real ) ).

% real_of_int_div3
thf(fact_5259_ceiling__divide__upper,axiom,
    ! [Q3: real,P6: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_eq_real @ P6 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P6 @ Q3 ) ) ) @ Q3 ) ) ) ).

% ceiling_divide_upper
thf(fact_5260_ceiling__divide__upper,axiom,
    ! [Q3: rat,P6: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q3 )
     => ( ord_less_eq_rat @ P6 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P6 @ Q3 ) ) ) @ Q3 ) ) ) ).

% ceiling_divide_upper
thf(fact_5261_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
thf(fact_5262_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Va: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B2 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(3)
thf(fact_5263_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
      = ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( X = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(1)
thf(fact_5264_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Vf2 )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
thf(fact_5265_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
    ! [Uu: $o,B2: $o] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uu @ B2 ) @ zero_zero_nat )
      = ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(1)
thf(fact_5266_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve2 )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
thf(fact_5267_ceiling__divide__lower,axiom,
    ! [Q3: real,P6: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P6 @ Q3 ) ) ) @ one_one_real ) @ Q3 ) @ P6 ) ) ).

% ceiling_divide_lower
thf(fact_5268_ceiling__divide__lower,axiom,
    ! [Q3: rat,P6: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q3 )
     => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P6 @ Q3 ) ) ) @ one_one_rat ) @ Q3 ) @ P6 ) ) ).

% ceiling_divide_lower
thf(fact_5269_ceiling__eq,axiom,
    ! [N3: int,X: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N3 ) @ X )
     => ( ( ord_less_eq_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N3 ) @ one_one_real ) )
       => ( ( archim7802044766580827645g_real @ X )
          = ( plus_plus_int @ N3 @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_5270_ceiling__eq,axiom,
    ! [N3: int,X: rat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ N3 ) @ X )
     => ( ( ord_less_eq_rat @ X @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N3 ) @ one_one_rat ) )
       => ( ( archim2889992004027027881ng_rat @ X )
          = ( plus_plus_int @ N3 @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_5271_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [A2: $o,Uw: $o] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ A2 @ Uw ) @ ( suc @ zero_zero_nat ) )
      = ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(2)
thf(fact_5272_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
thf(fact_5273_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
thf(fact_5274_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
thf(fact_5275_insersimp,axiom,
    ! [T: vEBT_VEBT,N3: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T @ Y ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ) ).

% insersimp
thf(fact_5276_insertsimp,axiom,
    ! [T: vEBT_VEBT,N3: nat,L2: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_VEBT_minNull @ T )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T @ L2 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ) ).

% insertsimp
thf(fact_5277_insert__bound__height,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T @ X ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% insert_bound_height
thf(fact_5278_norm__divide__numeral,axiom,
    ! [A2: real,W: num] :
      ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A2 @ ( numeral_numeral_real @ W ) ) )
      = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_divide_numeral
thf(fact_5279_norm__divide__numeral,axiom,
    ! [A2: complex,W: num] :
      ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A2 @ ( numera6690914467698888265omplex @ W ) ) )
      = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A2 ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_divide_numeral
thf(fact_5280_norm__mult__numeral2,axiom,
    ! [A2: real,W: num] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ A2 @ ( numeral_numeral_real @ W ) ) )
      = ( times_times_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_mult_numeral2
thf(fact_5281_norm__mult__numeral2,axiom,
    ! [A2: complex,W: num] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A2 @ ( numera6690914467698888265omplex @ W ) ) )
      = ( times_times_real @ ( real_V1022390504157884413omplex @ A2 ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_mult_numeral2
thf(fact_5282_norm__mult__numeral1,axiom,
    ! [W: num,A2: real] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A2 ) )
      = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A2 ) ) ) ).

% norm_mult_numeral1
thf(fact_5283_norm__mult__numeral1,axiom,
    ! [W: num,A2: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A2 ) )
      = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A2 ) ) ) ).

% norm_mult_numeral1
thf(fact_5284_norm__le__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% norm_le_zero_iff
thf(fact_5285_norm__le__zero__iff,axiom,
    ! [X: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
      = ( X = zero_zero_complex ) ) ).

% norm_le_zero_iff
thf(fact_5286_zero__less__norm__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
      = ( X != zero_zero_real ) ) ).

% zero_less_norm_iff
thf(fact_5287_zero__less__norm__iff,axiom,
    ! [X: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
      = ( X != zero_zero_complex ) ) ).

% zero_less_norm_iff
thf(fact_5288_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_5289_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_5290_norm__eq__zero,axiom,
    ! [X: real] :
      ( ( ( real_V7735802525324610683m_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% norm_eq_zero
thf(fact_5291_norm__eq__zero,axiom,
    ! [X: complex] :
      ( ( ( real_V1022390504157884413omplex @ X )
        = zero_zero_real )
      = ( X = zero_zero_complex ) ) ).

% norm_eq_zero
thf(fact_5292_norm__zero,axiom,
    ( ( real_V7735802525324610683m_real @ zero_zero_real )
    = zero_zero_real ) ).

% norm_zero
thf(fact_5293_norm__zero,axiom,
    ( ( real_V1022390504157884413omplex @ zero_zero_complex )
    = zero_zero_real ) ).

% norm_zero
thf(fact_5294_norm__minus__commute,axiom,
    ! [A2: real,B2: real] :
      ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A2 @ B2 ) )
      = ( real_V7735802525324610683m_real @ ( minus_minus_real @ B2 @ A2 ) ) ) ).

% norm_minus_commute
thf(fact_5295_norm__minus__commute,axiom,
    ! [A2: complex,B2: complex] :
      ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A2 @ B2 ) )
      = ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B2 @ A2 ) ) ) ).

% norm_minus_commute
thf(fact_5296_norm__mult,axiom,
    ! [X: real,Y: real] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) )
      = ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).

% norm_mult
thf(fact_5297_norm__mult,axiom,
    ! [X: complex,Y: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) )
      = ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).

% norm_mult
thf(fact_5298_norm__not__less__zero,axiom,
    ! [X: complex] :
      ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real ) ).

% norm_not_less_zero
thf(fact_5299_norm__ge__zero,axiom,
    ! [X: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) ) ).

% norm_ge_zero
thf(fact_5300_norm__divide,axiom,
    ! [A2: real,B2: real] :
      ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A2 @ B2 ) )
      = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) ).

% norm_divide
thf(fact_5301_norm__divide,axiom,
    ! [A2: complex,B2: complex] :
      ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A2 @ B2 ) )
      = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A2 ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) ).

% norm_divide
thf(fact_5302_norm__power,axiom,
    ! [X: real,N3: nat] :
      ( ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N3 ) )
      = ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N3 ) ) ).

% norm_power
thf(fact_5303_norm__power,axiom,
    ! [X: complex,N3: nat] :
      ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N3 ) )
      = ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N3 ) ) ).

% norm_power
thf(fact_5304_power__eq__imp__eq__norm,axiom,
    ! [W: real,N3: nat,Z: real] :
      ( ( ( power_power_real @ W @ N3 )
        = ( power_power_real @ Z @ N3 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( real_V7735802525324610683m_real @ W )
          = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_5305_power__eq__imp__eq__norm,axiom,
    ! [W: complex,N3: nat,Z: complex] :
      ( ( ( power_power_complex @ W @ N3 )
        = ( power_power_complex @ Z @ N3 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( real_V1022390504157884413omplex @ W )
          = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_5306_nonzero__norm__divide,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A2 @ B2 ) )
        = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) ) ).

% nonzero_norm_divide
thf(fact_5307_nonzero__norm__divide,axiom,
    ! [B2: complex,A2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A2 @ B2 ) )
        = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A2 ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) ) ).

% nonzero_norm_divide
thf(fact_5308_norm__mult__less,axiom,
    ! [X: real,R3: real,Y: real,S2: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R3 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S2 )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ R3 @ S2 ) ) ) ) ).

% norm_mult_less
thf(fact_5309_norm__mult__less,axiom,
    ! [X: complex,R3: real,Y: complex,S2: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R3 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S2 )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ R3 @ S2 ) ) ) ) ).

% norm_mult_less
thf(fact_5310_norm__mult__ineq,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).

% norm_mult_ineq
thf(fact_5311_norm__mult__ineq,axiom,
    ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).

% norm_mult_ineq
thf(fact_5312_norm__triangle__lt,axiom,
    ! [X: real,Y: real,E: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
     => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E ) ) ).

% norm_triangle_lt
thf(fact_5313_norm__triangle__lt,axiom,
    ! [X: complex,Y: complex,E: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
     => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E ) ) ).

% norm_triangle_lt
thf(fact_5314_norm__add__less,axiom,
    ! [X: real,R3: real,Y: real,S2: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R3 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S2 )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ R3 @ S2 ) ) ) ) ).

% norm_add_less
thf(fact_5315_norm__add__less,axiom,
    ! [X: complex,R3: real,Y: complex,S2: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R3 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S2 )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ R3 @ S2 ) ) ) ) ).

% norm_add_less
thf(fact_5316_norm__add__leD,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A2 @ B2 ) ) @ C2 )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B2 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A2 ) @ C2 ) ) ) ).

% norm_add_leD
thf(fact_5317_norm__add__leD,axiom,
    ! [A2: complex,B2: complex,C2: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A2 @ B2 ) ) @ C2 )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B2 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A2 ) @ C2 ) ) ) ).

% norm_add_leD
thf(fact_5318_norm__triangle__le,axiom,
    ! [X: real,Y: real,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E ) ) ).

% norm_triangle_le
thf(fact_5319_norm__triangle__le,axiom,
    ! [X: complex,Y: complex,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E ) ) ).

% norm_triangle_le
thf(fact_5320_norm__triangle__ineq,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).

% norm_triangle_ineq
thf(fact_5321_norm__triangle__ineq,axiom,
    ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).

% norm_triangle_ineq
thf(fact_5322_norm__triangle__mono,axiom,
    ! [A2: real,R3: real,B2: real,S2: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A2 ) @ R3 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B2 ) @ S2 )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A2 @ B2 ) ) @ ( plus_plus_real @ R3 @ S2 ) ) ) ) ).

% norm_triangle_mono
thf(fact_5323_norm__triangle__mono,axiom,
    ! [A2: complex,R3: real,B2: complex,S2: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A2 ) @ R3 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B2 ) @ S2 )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A2 @ B2 ) ) @ ( plus_plus_real @ R3 @ S2 ) ) ) ) ).

% norm_triangle_mono
thf(fact_5324_norm__diff__triangle__less,axiom,
    ! [X: real,Y: real,E1: real,Z: real,E22: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_5325_norm__diff__triangle__less,axiom,
    ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_5326_norm__power__ineq,axiom,
    ! [X: real,N3: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N3 ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N3 ) ) ).

% norm_power_ineq
thf(fact_5327_norm__power__ineq,axiom,
    ! [X: complex,N3: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N3 ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N3 ) ) ).

% norm_power_ineq
thf(fact_5328_norm__triangle__sub,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) ) ) ).

% norm_triangle_sub
thf(fact_5329_norm__triangle__sub,axiom,
    ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) ) ) ).

% norm_triangle_sub
thf(fact_5330_norm__triangle__ineq4,axiom,
    ! [A2: real,B2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A2 @ B2 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) ).

% norm_triangle_ineq4
thf(fact_5331_norm__triangle__ineq4,axiom,
    ! [A2: complex,B2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A2 @ B2 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A2 ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) ).

% norm_triangle_ineq4
thf(fact_5332_norm__diff__triangle__le,axiom,
    ! [X: real,Y: real,E1: real,Z: real,E22: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_5333_norm__diff__triangle__le,axiom,
    ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_5334_norm__triangle__le__diff,axiom,
    ! [X: real,Y: real,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E ) ) ).

% norm_triangle_le_diff
thf(fact_5335_norm__triangle__le__diff,axiom,
    ! [X: complex,Y: complex,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E ) ) ).

% norm_triangle_le_diff
thf(fact_5336_norm__diff__ineq,axiom,
    ! [A2: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( real_V7735802525324610683m_real @ B2 ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A2 @ B2 ) ) ) ).

% norm_diff_ineq
thf(fact_5337_norm__diff__ineq,axiom,
    ! [A2: complex,B2: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A2 ) @ ( real_V1022390504157884413omplex @ B2 ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A2 @ B2 ) ) ) ).

% norm_diff_ineq
thf(fact_5338_norm__triangle__ineq2,axiom,
    ! [A2: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( real_V7735802525324610683m_real @ B2 ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A2 @ B2 ) ) ) ).

% norm_triangle_ineq2
thf(fact_5339_norm__triangle__ineq2,axiom,
    ! [A2: complex,B2: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A2 ) @ ( real_V1022390504157884413omplex @ B2 ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A2 @ B2 ) ) ) ).

% norm_triangle_ineq2
thf(fact_5340_power__eq__1__iff,axiom,
    ! [W: real,N3: nat] :
      ( ( ( power_power_real @ W @ N3 )
        = one_one_real )
     => ( ( ( real_V7735802525324610683m_real @ W )
          = one_one_real )
        | ( N3 = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_5341_power__eq__1__iff,axiom,
    ! [W: complex,N3: nat] :
      ( ( ( power_power_complex @ W @ N3 )
        = one_one_complex )
     => ( ( ( real_V1022390504157884413omplex @ W )
          = one_one_real )
        | ( N3 = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_5342_norm__diff__triangle__ineq,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A2 @ B2 ) @ ( plus_plus_real @ C2 @ D2 ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A2 @ C2 ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B2 @ D2 ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_5343_norm__diff__triangle__ineq,axiom,
    ! [A2: complex,B2: complex,C2: complex,D2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A2 @ B2 ) @ ( plus_plus_complex @ C2 @ D2 ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A2 @ C2 ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B2 @ D2 ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_5344_square__norm__one,axiom,
    ! [X: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
     => ( ( real_V7735802525324610683m_real @ X )
        = one_one_real ) ) ).

% square_norm_one
thf(fact_5345_square__norm__one,axiom,
    ! [X: complex] :
      ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_complex )
     => ( ( real_V1022390504157884413omplex @ X )
        = one_one_real ) ) ).

% square_norm_one
thf(fact_5346_norm__power__diff,axiom,
    ! [Z: real,W: real,M: nat] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( power_power_real @ Z @ M ) @ ( power_power_real @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z @ W ) ) ) ) ) ) ).

% norm_power_diff
thf(fact_5347_norm__power__diff,axiom,
    ! [Z: complex,W: complex,M: nat] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W ) @ one_one_real )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( power_power_complex @ Z @ M ) @ ( power_power_complex @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z @ W ) ) ) ) ) ) ).

% norm_power_diff
thf(fact_5348_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_5349_del__x__mi,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L2: 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 ) ) ) )
                  = L2 )
               => ( ( 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 ) @ L2 ) )
                      @ ( 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 ) @ L2 ) ) @ ( 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 ) @ L2 ) )
                        @ ( 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 ) @ L2 ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
thf(fact_5350_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,L2: 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 ) ) ) )
                  = L2 )
               => ( ( 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 ) @ L2 ) )
                   => ( ( 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_5351_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,L2: 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 ) ) ) )
                  = L2 )
               => ( ( 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 ) @ L2 ) )
                   => ( ( 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_5352_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_5353_del__x__not__mi,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L2: 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 ) ) ) )
                = L2 )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
               => ( ( 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_5354_set__vebt_H__def,axiom,
    ( vEBT_VEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).

% set_vebt'_def
thf(fact_5355_succ__empty,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y2: nat] :
                ( ( vEBT_vebt_member @ T @ Y2 )
                & ( ord_less_nat @ X @ Y2 ) ) )
          = bot_bot_set_nat ) ) ) ).

% succ_empty
thf(fact_5356_pred__empty,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y2: nat] :
                ( ( vEBT_vebt_member @ T @ Y2 )
                & ( ord_less_nat @ Y2 @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% pred_empty
thf(fact_5357_singleton__conv2,axiom,
    ! [A2: vEBT_VEBT] :
      ( ( collect_VEBT_VEBT
        @ ( ^ [Y5: vEBT_VEBT,Z3: vEBT_VEBT] : Y5 = Z3
          @ A2 ) )
      = ( insert_VEBT_VEBT @ A2 @ bot_bo8194388402131092736T_VEBT ) ) ).

% singleton_conv2
thf(fact_5358_singleton__conv2,axiom,
    ! [A2: complex] :
      ( ( collect_complex
        @ ( ^ [Y5: complex,Z3: complex] : Y5 = Z3
          @ A2 ) )
      = ( insert_complex @ A2 @ bot_bot_set_complex ) ) ).

% singleton_conv2
thf(fact_5359_singleton__conv2,axiom,
    ! [A2: product_prod_int_int] :
      ( ( collec213857154873943460nt_int
        @ ( ^ [Y5: product_prod_int_int,Z3: product_prod_int_int] : Y5 = Z3
          @ A2 ) )
      = ( insert5033312907999012233nt_int @ A2 @ bot_bo1796632182523588997nt_int ) ) ).

% singleton_conv2
thf(fact_5360_singleton__conv2,axiom,
    ! [A2: real] :
      ( ( collect_real
        @ ( ^ [Y5: real,Z3: real] : Y5 = Z3
          @ A2 ) )
      = ( insert_real @ A2 @ bot_bot_set_real ) ) ).

% singleton_conv2
thf(fact_5361_singleton__conv2,axiom,
    ! [A2: $o] :
      ( ( collect_o
        @ ( ^ [Y5: $o,Z3: $o] : Y5 = Z3
          @ A2 ) )
      = ( insert_o @ A2 @ bot_bot_set_o ) ) ).

% singleton_conv2
thf(fact_5362_singleton__conv2,axiom,
    ! [A2: nat] :
      ( ( collect_nat
        @ ( ^ [Y5: nat,Z3: nat] : Y5 = Z3
          @ A2 ) )
      = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_5363_singleton__conv2,axiom,
    ! [A2: int] :
      ( ( collect_int
        @ ( ^ [Y5: int,Z3: int] : Y5 = Z3
          @ A2 ) )
      = ( insert_int @ A2 @ bot_bot_set_int ) ) ).

% singleton_conv2
thf(fact_5364_singleton__conv,axiom,
    ! [A2: vEBT_VEBT] :
      ( ( collect_VEBT_VEBT
        @ ^ [X3: vEBT_VEBT] : X3 = A2 )
      = ( insert_VEBT_VEBT @ A2 @ bot_bo8194388402131092736T_VEBT ) ) ).

% singleton_conv
thf(fact_5365_singleton__conv,axiom,
    ! [A2: complex] :
      ( ( collect_complex
        @ ^ [X3: complex] : X3 = A2 )
      = ( insert_complex @ A2 @ bot_bot_set_complex ) ) ).

% singleton_conv
thf(fact_5366_singleton__conv,axiom,
    ! [A2: product_prod_int_int] :
      ( ( collec213857154873943460nt_int
        @ ^ [X3: product_prod_int_int] : X3 = A2 )
      = ( insert5033312907999012233nt_int @ A2 @ bot_bo1796632182523588997nt_int ) ) ).

% singleton_conv
thf(fact_5367_singleton__conv,axiom,
    ! [A2: real] :
      ( ( collect_real
        @ ^ [X3: real] : X3 = A2 )
      = ( insert_real @ A2 @ bot_bot_set_real ) ) ).

% singleton_conv
thf(fact_5368_singleton__conv,axiom,
    ! [A2: $o] :
      ( ( collect_o
        @ ^ [X3: $o] : X3 = A2 )
      = ( insert_o @ A2 @ bot_bot_set_o ) ) ).

% singleton_conv
thf(fact_5369_singleton__conv,axiom,
    ! [A2: nat] :
      ( ( collect_nat
        @ ^ [X3: nat] : X3 = A2 )
      = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_5370_singleton__conv,axiom,
    ! [A2: int] :
      ( ( collect_int
        @ ^ [X3: int] : X3 = A2 )
      = ( insert_int @ A2 @ bot_bot_set_int ) ) ).

% singleton_conv
thf(fact_5371_del__x__not__mia,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L2: 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 ) ) ) )
                = L2 )
             => ( ( 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 ) @ L2 ) )
                    @ ( 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 ) @ L2 ) ) @ ( 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 ) @ L2 ) )
                      @ ( 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 ) @ L2 ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) ) @ Summary ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
thf(fact_5372_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L2: 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 ) ) ) )
                = L2 )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H2 ) @ L2 ) )
               => ( ( 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_5373_vebt__memberi_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] :
      ( hoare_hoare_triple_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_V854960066525838166emberi @ T @ Ti @ X )
      @ ^ [R5: $o] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_vebt_member @ T @ X ) ) ) ) ) ).

% vebt_memberi'_rf_abstr
thf(fact_5374_vebt__inserti_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( hoare_1429296392585015714_VEBTi @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_V3964819847710782039nserti @ T @ Ti @ X ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T @ X ) ) ) ).

% vebt_inserti'_rf_abstr
thf(fact_5375_vebt__succi_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,N3: nat,Ti: vEBT_VEBTi,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_VEBT_vebt_succi @ T @ Ti @ X )
        @ ^ [R5: option_nat] :
            ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
            @ ( pure_assn
              @ ( R5
                = ( vEBT_vebt_succ @ T @ X ) ) ) ) ) ) ).

% vebt_succi'_rf_abstr
thf(fact_5376_vebt__pred_H__rf__abstr,axiom,
    ! [T: vEBT_VEBT,N3: nat,Ti: vEBT_VEBTi,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_VEBT_vebt_predi @ T @ Ti @ X )
        @ ^ [R5: option_nat] :
            ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
            @ ( pure_assn
              @ ( R5
                = ( vEBT_vebt_pred @ T @ X ) ) ) ) ) ) ).

% vebt_pred'_rf_abstr
thf(fact_5377_Collect__conv__if,axiom,
    ! [P: vEBT_VEBT > $o,A2: vEBT_VEBT] :
      ( ( ( P @ A2 )
       => ( ( collect_VEBT_VEBT
            @ ^ [X3: vEBT_VEBT] :
                ( ( X3 = A2 )
                & ( P @ X3 ) ) )
          = ( insert_VEBT_VEBT @ A2 @ bot_bo8194388402131092736T_VEBT ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_VEBT_VEBT
            @ ^ [X3: vEBT_VEBT] :
                ( ( X3 = A2 )
                & ( P @ X3 ) ) )
          = bot_bo8194388402131092736T_VEBT ) ) ) ).

% Collect_conv_if
thf(fact_5378_Collect__conv__if,axiom,
    ! [P: complex > $o,A2: complex] :
      ( ( ( P @ A2 )
       => ( ( collect_complex
            @ ^ [X3: complex] :
                ( ( X3 = A2 )
                & ( P @ X3 ) ) )
          = ( insert_complex @ A2 @ bot_bot_set_complex ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_complex
            @ ^ [X3: complex] :
                ( ( X3 = A2 )
                & ( P @ X3 ) ) )
          = bot_bot_set_complex ) ) ) ).

% Collect_conv_if
thf(fact_5379_Collect__conv__if,axiom,
    ! [P: product_prod_int_int > $o,A2: product_prod_int_int] :
      ( ( ( P @ A2 )
       => ( ( collec213857154873943460nt_int
            @ ^ [X3: product_prod_int_int] :
                ( ( X3 = A2 )
                & ( P @ X3 ) ) )
          = ( insert5033312907999012233nt_int @ A2 @ bot_bo1796632182523588997nt_int ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collec213857154873943460nt_int
            @ ^ [X3: product_prod_int_int] :
                ( ( X3 = A2 )
                & ( P @ X3 ) ) )
          = bot_bo1796632182523588997nt_int ) ) ) ).

% Collect_conv_if
thf(fact_5380_Collect__conv__if,axiom,
    ! [P: real > $o,A2: real] :
      ( ( ( P @ A2 )
       => ( ( collect_real
            @ ^ [X3: real] :
                ( ( X3 = A2 )
                & ( P @ X3 ) ) )
          = ( insert_real @ A2 @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_real
            @ ^ [X3: real] :
                ( ( X3 = A2 )
                & ( P @ X3 ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if
thf(fact_5381_Collect__conv__if,axiom,
    ! [P: $o > $o,A2: $o] :
      ( ( ( P @ A2 )
       => ( ( collect_o
            @ ^ [X3: $o] :
                ( ( X3 = A2 )
                & ( P @ X3 ) ) )
          = ( insert_o @ A2 @ bot_bot_set_o ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_o
            @ ^ [X3: $o] :
                ( ( X3 = A2 )
                & ( P @ X3 ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if
thf(fact_5382_Collect__conv__if,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ( ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X3: nat] :
                ( ( X3 = A2 )
                & ( P @ X3 ) ) )
          = ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X3: nat] :
                ( ( X3 = A2 )
                & ( P @ X3 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_5383_Collect__conv__if,axiom,
    ! [P: int > $o,A2: int] :
      ( ( ( P @ A2 )
       => ( ( collect_int
            @ ^ [X3: int] :
                ( ( X3 = A2 )
                & ( P @ X3 ) ) )
          = ( insert_int @ A2 @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_int
            @ ^ [X3: int] :
                ( ( X3 = A2 )
                & ( P @ X3 ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if
thf(fact_5384_Collect__conv__if2,axiom,
    ! [P: vEBT_VEBT > $o,A2: vEBT_VEBT] :
      ( ( ( P @ A2 )
       => ( ( collect_VEBT_VEBT
            @ ^ [X3: vEBT_VEBT] :
                ( ( A2 = X3 )
                & ( P @ X3 ) ) )
          = ( insert_VEBT_VEBT @ A2 @ bot_bo8194388402131092736T_VEBT ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_VEBT_VEBT
            @ ^ [X3: vEBT_VEBT] :
                ( ( A2 = X3 )
                & ( P @ X3 ) ) )
          = bot_bo8194388402131092736T_VEBT ) ) ) ).

% Collect_conv_if2
thf(fact_5385_Collect__conv__if2,axiom,
    ! [P: complex > $o,A2: complex] :
      ( ( ( P @ A2 )
       => ( ( collect_complex
            @ ^ [X3: complex] :
                ( ( A2 = X3 )
                & ( P @ X3 ) ) )
          = ( insert_complex @ A2 @ bot_bot_set_complex ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_complex
            @ ^ [X3: complex] :
                ( ( A2 = X3 )
                & ( P @ X3 ) ) )
          = bot_bot_set_complex ) ) ) ).

% Collect_conv_if2
thf(fact_5386_Collect__conv__if2,axiom,
    ! [P: product_prod_int_int > $o,A2: product_prod_int_int] :
      ( ( ( P @ A2 )
       => ( ( collec213857154873943460nt_int
            @ ^ [X3: product_prod_int_int] :
                ( ( A2 = X3 )
                & ( P @ X3 ) ) )
          = ( insert5033312907999012233nt_int @ A2 @ bot_bo1796632182523588997nt_int ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collec213857154873943460nt_int
            @ ^ [X3: product_prod_int_int] :
                ( ( A2 = X3 )
                & ( P @ X3 ) ) )
          = bot_bo1796632182523588997nt_int ) ) ) ).

% Collect_conv_if2
thf(fact_5387_Collect__conv__if2,axiom,
    ! [P: real > $o,A2: real] :
      ( ( ( P @ A2 )
       => ( ( collect_real
            @ ^ [X3: real] :
                ( ( A2 = X3 )
                & ( P @ X3 ) ) )
          = ( insert_real @ A2 @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_real
            @ ^ [X3: real] :
                ( ( A2 = X3 )
                & ( P @ X3 ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if2
thf(fact_5388_Collect__conv__if2,axiom,
    ! [P: $o > $o,A2: $o] :
      ( ( ( P @ A2 )
       => ( ( collect_o
            @ ^ [X3: $o] :
                ( ( A2 = X3 )
                & ( P @ X3 ) ) )
          = ( insert_o @ A2 @ bot_bot_set_o ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_o
            @ ^ [X3: $o] :
                ( ( A2 = X3 )
                & ( P @ X3 ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if2
thf(fact_5389_Collect__conv__if2,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ( ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X3: nat] :
                ( ( A2 = X3 )
                & ( P @ X3 ) ) )
          = ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X3: nat] :
                ( ( A2 = X3 )
                & ( P @ X3 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_5390_Collect__conv__if2,axiom,
    ! [P: int > $o,A2: int] :
      ( ( ( P @ A2 )
       => ( ( collect_int
            @ ^ [X3: int] :
                ( ( A2 = X3 )
                & ( P @ X3 ) ) )
          = ( insert_int @ A2 @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_int
            @ ^ [X3: int] :
                ( ( A2 = X3 )
                & ( P @ X3 ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if2
thf(fact_5391_max__def__raw,axiom,
    ( ord_max_set_nat
    = ( ^ [A7: set_nat,B7: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A7 @ B7 ) @ B7 @ A7 ) ) ) ).

% max_def_raw
thf(fact_5392_max__def__raw,axiom,
    ( ord_max_rat
    = ( ^ [A7: rat,B7: rat] : ( if_rat @ ( ord_less_eq_rat @ A7 @ B7 ) @ B7 @ A7 ) ) ) ).

% max_def_raw
thf(fact_5393_max__def__raw,axiom,
    ( ord_max_num
    = ( ^ [A7: num,B7: num] : ( if_num @ ( ord_less_eq_num @ A7 @ B7 ) @ B7 @ A7 ) ) ) ).

% max_def_raw
thf(fact_5394_max__def__raw,axiom,
    ( ord_max_nat
    = ( ^ [A7: nat,B7: nat] : ( if_nat @ ( ord_less_eq_nat @ A7 @ B7 ) @ B7 @ A7 ) ) ) ).

% max_def_raw
thf(fact_5395_max__def__raw,axiom,
    ( ord_max_int
    = ( ^ [A7: int,B7: int] : ( if_int @ ( ord_less_eq_int @ A7 @ B7 ) @ B7 @ A7 ) ) ) ).

% max_def_raw
thf(fact_5396_Collect__subset,axiom,
    ! [A: set_real,P: real > $o] :
      ( ord_less_eq_set_real
      @ ( collect_real
        @ ^ [X3: real] :
            ( ( member_real @ X3 @ A )
            & ( P @ X3 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_5397_Collect__subset,axiom,
    ! [A: set_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ord_le4337996190870823476T_VEBT
      @ ( collect_VEBT_VEBT
        @ ^ [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A )
            & ( P @ X3 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_5398_Collect__subset,axiom,
    ! [A: set_int,P: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X3: int] :
            ( ( member_int @ X3 @ A )
            & ( P @ X3 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_5399_Collect__subset,axiom,
    ! [A: set_complex,P: complex > $o] :
      ( ord_le211207098394363844omplex
      @ ( collect_complex
        @ ^ [X3: complex] :
            ( ( member_complex @ X3 @ A )
            & ( P @ X3 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_5400_Collect__subset,axiom,
    ! [A: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
      ( ord_le2843351958646193337nt_int
      @ ( collec213857154873943460nt_int
        @ ^ [X3: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X3 @ A )
            & ( P @ X3 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_5401_Collect__subset,axiom,
    ! [A: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X3: nat] :
            ( ( member_nat @ X3 @ A )
            & ( P @ X3 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_5402_less__eq__set__def,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B5: set_real] :
          ( ord_less_eq_real_o
          @ ^ [X3: real] : ( member_real @ X3 @ A6 )
          @ ^ [X3: real] : ( member_real @ X3 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_5403_less__eq__set__def,axiom,
    ( ord_le4337996190870823476T_VEBT
    = ( ^ [A6: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
          ( ord_le418104280809901481VEBT_o
          @ ^ [X3: vEBT_VEBT] : ( member_VEBT_VEBT @ X3 @ A6 )
          @ ^ [X3: vEBT_VEBT] : ( member_VEBT_VEBT @ X3 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_5404_less__eq__set__def,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B5: set_int] :
          ( ord_less_eq_int_o
          @ ^ [X3: int] : ( member_int @ X3 @ A6 )
          @ ^ [X3: int] : ( member_int @ X3 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_5405_less__eq__set__def,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A6: set_complex,B5: set_complex] :
          ( ord_le4573692005234683329plex_o
          @ ^ [X3: complex] : ( member_complex @ X3 @ A6 )
          @ ^ [X3: complex] : ( member_complex @ X3 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_5406_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B5: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
          @ ^ [X3: nat] : ( member_nat @ X3 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_5407_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A6: set_nat,B5: set_nat] :
          ( ord_less_nat_o
          @ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
          @ ^ [X3: nat] : ( member_nat @ X3 @ B5 ) ) ) ) ).

% less_set_def
thf(fact_5408_less__set__def,axiom,
    ( ord_less_set_real
    = ( ^ [A6: set_real,B5: set_real] :
          ( ord_less_real_o
          @ ^ [X3: real] : ( member_real @ X3 @ A6 )
          @ ^ [X3: real] : ( member_real @ X3 @ B5 ) ) ) ) ).

% less_set_def
thf(fact_5409_less__set__def,axiom,
    ( ord_le3480810397992357184T_VEBT
    = ( ^ [A6: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
          ( ord_less_VEBT_VEBT_o
          @ ^ [X3: vEBT_VEBT] : ( member_VEBT_VEBT @ X3 @ A6 )
          @ ^ [X3: vEBT_VEBT] : ( member_VEBT_VEBT @ X3 @ B5 ) ) ) ) ).

% less_set_def
thf(fact_5410_less__set__def,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B5: set_int] :
          ( ord_less_int_o
          @ ^ [X3: int] : ( member_int @ X3 @ A6 )
          @ ^ [X3: int] : ( member_int @ X3 @ B5 ) ) ) ) ).

% less_set_def
thf(fact_5411_less__set__def,axiom,
    ( ord_less_set_complex
    = ( ^ [A6: set_complex,B5: set_complex] :
          ( ord_less_complex_o
          @ ^ [X3: complex] : ( member_complex @ X3 @ A6 )
          @ ^ [X3: complex] : ( member_complex @ X3 @ B5 ) ) ) ) ).

% less_set_def
thf(fact_5412_minus__set__def,axiom,
    ( minus_minus_set_real
    = ( ^ [A6: set_real,B5: set_real] :
          ( collect_real
          @ ( minus_minus_real_o
            @ ^ [X3: real] : ( member_real @ X3 @ A6 )
            @ ^ [X3: real] : ( member_real @ X3 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_5413_minus__set__def,axiom,
    ( minus_5127226145743854075T_VEBT
    = ( ^ [A6: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ( minus_2794559001203777698VEBT_o
            @ ^ [X3: vEBT_VEBT] : ( member_VEBT_VEBT @ X3 @ A6 )
            @ ^ [X3: vEBT_VEBT] : ( member_VEBT_VEBT @ X3 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_5414_minus__set__def,axiom,
    ( minus_minus_set_int
    = ( ^ [A6: set_int,B5: set_int] :
          ( collect_int
          @ ( minus_minus_int_o
            @ ^ [X3: int] : ( member_int @ X3 @ A6 )
            @ ^ [X3: int] : ( member_int @ X3 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_5415_minus__set__def,axiom,
    ( minus_811609699411566653omplex
    = ( ^ [A6: set_complex,B5: set_complex] :
          ( collect_complex
          @ ( minus_8727706125548526216plex_o
            @ ^ [X3: complex] : ( member_complex @ X3 @ A6 )
            @ ^ [X3: complex] : ( member_complex @ X3 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_5416_minus__set__def,axiom,
    ( minus_1052850069191792384nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int,B5: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ( minus_711738161318947805_int_o
            @ ^ [X3: product_prod_int_int] : ( member5262025264175285858nt_int @ X3 @ A6 )
            @ ^ [X3: product_prod_int_int] : ( member5262025264175285858nt_int @ X3 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_5417_minus__set__def,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B5: set_nat] :
          ( collect_nat
          @ ( minus_minus_nat_o
            @ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
            @ ^ [X3: nat] : ( member_nat @ X3 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_5418_set__diff__eq,axiom,
    ( minus_minus_set_real
    = ( ^ [A6: set_real,B5: set_real] :
          ( collect_real
          @ ^ [X3: real] :
              ( ( member_real @ X3 @ A6 )
              & ~ ( member_real @ X3 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_5419_set__diff__eq,axiom,
    ( minus_5127226145743854075T_VEBT
    = ( ^ [A6: set_VEBT_VEBT,B5: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ^ [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A6 )
              & ~ ( member_VEBT_VEBT @ X3 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_5420_set__diff__eq,axiom,
    ( minus_minus_set_int
    = ( ^ [A6: set_int,B5: set_int] :
          ( collect_int
          @ ^ [X3: int] :
              ( ( member_int @ X3 @ A6 )
              & ~ ( member_int @ X3 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_5421_set__diff__eq,axiom,
    ( minus_811609699411566653omplex
    = ( ^ [A6: set_complex,B5: set_complex] :
          ( collect_complex
          @ ^ [X3: complex] :
              ( ( member_complex @ X3 @ A6 )
              & ~ ( member_complex @ X3 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_5422_set__diff__eq,axiom,
    ( minus_1052850069191792384nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int,B5: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ^ [X3: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X3 @ A6 )
              & ~ ( member5262025264175285858nt_int @ X3 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_5423_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B5: set_nat] :
          ( collect_nat
          @ ^ [X3: nat] :
              ( ( member_nat @ X3 @ A6 )
              & ~ ( member_nat @ X3 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_5424_insert__compr,axiom,
    ( insert_o
    = ( ^ [A7: $o,B5: set_o] :
          ( collect_o
          @ ^ [X3: $o] :
              ( ( X3 = A7 )
              | ( member_o @ X3 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_5425_insert__compr,axiom,
    ( insert_real
    = ( ^ [A7: real,B5: set_real] :
          ( collect_real
          @ ^ [X3: real] :
              ( ( X3 = A7 )
              | ( member_real @ X3 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_5426_insert__compr,axiom,
    ( insert_VEBT_VEBT
    = ( ^ [A7: vEBT_VEBT,B5: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ^ [X3: vEBT_VEBT] :
              ( ( X3 = A7 )
              | ( member_VEBT_VEBT @ X3 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_5427_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A7: nat,B5: set_nat] :
          ( collect_nat
          @ ^ [X3: nat] :
              ( ( X3 = A7 )
              | ( member_nat @ X3 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_5428_insert__compr,axiom,
    ( insert_int
    = ( ^ [A7: int,B5: set_int] :
          ( collect_int
          @ ^ [X3: int] :
              ( ( X3 = A7 )
              | ( member_int @ X3 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_5429_insert__compr,axiom,
    ( insert_complex
    = ( ^ [A7: complex,B5: set_complex] :
          ( collect_complex
          @ ^ [X3: complex] :
              ( ( X3 = A7 )
              | ( member_complex @ X3 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_5430_insert__compr,axiom,
    ( insert5033312907999012233nt_int
    = ( ^ [A7: product_prod_int_int,B5: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ^ [X3: product_prod_int_int] :
              ( ( X3 = A7 )
              | ( member5262025264175285858nt_int @ X3 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_5431_insert__Collect,axiom,
    ! [A2: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( insert_VEBT_VEBT @ A2 @ ( collect_VEBT_VEBT @ P ) )
      = ( collect_VEBT_VEBT
        @ ^ [U2: vEBT_VEBT] :
            ( ( U2 != A2 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_5432_insert__Collect,axiom,
    ! [A2: real,P: real > $o] :
      ( ( insert_real @ A2 @ ( collect_real @ P ) )
      = ( collect_real
        @ ^ [U2: real] :
            ( ( U2 != A2 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_5433_insert__Collect,axiom,
    ! [A2: $o,P: $o > $o] :
      ( ( insert_o @ A2 @ ( collect_o @ P ) )
      = ( collect_o
        @ ^ [U2: $o] :
            ( ( U2 != A2 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_5434_insert__Collect,axiom,
    ! [A2: nat,P: nat > $o] :
      ( ( insert_nat @ A2 @ ( collect_nat @ P ) )
      = ( collect_nat
        @ ^ [U2: nat] :
            ( ( U2 != A2 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_5435_insert__Collect,axiom,
    ! [A2: int,P: int > $o] :
      ( ( insert_int @ A2 @ ( collect_int @ P ) )
      = ( collect_int
        @ ^ [U2: int] :
            ( ( U2 != A2 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_5436_insert__Collect,axiom,
    ! [A2: complex,P: complex > $o] :
      ( ( insert_complex @ A2 @ ( collect_complex @ P ) )
      = ( collect_complex
        @ ^ [U2: complex] :
            ( ( U2 != A2 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_5437_insert__Collect,axiom,
    ! [A2: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( insert5033312907999012233nt_int @ A2 @ ( collec213857154873943460nt_int @ P ) )
      = ( collec213857154873943460nt_int
        @ ^ [U2: product_prod_int_int] :
            ( ( U2 != A2 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_5438_Set_Oempty__def,axiom,
    ( bot_bot_set_complex
    = ( collect_complex
      @ ^ [X3: complex] : $false ) ) ).

% Set.empty_def
thf(fact_5439_Set_Oempty__def,axiom,
    ( bot_bo1796632182523588997nt_int
    = ( collec213857154873943460nt_int
      @ ^ [X3: product_prod_int_int] : $false ) ) ).

% Set.empty_def
thf(fact_5440_Set_Oempty__def,axiom,
    ( bot_bot_set_real
    = ( collect_real
      @ ^ [X3: real] : $false ) ) ).

% Set.empty_def
thf(fact_5441_Set_Oempty__def,axiom,
    ( bot_bot_set_o
    = ( collect_o
      @ ^ [X3: $o] : $false ) ) ).

% Set.empty_def
thf(fact_5442_Set_Oempty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X3: nat] : $false ) ) ).

% Set.empty_def
thf(fact_5443_Set_Oempty__def,axiom,
    ( bot_bot_set_int
    = ( collect_int
      @ ^ [X3: int] : $false ) ) ).

% Set.empty_def
thf(fact_5444_lambda__one,axiom,
    ( ( ^ [X3: real] : X3 )
    = ( times_times_real @ one_one_real ) ) ).

% lambda_one
thf(fact_5445_lambda__one,axiom,
    ( ( ^ [X3: rat] : X3 )
    = ( times_times_rat @ one_one_rat ) ) ).

% lambda_one
thf(fact_5446_lambda__one,axiom,
    ( ( ^ [X3: nat] : X3 )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_5447_lambda__one,axiom,
    ( ( ^ [X3: int] : X3 )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_5448_lambda__one,axiom,
    ( ( ^ [X3: assn] : X3 )
    = ( times_times_assn @ one_one_assn ) ) ).

% lambda_one
thf(fact_5449_lambda__zero,axiom,
    ( ( ^ [H: complex] : zero_zero_complex )
    = ( times_times_complex @ zero_zero_complex ) ) ).

% lambda_zero
thf(fact_5450_lambda__zero,axiom,
    ( ( ^ [H: real] : zero_zero_real )
    = ( times_times_real @ zero_zero_real ) ) ).

% lambda_zero
thf(fact_5451_lambda__zero,axiom,
    ( ( ^ [H: rat] : zero_zero_rat )
    = ( times_times_rat @ zero_zero_rat ) ) ).

% lambda_zero
thf(fact_5452_lambda__zero,axiom,
    ( ( ^ [H: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_5453_lambda__zero,axiom,
    ( ( ^ [H: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_5454_mult__commute__abs,axiom,
    ! [C2: real] :
      ( ( ^ [X3: real] : ( times_times_real @ X3 @ C2 ) )
      = ( times_times_real @ C2 ) ) ).

% mult_commute_abs
thf(fact_5455_mult__commute__abs,axiom,
    ! [C2: rat] :
      ( ( ^ [X3: rat] : ( times_times_rat @ X3 @ C2 ) )
      = ( times_times_rat @ C2 ) ) ).

% mult_commute_abs
thf(fact_5456_mult__commute__abs,axiom,
    ! [C2: nat] :
      ( ( ^ [X3: nat] : ( times_times_nat @ X3 @ C2 ) )
      = ( times_times_nat @ C2 ) ) ).

% mult_commute_abs
thf(fact_5457_mult__commute__abs,axiom,
    ! [C2: int] :
      ( ( ^ [X3: int] : ( times_times_int @ X3 @ C2 ) )
      = ( times_times_int @ C2 ) ) ).

% mult_commute_abs
thf(fact_5458_mult__commute__abs,axiom,
    ! [C2: assn] :
      ( ( ^ [X3: assn] : ( times_times_assn @ X3 @ C2 ) )
      = ( times_times_assn @ C2 ) ) ).

% mult_commute_abs
thf(fact_5459_frame__rule,axiom,
    ! [P: assn,C2: heap_Time_Heap_o,Q: $o > assn,R: assn] :
      ( ( hoare_hoare_triple_o @ P @ C2 @ Q )
     => ( hoare_hoare_triple_o @ ( times_times_assn @ P @ R ) @ C2
        @ ^ [X3: $o] : ( times_times_assn @ ( Q @ X3 ) @ R ) ) ) ).

% frame_rule
thf(fact_5460_frame__rule,axiom,
    ! [P: assn,C2: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,R: assn] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C2 @ Q )
     => ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ P @ R ) @ C2
        @ ^ [X3: vEBT_VEBTi] : ( times_times_assn @ ( Q @ X3 ) @ R ) ) ) ).

% frame_rule
thf(fact_5461_frame__rule,axiom,
    ! [P: assn,C2: heap_T2636463487746394924on_nat,Q: option_nat > assn,R: assn] :
      ( ( hoare_7629718768684598413on_nat @ P @ C2 @ Q )
     => ( hoare_7629718768684598413on_nat @ ( times_times_assn @ P @ R ) @ C2
        @ ^ [X3: option_nat] : ( times_times_assn @ ( Q @ X3 ) @ R ) ) ) ).

% frame_rule
thf(fact_5462_frame__rule,axiom,
    ! [P: assn,C2: heap_Time_Heap_nat,Q: nat > assn,R: assn] :
      ( ( hoare_3067605981109127869le_nat @ P @ C2 @ Q )
     => ( hoare_3067605981109127869le_nat @ ( times_times_assn @ P @ R ) @ C2
        @ ^ [X3: nat] : ( times_times_assn @ ( Q @ X3 ) @ R ) ) ) ).

% frame_rule
thf(fact_5463_cons__pre__rule,axiom,
    ! [P: assn,P2: assn,C2: heap_Time_Heap_o,Q: $o > assn] :
      ( ( entails @ P @ P2 )
     => ( ( hoare_hoare_triple_o @ P2 @ C2 @ Q )
       => ( hoare_hoare_triple_o @ P @ C2 @ Q ) ) ) ).

% cons_pre_rule
thf(fact_5464_cons__pre__rule,axiom,
    ! [P: assn,P2: assn,C2: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn] :
      ( ( entails @ P @ P2 )
     => ( ( hoare_1429296392585015714_VEBTi @ P2 @ C2 @ Q )
       => ( hoare_1429296392585015714_VEBTi @ P @ C2 @ Q ) ) ) ).

% cons_pre_rule
thf(fact_5465_cons__pre__rule,axiom,
    ! [P: assn,P2: assn,C2: heap_T2636463487746394924on_nat,Q: option_nat > assn] :
      ( ( entails @ P @ P2 )
     => ( ( hoare_7629718768684598413on_nat @ P2 @ C2 @ Q )
       => ( hoare_7629718768684598413on_nat @ P @ C2 @ Q ) ) ) ).

% cons_pre_rule
thf(fact_5466_cons__pre__rule,axiom,
    ! [P: assn,P2: assn,C2: heap_Time_Heap_nat,Q: nat > assn] :
      ( ( entails @ P @ P2 )
     => ( ( hoare_3067605981109127869le_nat @ P2 @ C2 @ Q )
       => ( hoare_3067605981109127869le_nat @ P @ C2 @ Q ) ) ) ).

% cons_pre_rule
thf(fact_5467_set__vebt__def,axiom,
    ( vEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).

% set_vebt_def
thf(fact_5468_numeral__code_I2_J,axiom,
    ! [N3: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N3 ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N3 ) @ ( numera6690914467698888265omplex @ N3 ) ) ) ).

% numeral_code(2)
thf(fact_5469_numeral__code_I2_J,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N3 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N3 ) @ ( numeral_numeral_real @ N3 ) ) ) ).

% numeral_code(2)
thf(fact_5470_numeral__code_I2_J,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N3 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N3 ) @ ( numeral_numeral_rat @ N3 ) ) ) ).

% numeral_code(2)
thf(fact_5471_numeral__code_I2_J,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N3 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N3 ) @ ( numeral_numeral_nat @ N3 ) ) ) ).

% numeral_code(2)
thf(fact_5472_numeral__code_I2_J,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N3 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N3 ) @ ( numeral_numeral_int @ N3 ) ) ) ).

% numeral_code(2)
thf(fact_5473_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A7: nat,B7: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A7 ) @ ( semiri1314217659103216013at_int @ B7 ) ) ) ) ).

% nat_less_as_int
thf(fact_5474_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A7: nat,B7: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A7 ) @ ( semiri1314217659103216013at_int @ B7 ) ) ) ) ).

% nat_leq_as_int
thf(fact_5475_numeral__code_I3_J,axiom,
    ! [N3: num] :
      ( ( numera6690914467698888265omplex @ ( bit1 @ N3 ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N3 ) @ ( numera6690914467698888265omplex @ N3 ) ) @ one_one_complex ) ) ).

% numeral_code(3)
thf(fact_5476_numeral__code_I3_J,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N3 ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N3 ) @ ( numeral_numeral_real @ N3 ) ) @ one_one_real ) ) ).

% numeral_code(3)
thf(fact_5477_numeral__code_I3_J,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N3 ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N3 ) @ ( numeral_numeral_rat @ N3 ) ) @ one_one_rat ) ) ).

% numeral_code(3)
thf(fact_5478_numeral__code_I3_J,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N3 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N3 ) @ ( numeral_numeral_nat @ N3 ) ) @ one_one_nat ) ) ).

% numeral_code(3)
thf(fact_5479_numeral__code_I3_J,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N3 ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N3 ) @ ( numeral_numeral_int @ N3 ) ) @ one_one_int ) ) ).

% numeral_code(3)
thf(fact_5480_power__numeral__even,axiom,
    ! [Z: complex,W: num] :
      ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_complex @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_5481_power__numeral__even,axiom,
    ! [Z: code_integer,W: num] :
      ( ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_5482_power__numeral__even,axiom,
    ! [Z: real,W: num] :
      ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_real @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_5483_power__numeral__even,axiom,
    ! [Z: rat,W: num] :
      ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_rat @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_5484_power__numeral__even,axiom,
    ! [Z: nat,W: num] :
      ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_5485_power__numeral__even,axiom,
    ! [Z: int,W: num] :
      ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_5486_power__numeral__even,axiom,
    ! [Z: assn,W: num] :
      ( ( power_power_assn @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_assn @ ( power_power_assn @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_assn @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_5487_power__numeral__odd,axiom,
    ! [Z: complex,W: num] :
      ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_complex @ ( times_times_complex @ Z @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_5488_power__numeral__odd,axiom,
    ! [Z: code_integer,W: num] :
      ( ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ Z @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_5489_power__numeral__odd,axiom,
    ! [Z: real,W: num] :
      ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_real @ ( times_times_real @ Z @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_5490_power__numeral__odd,axiom,
    ! [Z: rat,W: num] :
      ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_rat @ ( times_times_rat @ Z @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_5491_power__numeral__odd,axiom,
    ! [Z: nat,W: num] :
      ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_nat @ ( times_times_nat @ Z @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_5492_power__numeral__odd,axiom,
    ! [Z: int,W: num] :
      ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_int @ ( times_times_int @ Z @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_5493_power__numeral__odd,axiom,
    ! [Z: assn,W: num] :
      ( ( power_power_assn @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_assn @ ( times_times_assn @ Z @ ( power_power_assn @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_assn @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_5494_rule__at__index,axiom,
    ! [P: assn,A: vEBT_VEBT > vEBT_VEBT > assn,Xs2: list_VEBT_VEBT,Xsi: list_VEBT_VEBT,F: assn,I: nat,C2: heap_Time_Heap_o,Q2: $o > assn,F4: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L1279224858307276611T_VEBT @ A @ Xs2 @ Xsi ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) @ C2 @ Q2 )
         => ( ! [R2: $o] : ( entails @ ( Q2 @ R2 ) @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3204528365124325536T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ ( F4 @ R2 ) ) )
           => ( hoare_hoare_triple_o @ P @ C2
              @ ^ [R5: $o] : ( times_times_assn @ ( vEBT_L1279224858307276611T_VEBT @ A @ Xs2 @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_5495_rule__at__index,axiom,
    ! [P: assn,A: vEBT_VEBT > nat > assn,Xs2: list_VEBT_VEBT,Xsi: list_nat,F: assn,I: nat,C2: heap_Time_Heap_o,Q2: $o > assn,F4: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L8296926524756676353BT_nat @ A @ Xs2 @ Xsi ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8650695023172932196BT_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) @ C2 @ Q2 )
         => ( ! [R2: $o] : ( entails @ ( Q2 @ R2 ) @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_VEBT_VEBT @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L8650695023172932196BT_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ ( F4 @ R2 ) ) )
           => ( hoare_hoare_triple_o @ P @ C2
              @ ^ [R5: $o] : ( times_times_assn @ ( vEBT_L8296926524756676353BT_nat @ A @ Xs2 @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_5496_rule__at__index,axiom,
    ! [P: assn,A: real > vEBT_VEBT > assn,Xs2: list_real,Xsi: list_VEBT_VEBT,F: assn,I: nat,C2: heap_Time_Heap_o,Q2: $o > assn,F4: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L4595930785310033027T_VEBT @ A @ Xs2 @ Xsi ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_real @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3095048238742455910T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) @ C2 @ Q2 )
         => ( ! [R2: $o] : ( entails @ ( Q2 @ R2 ) @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_real @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L3095048238742455910T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ ( F4 @ R2 ) ) )
           => ( hoare_hoare_triple_o @ P @ C2
              @ ^ [R5: $o] : ( times_times_assn @ ( vEBT_L4595930785310033027T_VEBT @ A @ Xs2 @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_5497_rule__at__index,axiom,
    ! [P: assn,A: real > vEBT_VEBTi > assn,Xs2: list_real,Xsi: list_VEBT_VEBTi,F: assn,I: nat,C2: heap_Time_Heap_o,Q2: $o > assn,F4: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L9060850011106065574_VEBTi @ A @ Xs2 @ Xsi ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_real @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L7851252805511451907_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) @ C2 @ Q2 )
         => ( ! [R2: $o] : ( entails @ ( Q2 @ R2 ) @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_real @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L7851252805511451907_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ ( F4 @ R2 ) ) )
           => ( hoare_hoare_triple_o @ P @ C2
              @ ^ [R5: $o] : ( times_times_assn @ ( vEBT_L9060850011106065574_VEBTi @ A @ Xs2 @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_5498_rule__at__index,axiom,
    ! [P: assn,A: real > nat > assn,Xs2: list_real,Xsi: list_nat,F: assn,I: nat,C2: heap_Time_Heap_o,Q2: $o > assn,F4: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L1446010312343316929al_nat @ A @ Xs2 @ Xsi ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_real @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L234762979517870878al_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) @ C2 @ Q2 )
         => ( ! [R2: $o] : ( entails @ ( Q2 @ R2 ) @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_real @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L234762979517870878al_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ ( F4 @ R2 ) ) )
           => ( hoare_hoare_triple_o @ P @ C2
              @ ^ [R5: $o] : ( times_times_assn @ ( vEBT_L1446010312343316929al_nat @ A @ Xs2 @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_5499_rule__at__index,axiom,
    ! [P: assn,A: $o > vEBT_VEBT > assn,Xs2: list_o,Xsi: list_VEBT_VEBT,F: assn,I: nat,C2: heap_Time_Heap_o,Q2: $o > assn,F4: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L1750719106661372127T_VEBT @ A @ Xs2 @ Xsi ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_o @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L1319876754960170684T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) @ C2 @ Q2 )
         => ( ! [R2: $o] : ( entails @ ( Q2 @ R2 ) @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_o @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L1319876754960170684T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ ( F4 @ R2 ) ) )
           => ( hoare_hoare_triple_o @ P @ C2
              @ ^ [R5: $o] : ( times_times_assn @ ( vEBT_L1750719106661372127T_VEBT @ A @ Xs2 @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_5500_rule__at__index,axiom,
    ! [P: assn,A: $o > vEBT_VEBTi > assn,Xs2: list_o,Xsi: list_VEBT_VEBTi,F: assn,I: nat,C2: heap_Time_Heap_o,Q2: $o > assn,F4: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L3704169666673096010_VEBTi @ A @ Xs2 @ Xsi ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_o @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L6286945158656146733_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) @ C2 @ Q2 )
         => ( ! [R2: $o] : ( entails @ ( Q2 @ R2 ) @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_o @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L6286945158656146733_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ ( F4 @ R2 ) ) )
           => ( hoare_hoare_triple_o @ P @ C2
              @ ^ [R5: $o] : ( times_times_assn @ ( vEBT_L3704169666673096010_VEBTi @ A @ Xs2 @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_5501_rule__at__index,axiom,
    ! [P: assn,A: $o > nat > assn,Xs2: list_o,Xsi: list_nat,F: assn,I: nat,C2: heap_Time_Heap_o,Q2: $o > assn,F4: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L4785011123346445925_o_nat @ A @ Xs2 @ Xsi ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_o @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L2281750874075065672_o_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) @ C2 @ Q2 )
         => ( ! [R2: $o] : ( entails @ ( Q2 @ R2 ) @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_o @ Xs2 @ I ) @ ( nth_nat @ Xsi @ I ) ) @ ( vEBT_L2281750874075065672_o_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ ( F4 @ R2 ) ) )
           => ( hoare_hoare_triple_o @ P @ C2
              @ ^ [R5: $o] : ( times_times_assn @ ( vEBT_L4785011123346445925_o_nat @ A @ Xs2 @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_5502_rule__at__index,axiom,
    ! [P: assn,A: nat > vEBT_VEBT > assn,Xs2: list_nat,Xsi: list_VEBT_VEBT,F: assn,I: nat,C2: heap_Time_Heap_o,Q2: $o > assn,F4: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L8158188754432654943T_VEBT @ A @ Xs2 @ Xsi ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_nat @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L8511957252848910786T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) @ C2 @ Q2 )
         => ( ! [R2: $o] : ( entails @ ( Q2 @ R2 ) @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_nat @ Xs2 @ I ) @ ( nth_VEBT_VEBT @ Xsi @ I ) ) @ ( vEBT_L8511957252848910786T_VEBT @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ ( F4 @ R2 ) ) )
           => ( hoare_hoare_triple_o @ P @ C2
              @ ^ [R5: $o] : ( times_times_assn @ ( vEBT_L8158188754432654943T_VEBT @ A @ Xs2 @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_5503_rule__at__index,axiom,
    ! [P: assn,A: nat > vEBT_VEBTi > assn,Xs2: list_nat,Xsi: list_VEBT_VEBTi,F: assn,I: nat,C2: heap_Time_Heap_o,Q2: $o > assn,F4: $o > assn] :
      ( ( entails @ P @ ( times_times_assn @ ( vEBT_L4387162340545308618_VEBTi @ A @ Xs2 @ Xsi ) @ F ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
       => ( ( hoare_hoare_triple_o @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_nat @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L7489483478785760935_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ F ) @ C2 @ Q2 )
         => ( ! [R2: $o] : ( entails @ ( Q2 @ R2 ) @ ( times_times_assn @ ( times_times_assn @ ( A @ ( nth_nat @ Xs2 @ I ) @ ( nth_VEBT_VEBTi @ Xsi @ I ) ) @ ( vEBT_L7489483478785760935_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ A @ Xs2 @ Xsi ) ) @ ( F4 @ R2 ) ) )
           => ( hoare_hoare_triple_o @ P @ C2
              @ ^ [R5: $o] : ( times_times_assn @ ( vEBT_L4387162340545308618_VEBTi @ A @ Xs2 @ Xsi ) @ ( F4 @ R5 ) ) ) ) ) ) ) ).

% rule_at_index
thf(fact_5504_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: 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 @ Va ) ) @ 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
thf(fact_5505_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_vebt_member @ X @ Xa )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => A3 )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => B3 )
                    & ( Xa = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
             => ~ ( ( Xa != Mi2 )
                 => ( ( Xa != Ma2 )
                   => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                      & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                       => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                          & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                           => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                               => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
thf(fact_5506_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( X = Mi ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( X = Ma ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma @ X ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
thf(fact_5507_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: 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 @ Va ) ) @ 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ).

% vebt_insert.simps(5)
thf(fact_5508_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X @ Xa )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( Y
              = ( ~ ( ( ( Xa = zero_zero_nat )
                     => A3 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B3 )
                        & ( Xa = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) )
           => Y )
         => ( ( ? [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) )
             => Y )
           => ( ( ? [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) )
               => Y )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                   => ( Y
                      = ( ~ ( ( Xa != Mi2 )
                           => ( ( Xa != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
thf(fact_5509_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_vebt_member @ X @ Xa )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( ( ( Xa = zero_zero_nat )
               => A3 )
              & ( ( Xa != zero_zero_nat )
               => ( ( ( Xa = one_one_nat )
                   => B3 )
                  & ( Xa = one_one_nat ) ) ) ) )
       => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) )
         => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) )
           => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                  ( X
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                   => ( ( Xa != Mi2 )
                     => ( ( Xa != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
thf(fact_5510_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r @ X @ Xa )
        = Y )
     => ( ( ? [A3: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( Y
           != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) )
           => ( Y
             != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) )
             => ( Y
               != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) )
               => ( Y
                 != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                   => ( Y
                     != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa = Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa = Ma2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
thf(fact_5511_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => ( Y
                    = ( vEBT_Leaf @ $true @ B3 ) ) )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A3 @ $true ) ) )
                    & ( ( Xa != one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) )
             => ( Y
               != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
               => ( Y
                 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) ) )
           => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
                 => ( Y
                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                   => ( Y
                     != ( if_VEBT_VEBT
                        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                          & ~ ( ( Xa = Mi2 )
                              | ( Xa = Ma2 ) ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
thf(fact_5512_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I7_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ( ord_less_nat @ X @ Mi )
          | ( ord_less_nat @ Ma @ X ) )
       => ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
          = one_one_nat ) )
      & ( ~ ( ( ord_less_nat @ X @ Mi )
            | ( ord_less_nat @ Ma @ X ) )
       => ( ( ( ( X = Mi )
              & ( X = Ma ) )
           => ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
              = one_one_nat ) )
          & ( ~ ( ( X = Mi )
                & ( X = Ma ) )
           => ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
              = ( if_nat @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( plus_plus_nat @ ( vEBT_V1232361888498592333_e_t_e @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) ) @ one_one_nat ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(7)
thf(fact_5513_vebt__delete_Osimps_I7_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va: 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 @ Va ) ) @ TreeList @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ 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 @ Va ) ) @ TreeList @ Summary ) @ X )
              = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) )
          & ( ~ ( ( X = Mi )
                & ( X = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va ) )
                    @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ 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 @ 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va ) )
                    @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ Summary ) )
                @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
thf(fact_5514_vebt__delete_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X @ Xa )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( ( Xa = zero_zero_nat )
             => ( Y
               != ( vEBT_Leaf @ $false @ B3 ) ) ) )
       => ( ! [A3: $o] :
              ( ? [B3: $o] :
                  ( X
                  = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Xa
                  = ( suc @ zero_zero_nat ) )
               => ( Y
                 != ( vEBT_Leaf @ A3 @ $false ) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ B3 ) )
               => ( ? [N: nat] :
                      ( Xa
                      = ( suc @ ( suc @ N ) ) )
                 => ( Y
                   != ( vEBT_Leaf @ A3 @ B3 ) ) ) )
           => ( ! [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
                 => ( Y
                   != ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                   => ( Y
                     != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                     => ( Y
                       != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                       => ~ ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa ) )
                             => ( Y
                                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) ) )
                            & ( ~ ( ( ord_less_nat @ Xa @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa ) )
                             => ( ( ( ( Xa = Mi2 )
                                    & ( Xa = Ma2 ) )
                                 => ( Y
                                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) ) )
                                & ( ~ ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                 => ( Y
                                    = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                      @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some_P7363390416028606310at_nat
                                            @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa = Mi2 ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa != Mi2 )
                                                   => ( Xa = Ma2 ) ) )
                                                @ ( if_nat
                                                  @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                    = none_nat )
                                                  @ ( if_nat @ ( Xa = Mi2 ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va3 ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some_P7363390416028606310at_nat
                                            @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa = Mi2 ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa != Mi2 )
                                                   => ( Xa = Ma2 ) ) )
                                                @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va3 ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ Summary2 ) )
                                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
thf(fact_5515_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_V1232361888498592333_e_t_e @ X @ Xa )
        = Y )
     => ( ( ? [A3: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A3: $o,B3: $o] :
                ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( ( Xa
                = ( suc @ zero_zero_nat ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A3: $o,B3: $o] :
                  ( X
                  = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ? [N: nat] :
                    ( Xa
                    = ( suc @ ( suc @ N ) ) )
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList3 @ Summary2 ) )
                 => ( Y != one_one_nat ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList3 @ Summary2 ) )
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                       => ~ ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa ) )
                             => ( Y = one_one_nat ) )
                            & ( ~ ( ( ord_less_nat @ Xa @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa ) )
                             => ( ( ( ( Xa = Mi2 )
                                    & ( Xa = Ma2 ) )
                                 => ( Y = one_one_nat ) )
                                & ( ~ ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                 => ( Y
                                    = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) @ ( plus_plus_nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.elims
thf(fact_5516__C7_OIH_C_I2_J,axiom,
    ! [Xa: nat,Xb: nat,Xc: nat,Xd: nat,Xe: vEBT_VEBT,Xf: list_VEBT_VEBT,N3: nat,Ti: vEBT_VEBTi] :
      ( ~ ( ( ord_less_nat @ xa @ mi )
          | ( ord_less_nat @ ma @ xa ) )
     => ( ~ ( ( xa = mi )
            & ( xa = ma ) )
       => ( ( ( ( xa = mi )
             => ( Xa
                = ( 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 @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) ) )
            & ( ( xa != mi )
             => ( Xa = xa ) ) )
         => ( ( ( ( xa = mi )
               => ( Xb = Xa ) )
              & ( ( xa != mi )
               => ( Xb = mi ) ) )
           => ( ( Xc
                = ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
             => ( ( Xd
                  = ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
               => ( ( ord_less_nat @ Xd @ ( size_s6755466524823107622T_VEBT @ treeList ) )
                 => ( ( Xe
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ Xd ) @ Xc ) )
                   => ( ( Xf
                        = ( list_u1324408373059187874T_VEBT @ treeList @ Xd @ Xe ) )
                     => ( ( vEBT_VEBT_minNull @ Xe )
                       => ( ( vEBT_invar_vebt @ summary @ N3 )
                         => ( hoare_1429296392585015714_VEBTi @ ( vEBT_vebt_assn_raw @ summary @ Ti ) @ ( vEBT_V1365221501068881998eletei @ summary @ Ti @ Xd ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ summary @ Xd ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% "7.IH"(2)
thf(fact_5517__C7_OIH_C_I1_J,axiom,
    ! [Xa: nat,Xb: nat,Xc: nat,Xd: nat,N3: nat,Ti: vEBT_VEBTi] :
      ( ~ ( ( ord_less_nat @ xa @ mi )
          | ( ord_less_nat @ ma @ xa ) )
     => ( ~ ( ( xa = mi )
            & ( xa = ma ) )
       => ( ( ( ( xa = mi )
             => ( Xa
                = ( 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 @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) ) )
            & ( ( xa != mi )
             => ( Xa = xa ) ) )
         => ( ( ( ( xa = mi )
               => ( Xb = Xa ) )
              & ( ( xa != mi )
               => ( Xb = mi ) ) )
           => ( ( Xc
                = ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
             => ( ( Xd
                  = ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
               => ( ( ord_less_nat @ Xd @ ( size_s6755466524823107622T_VEBT @ treeList ) )
                 => ( ( vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ Xd ) @ N3 )
                   => ( hoare_1429296392585015714_VEBTi @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ treeList @ Xd ) @ Ti ) @ ( vEBT_V1365221501068881998eletei @ ( nth_VEBT_VEBT @ treeList @ Xd ) @ Ti @ Xc ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ Xd ) @ Xc ) ) ) ) ) ) ) ) ) ) ) ).

% "7.IH"(1)
thf(fact_5518_minNulli__rule,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( hoare_hoare_triple_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_VEBT_minNulli @ Ti )
      @ ^ [R5: $o] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_VEBT_minNull @ T ) ) ) ) ) ).

% minNulli_rule
thf(fact_5519_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_s_u_c_c2 @ X @ Xa )
        = Y )
     => ( ( ? [Uu2: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ B3 ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [Uv: $o,Uw2: $o] :
                ( X
                = ( vEBT_Leaf @ Uv @ Uw2 ) )
           => ( ? [N: nat] :
                  ( Xa
                  = ( suc @ N ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc @ Vd ) )
               => ( Y != one_one_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 != one_one_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                     => ~ ( ( ( ord_less_nat @ Xa @ Mi2 )
                           => ( Y = one_one_nat ) )
                          & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                           => ( Y
                              = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                @ ( if_nat
                                  @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                     != none_nat )
                                    & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                  @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c2 @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  @ ( plus_plus_nat @ ( vEBT_T_s_u_c_c2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                                @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
thf(fact_5520_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_p_r_e_d2 @ X @ Xa )
        = Y )
     => ( ( ? [Uu2: $o,Uv: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A3: $o,Uw2: $o] :
                ( X
                = ( vEBT_Leaf @ A3 @ Uw2 ) )
           => ( ( Xa
                = ( suc @ zero_zero_nat ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A3: $o,B3: $o] :
                  ( X
                  = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ? [Va3: nat] :
                    ( Xa
                    = ( suc @ ( suc @ Va3 ) ) )
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Uy: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va2 ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd @ Ve ) )
                 => ( Y != one_one_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 != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                       => ~ ( ( ( ord_less_nat @ Ma2 @ Xa )
                             => ( Y = one_one_nat ) )
                            & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                             => ( Y
                                = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                  @ ( if_nat
                                    @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d2 @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( plus_plus_nat @ ( vEBT_T_p_r_e_d2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                                  @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
thf(fact_5521_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t @ X @ Xa )
        = Y )
     => ( ( ? [A3: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( Y
           != ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) )
           => ( Y != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
               => ( Y
                 != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                   => ( Y
                     != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
                        @ ( if_nat
                          @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                            & ~ ( ( Xa = Mi2 )
                                | ( Xa = Ma2 ) ) )
                          @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                          @ one_one_nat ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
thf(fact_5522_minNull__bound,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_eq_nat @ ( vEBT_T_m_i_n_N_u_l_l @ T ) @ one_one_nat ) ).

% minNull_bound
thf(fact_5523_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Va: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( suc @ Va ) ) )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
thf(fact_5524_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv2: $o] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ Uu @ Uv2 ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
thf(fact_5525_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw: $o,N3: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uv2 @ Uw ) @ ( suc @ N3 ) )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
thf(fact_5526_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I4_J,axiom,
    ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va: vEBT_VEBT,Vb2: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va ) @ Vb2 )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(4)
thf(fact_5527_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
    ! [Uu: $o,B2: $o] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uu @ B2 ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
thf(fact_5528_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I3_J,axiom,
    ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
thf(fact_5529_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I5_J,axiom,
    ! [Uz2: product_prod_nat_nat,Va: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va @ Vb2 @ Vc2 ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
thf(fact_5530_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
thf(fact_5531_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
    ! [A2: $o,Uw: $o] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ A2 @ Uw ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
thf(fact_5532_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Vf2 )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(5)
thf(fact_5533_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve2 )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
thf(fact_5534_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
thf(fact_5535_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
thf(fact_5536_pred__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_nat @ ( vEBT_T_p_r_e_d2 @ T @ X ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% pred_bound_height'
thf(fact_5537_succ_H__bound__height,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_nat @ ( vEBT_T_s_u_c_c2 @ T @ X ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% succ'_bound_height
thf(fact_5538_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_i_n_N_u_l_l @ X )
        = Y )
     => ( ( ( X
            = ( vEBT_Leaf @ $false @ $false ) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Uv: $o] :
                ( X
                = ( vEBT_Leaf @ $true @ Uv ) )
           => ( Y != one_one_nat ) )
         => ( ( ? [Uu2: $o] :
                  ( X
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [Uw2: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux @ Uy ) )
               => ( Y != one_one_nat ) )
             => ~ ( ? [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va2 @ Vb @ Vc ) )
                 => ( Y != one_one_nat ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
thf(fact_5539_pred__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N3: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_p_r_e_d2 @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ).

% pred_bound_size_univ'
thf(fact_5540_succ__bound__size__univ_H,axiom,
    ! [T: vEBT_VEBT,N3: nat,U: real,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_s_u_c_c2 @ T @ X ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ).

% succ_bound_size_univ'
thf(fact_5541_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
        @ ( if_nat
          @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            & ~ ( ( X = Mi )
                | ( X = Ma ) ) )
          @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
          @ one_one_nat ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
thf(fact_5542_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
    ! [Ma: nat,X: nat,Mi: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
          = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( plus_plus_nat @ ( vEBT_T_p_r_e_d2 @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
            @ one_one_nat ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
thf(fact_5543_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
          = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( plus_plus_nat @ ( vEBT_T_s_u_c_c2 @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
            @ one_one_nat ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
thf(fact_5544_minNrulli__ruleT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( time_htt_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_VEBT_minNulli @ Ti )
      @ ^ [R5: $o] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_VEBT_minNull @ T ) ) ) )
      @ one_one_nat ) ).

% minNrulli_ruleT
thf(fact_5545__092_060open_062_092_060And_062x22_Ax21_O_Ati_A_061_ALeafi_Ax21_Ax22_A_092_060Longrightarrow_062_A_060vebt__assn__raw_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ati_062_Avebt__deletei_H_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ati_Ax_A_060vebt__assn__raw_A_Ivebt__delete_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ax_J_062_092_060close_062,axiom,
    ! [X21: $o,X222: $o] :
      ( ( tia
        = ( vEBT_Leafi @ X21 @ X222 ) )
     => ( hoare_1429296392585015714_VEBTi @ ( vEBT_vebt_assn_raw @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ treeList @ summary ) @ tia ) @ ( vEBT_V1365221501068881998eletei @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ treeList @ summary ) @ tia @ xa ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ ( suc @ ( suc @ va ) ) @ treeList @ summary ) @ xa ) ) ) ) ).

% \<open>\<And>x22 x21. ti = Leafi x21 x22 \<Longrightarrow> <vebt_assn_raw (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) ti> vebt_deletei' (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) ti x <vebt_assn_raw (vebt_delete (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) x)>\<close>
thf(fact_5546_builupi_Hcorr,axiom,
    ! [N3: nat] : ( hoare_1429296392585015714_VEBTi @ one_one_assn @ ( vEBT_V739175172307565963ildupi @ N3 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N3 ) ) ) ).

% builupi'corr
thf(fact_5547_builupicorr,axiom,
    ! [N3: nat] : ( hoare_1429296392585015714_VEBTi @ one_one_assn @ ( vEBT_vebt_buildupi @ N3 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N3 ) ) ) ).

% builupicorr
thf(fact_5548_vebt__mintilist,axiom,
    ! [I: nat,Ts2: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ts2 ) )
     => ( hoare_7629718768684598413on_nat @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Ts2 @ Tsi ) @ ( vEBT_vebt_minti @ ( nth_VEBT_VEBTi @ Tsi @ I ) )
        @ ^ [R5: option_nat] :
            ( times_times_assn
            @ ( pure_assn
              @ ( R5
                = ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ Ts2 @ I ) ) ) )
            @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Ts2 @ Tsi ) ) ) ) ).

% vebt_mintilist
thf(fact_5549_vebt__maxtilist,axiom,
    ! [I: nat,Ts2: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ts2 ) )
     => ( hoare_7629718768684598413on_nat @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Ts2 @ Tsi ) @ ( vEBT_vebt_maxti @ ( nth_VEBT_VEBTi @ Tsi @ I ) )
        @ ^ [R5: option_nat] :
            ( times_times_assn
            @ ( pure_assn
              @ ( R5
                = ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Ts2 @ I ) ) ) )
            @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Ts2 @ Tsi ) ) ) ) ).

% vebt_maxtilist
thf(fact_5550_vebt__minti__h,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_minti @ Ti )
      @ ^ [R5: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_vebt_mint @ T ) ) ) ) ) ).

% vebt_minti_h
thf(fact_5551_vebt__maxti__h,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_maxti @ Ti )
      @ ^ [R5: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_vebt_maxt @ T ) ) ) ) ) ).

% vebt_maxti_h
thf(fact_5552_htt__vebt__memberi,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] :
      ( time_htt_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X )
      @ ^ [R5: $o] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_vebt_member @ T @ X ) ) ) )
      @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% htt_vebt_memberi
thf(fact_5553_VEBTi_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] :
      ( ( size_size_VEBT_VEBTi @ ( vEBT_Leafi @ X21 @ X222 ) )
      = zero_zero_nat ) ).

% VEBTi.size(4)
thf(fact_5554_VEBTi_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi,X21: $o,X222: $o] :
      ( ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 )
     != ( vEBT_Leafi @ X21 @ X222 ) ) ).

% VEBTi.distinct(1)
thf(fact_5555_VEBTi_Oexhaust,axiom,
    ! [Y: vEBT_VEBTi] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: array_VEBT_VEBTi,X142: vEBT_VEBTi] :
          ( Y
         != ( vEBT_Nodei @ X112 @ X122 @ X132 @ X142 ) )
     => ~ ! [X212: $o,X223: $o] :
            ( Y
           != ( vEBT_Leafi @ X212 @ X223 ) ) ) ).

% VEBTi.exhaust
thf(fact_5556_vebt__assn__raw_Ocases,axiom,
    ! [X: produc3625547720036274456_VEBTi] :
      ( ! [A3: $o,B3: $o,Ai: $o,Bi: $o] :
          ( X
         != ( produc6084888613844515218_VEBTi @ ( vEBT_Leaf @ A3 @ B3 ) @ ( vEBT_Leafi @ Ai @ Bi ) ) )
     => ( ! [Mmo: option4927543243414619207at_nat,Deg2: nat,Tree_list: list_VEBT_VEBT,Summary2: vEBT_VEBT,Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] :
            ( X
           != ( produc6084888613844515218_VEBTi @ ( vEBT_Node @ Mmo @ Deg2 @ Tree_list @ Summary2 ) @ ( vEBT_Nodei @ Mmoi @ Degi @ Tree_array @ Summaryi ) ) )
       => ( ! [V2: option4927543243414619207at_nat,Va3: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT,Vd3: $o,Ve3: $o] :
              ( X
             != ( produc6084888613844515218_VEBTi @ ( vEBT_Node @ V2 @ Va3 @ Vb3 @ Vc3 ) @ ( vEBT_Leafi @ Vd3 @ Ve3 ) ) )
         => ~ ! [Vd3: $o,Ve3: $o,V2: option4927543243414619207at_nat,Va3: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] :
                ( X
               != ( produc6084888613844515218_VEBTi @ ( vEBT_Leaf @ Vd3 @ Ve3 ) @ ( vEBT_Nodei @ V2 @ Va3 @ Vb3 @ Vc3 ) ) ) ) ) ) ).

% vebt_assn_raw.cases
thf(fact_5557_VEBT__internal_OminNulli_Ocases,axiom,
    ! [X: vEBT_VEBTi] :
      ( ( X
       != ( vEBT_Leafi @ $false @ $false ) )
     => ( ! [Uv: $o] :
            ( X
           != ( vEBT_Leafi @ $true @ Uv ) )
       => ( ! [Uu2: $o] :
              ( X
             != ( vEBT_Leafi @ Uu2 @ $true ) )
         => ( ! [Uw2: nat,Ux: array_VEBT_VEBTi,Uy: vEBT_VEBTi] :
                ( X
               != ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uw2 @ Ux @ Uy ) )
           => ~ ! [Uz: product_prod_nat_nat,Va2: nat,Vb: array_VEBT_VEBTi,Vc: vEBT_VEBTi] :
                  ( X
                 != ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ Uz ) @ Va2 @ Vb @ Vc ) ) ) ) ) ) ).

% VEBT_internal.minNulli.cases
thf(fact_5558_vebt__assn__raw_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Ai2: $o,Bi2: $o] :
      ( ( vEBT_vebt_assn_raw @ ( vEBT_Leaf @ A2 @ B2 ) @ ( vEBT_Leafi @ Ai2 @ Bi2 ) )
      = ( pure_assn
        @ ( ( Ai2 = A2 )
          & ( Bi2 = B2 ) ) ) ) ).

% vebt_assn_raw.simps(1)
thf(fact_5559_vebt__assn__raw_Osimps_I3_J,axiom,
    ! [V: option4927543243414619207at_nat,Va: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,Vd2: $o,Ve2: $o] :
      ( ( vEBT_vebt_assn_raw @ ( vEBT_Node @ V @ Va @ Vb2 @ Vc2 ) @ ( vEBT_Leafi @ Vd2 @ Ve2 ) )
      = bot_bot_assn ) ).

% vebt_assn_raw.simps(3)
thf(fact_5560_vebt__minti_Ocases,axiom,
    ! [X: vEBT_VEBTi] :
      ( ! [A3: $o,B3: $o] :
          ( X
         != ( vEBT_Leafi @ A3 @ B3 ) )
     => ( ! [Uu2: nat,Uv: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
            ( X
           != ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] :
              ( X
             != ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) ) ) ) ).

% vebt_minti.cases
thf(fact_5561_vebt__buildupi__rule,axiom,
    ! [N3: nat] : ( time_htt_VEBT_VEBTi @ ( pure_assn @ ( ord_less_nat @ zero_zero_nat @ N3 ) ) @ ( vEBT_vebt_buildupi @ N3 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N3 ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% vebt_buildupi_rule
thf(fact_5562_htt__vebt__buildupi__univ,axiom,
    ! [U: nat,N3: nat] :
      ( ( U
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
     => ( time_htt_VEBT_VEBTi @ one_one_assn @ ( vEBT_vebt_buildupi @ N3 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N3 ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ U ) ) ) ).

% htt_vebt_buildupi_univ
thf(fact_5563_htt__vebt__buildupi_H__univ,axiom,
    ! [U: nat,N3: nat] :
      ( ( U
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
     => ( time_htt_VEBT_VEBTi @ one_one_assn @ ( vEBT_V739175172307565963ildupi @ N3 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N3 ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ U ) ) ) ).

% htt_vebt_buildupi'_univ
thf(fact_5564_vebt__maxti__hT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( time_htt_option_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_maxti @ Ti )
      @ ^ [R5: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_vebt_maxt @ T ) ) ) )
      @ one_one_nat ) ).

% vebt_maxti_hT
thf(fact_5565_vebt__minti__hT,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( time_htt_option_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_minti @ Ti )
      @ ^ [R5: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
          @ ( pure_assn
            @ ( R5
              = ( vEBT_vebt_mint @ T ) ) ) )
      @ one_one_nat ) ).

% vebt_minti_hT
thf(fact_5566_T__vebt__buildupi,axiom,
    ! [N3: nat,H2: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ ( vEBT_V739175172307565963ildupi @ N3 ) @ H2 ) @ ( vEBT_V441764108873111860ildupi @ N3 ) ) ).

% T_vebt_buildupi
thf(fact_5567_htt__vebt__buildupi_H,axiom,
    ! [N3: nat] : ( time_htt_VEBT_VEBTi @ one_one_assn @ ( vEBT_V739175172307565963ildupi @ N3 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N3 ) ) @ ( vEBT_V441764108873111860ildupi @ N3 ) ) ).

% htt_vebt_buildupi'
thf(fact_5568_htt__vebt__buildupi,axiom,
    ! [N3: nat] : ( time_htt_VEBT_VEBTi @ one_one_assn @ ( vEBT_vebt_buildupi @ N3 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N3 ) ) @ ( vEBT_V441764108873111860ildupi @ N3 ) ) ).

% htt_vebt_buildupi
thf(fact_5569_htt__vebt__inserti,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_htt_VEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T @ X ) ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% htt_vebt_inserti
thf(fact_5570_time__replicate,axiom,
    ! [X: heap_T8145700208782473153_VEBTi,C2: nat,N3: nat,H2: heap_e7401611519738050253t_unit] :
      ( ! [H4: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ X @ H4 ) @ C2 )
     => ( ord_less_eq_nat @ ( time_t3534373299052942712_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ N3 @ X ) @ H2 ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C2 ) @ N3 ) ) ) ) ).

% time_replicate
thf(fact_5571_htt__vebt__memberi__invar__vebt,axiom,
    ! [T: vEBT_VEBT,N3: nat,Ti: vEBT_VEBTi,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( time_htt_o @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X )
        @ ^ [R5: $o] :
            ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
            @ ( pure_assn
              @ ( R5
                = ( vEBT_vebt_member @ T @ X ) ) ) )
        @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( nat2 @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ) ) ) ).

% htt_vebt_memberi_invar_vebt
thf(fact_5572_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r2 @ X @ Xa )
        = Y )
     => ( ( ? [A3: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) )
           => ( Y != one_one_nat ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) )
               => ( Y != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                   => ( Y
                     != ( plus_plus_nat @ one_one_nat
                        @ ( if_nat @ ( Xa = Mi2 ) @ zero_zero_nat
                          @ ( if_nat @ ( Xa = Ma2 ) @ zero_zero_nat
                            @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ zero_zero_nat
                              @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa ) @ zero_zero_nat
                                @ ( if_nat
                                  @ ( ( ord_less_nat @ Mi2 @ Xa )
                                    & ( ord_less_nat @ Xa @ Ma2 ) )
                                  @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ zero_zero_nat )
                                  @ zero_zero_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
thf(fact_5573_TBOUND__vebt__inserti,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_T5737551269749752165_VEBTi @ ( vEBT_V3964819847710782039nserti @ T @ Ti @ X ) @ ( if_nat @ ( vEBT_VEBT_minNull @ T ) @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ) ).

% TBOUND_vebt_inserti
thf(fact_5574_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t2 @ X @ Xa )
        = Y )
     => ( ( ? [A3: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( Y != one_one_nat ) )
       => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) )
           => ( Y != one_one_nat ) )
         => ( ( ? [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
               => ( Y != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                   => ( Y
                     != ( if_nat
                        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                          & ~ ( ( Xa = Mi2 )
                              | ( Xa = Ma2 ) ) )
                        @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                        @ one_one_nat ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
thf(fact_5575_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ one_one_nat
        @ ( if_nat @ ( X = Mi ) @ zero_zero_nat
          @ ( if_nat @ ( X = Ma ) @ zero_zero_nat
            @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ zero_zero_nat
              @ ( if_nat @ ( ord_less_nat @ Ma @ X ) @ zero_zero_nat
                @ ( if_nat
                  @ ( ( ord_less_nat @ Mi @ X )
                    & ( ord_less_nat @ X @ Ma ) )
                  @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ zero_zero_nat )
                  @ zero_zero_nat ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
thf(fact_5576_TBOUND__vebt__buildupi,axiom,
    ! [N3: nat] : ( time_T5737551269749752165_VEBTi @ ( vEBT_V739175172307565963ildupi @ N3 ) @ ( vEBT_V441764108873111860ildupi @ N3 ) ) ).

% TBOUND_vebt_buildupi
thf(fact_5577_TBOUND__minNull,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ( time_T5737551269749752165_VEBTi @ ( vEBT_V3964819847710782039nserti @ T @ Ti @ X ) @ one_one_nat ) ) ).

% TBOUND_minNull
thf(fact_5578_TBOUND__replicate,axiom,
    ! [X: heap_T8145700208782473153_VEBTi,C2: nat,N3: nat] :
      ( ( time_T5737551269749752165_VEBTi @ X @ C2 )
     => ( time_T8149879359713347829_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ N3 @ X ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C2 ) @ N3 ) ) ) ) ).

% TBOUND_replicate
thf(fact_5579_TBOUND__replicate,axiom,
    ! [X: heap_Time_Heap_o,C2: nat,N3: nat] :
      ( ( time_TBOUND_o @ X @ C2 )
     => ( time_TBOUND_list_o @ ( vEBT_V2326993469660664182atei_o @ N3 @ X ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C2 ) @ N3 ) ) ) ) ).

% TBOUND_replicate
thf(fact_5580_TBOUND__replicate,axiom,
    ! [X: heap_T2636463487746394924on_nat,C2: nat,N3: nat] :
      ( ( time_T8353473612707095248on_nat @ X @ C2 )
     => ( time_T3808005469503390304on_nat @ ( vEBT_V792416675989592002on_nat @ N3 @ X ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C2 ) @ N3 ) ) ) ) ).

% TBOUND_replicate
thf(fact_5581_TBOUND__replicate,axiom,
    ! [X: heap_Time_Heap_nat,C2: nat,N3: nat] :
      ( ( time_TBOUND_nat @ X @ C2 )
     => ( time_TBOUND_list_nat @ ( vEBT_V7726092123322077554ei_nat @ N3 @ X ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C2 ) @ N3 ) ) ) ) ).

% TBOUND_replicate
thf(fact_5582_TBOUND__buildupi,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( time_T5737551269749752165_VEBTi @ ( vEBT_vebt_buildupi @ N3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% TBOUND_buildupi
thf(fact_5583_nat__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_numeral
thf(fact_5584_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_5585_nat__0__iff,axiom,
    ! [I: int] :
      ( ( ( nat2 @ I )
        = zero_zero_nat )
      = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).

% nat_0_iff
thf(fact_5586_nat__le__0,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ Z )
        = zero_zero_nat ) ) ).

% nat_le_0
thf(fact_5587_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
      = ( ( ord_less_int @ zero_zero_int @ Z )
        & ( ord_less_int @ W @ Z ) ) ) ).

% zless_nat_conj
thf(fact_5588_htt__vebt__inserti__invar__vebt,axiom,
    ! [T: vEBT_VEBT,N3: nat,Ti: vEBT_VEBTi,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( time_htt_VEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T @ X ) ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( nat2 @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ) ) ) ).

% htt_vebt_inserti_invar_vebt
thf(fact_5589_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% zero_less_nat_eq
thf(fact_5590_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_5591_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: int] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 )
        = ( nat2 @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 )
        = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
thf(fact_5592_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N3: nat] :
      ( ( ( nat2 @ Y )
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% nat_eq_numeral_power_cancel_iff
thf(fact_5593_nat__ceiling__le__eq,axiom,
    ! [X: real,A2: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A2 )
      = ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A2 ) ) ) ).

% nat_ceiling_le_eq
thf(fact_5594_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% one_less_nat_eq
thf(fact_5595_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_5596_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N3: nat,A2: int] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 ) @ ( nat2 @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) @ A2 ) ) ).

% numeral_power_less_nat_cancel_iff
thf(fact_5597_nat__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N3: nat] :
      ( ( ord_less_nat @ ( nat2 @ A2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% nat_less_numeral_power_cancel_iff
thf(fact_5598_nat__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N3: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ A2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) ) ) ).

% nat_le_numeral_power_cancel_iff
thf(fact_5599_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N3: nat,A2: int] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N3 ) @ ( nat2 @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N3 ) @ A2 ) ) ).

% numeral_power_le_nat_cancel_iff
thf(fact_5600_htt__vebt__succi,axiom,
    ! [T: vEBT_VEBT,N3: nat,Ti: vEBT_VEBTi,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( time_htt_option_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_succi @ Ti @ X )
        @ ^ [R5: option_nat] :
            ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
            @ ( pure_assn
              @ ( R5
                = ( vEBT_vebt_succ @ T @ X ) ) ) )
        @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( nat2 @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ) ) ) ).

% htt_vebt_succi
thf(fact_5601_htt__vebt__predi,axiom,
    ! [T: vEBT_VEBT,N3: nat,Ti: vEBT_VEBTi,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( time_htt_option_nat @ ( vEBT_vebt_assn_raw @ T @ Ti ) @ ( vEBT_vebt_predi @ Ti @ X )
        @ ^ [R5: option_nat] :
            ( times_times_assn @ ( vEBT_vebt_assn_raw @ T @ Ti )
            @ ( pure_assn
              @ ( R5
                = ( vEBT_vebt_pred @ T @ X ) ) ) )
        @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( nat2 @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ) ) ) ).

% htt_vebt_predi
thf(fact_5602_nat__numeral__as__int,axiom,
    ( numeral_numeral_nat
    = ( ^ [I2: num] : ( nat2 @ ( numeral_numeral_int @ I2 ) ) ) ) ).

% nat_numeral_as_int
thf(fact_5603_nat__zero__as__int,axiom,
    ( zero_zero_nat
    = ( nat2 @ zero_zero_int ) ) ).

% nat_zero_as_int
thf(fact_5604_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).

% nat_mono
thf(fact_5605_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
        = ( ord_less_int @ W @ Z ) ) ) ).

% nat_mono_iff
thf(fact_5606_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( ( ord_less_nat @ M @ ( nat2 @ Z ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).

% zless_nat_eq_int_zless
thf(fact_5607_nat__le__iff,axiom,
    ! [X: int,N3: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N3 )
      = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% nat_le_iff
thf(fact_5608_of__nat__ceiling,axiom,
    ! [R3: real] : ( ord_less_eq_real @ R3 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ R3 ) ) ) ) ).

% of_nat_ceiling
thf(fact_5609_of__nat__ceiling,axiom,
    ! [R3: rat] : ( ord_less_eq_rat @ R3 @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim2889992004027027881ng_rat @ R3 ) ) ) ) ).

% of_nat_ceiling
thf(fact_5610_nat__int__add,axiom,
    ! [A2: nat,B2: nat] :
      ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) )
      = ( plus_plus_nat @ A2 @ B2 ) ) ).

% nat_int_add
thf(fact_5611_int__minus,axiom,
    ! [N3: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N3 @ M ) )
      = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N3 ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).

% int_minus
thf(fact_5612_real__nat__ceiling__ge,axiom,
    ! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).

% real_nat_ceiling_ge
thf(fact_5613_nat__plus__as__int,axiom,
    ( plus_plus_nat
    = ( ^ [A7: nat,B7: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A7 ) @ ( semiri1314217659103216013at_int @ B7 ) ) ) ) ) ).

% nat_plus_as_int
thf(fact_5614_nat__times__as__int,axiom,
    ( times_times_nat
    = ( ^ [A7: nat,B7: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A7 ) @ ( semiri1314217659103216013at_int @ B7 ) ) ) ) ) ).

% nat_times_as_int
thf(fact_5615_nat__minus__as__int,axiom,
    ( minus_minus_nat
    = ( ^ [A7: nat,B7: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A7 ) @ ( semiri1314217659103216013at_int @ B7 ) ) ) ) ) ).

% nat_minus_as_int
thf(fact_5616_nat__div__as__int,axiom,
    ( divide_divide_nat
    = ( ^ [A7: nat,B7: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A7 ) @ ( semiri1314217659103216013at_int @ B7 ) ) ) ) ) ).

% nat_div_as_int
thf(fact_5617_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ W )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
        = ( ord_less_int @ W @ Z ) ) ) ).

% nat_less_eq_zless
thf(fact_5618_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_5619_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_5620_split__nat,axiom,
    ! [P: nat > $o,I: int] :
      ( ( P @ ( nat2 @ I ) )
      = ( ! [N2: nat] :
            ( ( I
              = ( semiri1314217659103216013at_int @ N2 ) )
           => ( P @ N2 ) )
        & ( ( ord_less_int @ I @ zero_zero_int )
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_nat
thf(fact_5621_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( ( ord_less_int @ zero_zero_int @ W )
        | ( ord_less_eq_int @ zero_zero_int @ Z ) )
     => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
        = ( ord_less_eq_int @ W @ Z ) ) ) ).

% nat_le_eq_zle
thf(fact_5622_nat__add__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
       => ( ( nat2 @ ( plus_plus_int @ Z @ Z6 ) )
          = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_5623_le__nat__iff,axiom,
    ! [K: int,N3: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_nat @ N3 @ ( nat2 @ K ) )
        = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N3 ) @ K ) ) ) ).

% le_nat_iff
thf(fact_5624_Suc__as__int,axiom,
    ( suc
    = ( ^ [A7: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A7 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_5625_nat__mult__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
        = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ).

% nat_mult_distrib
thf(fact_5626_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_5627_nat__diff__distrib,axiom,
    ! [Z6: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
     => ( ( ord_less_eq_int @ Z6 @ Z )
       => ( ( nat2 @ ( minus_minus_int @ Z @ Z6 ) )
          = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).

% nat_diff_distrib
thf(fact_5628_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_5629_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_5630_nat__power__eq,axiom,
    ! [Z: int,N3: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( power_power_int @ Z @ N3 ) )
        = ( power_power_nat @ ( nat2 @ Z ) @ N3 ) ) ) ).

% nat_power_eq
thf(fact_5631_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S2 ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
thf(fact_5632_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_5633_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( suc @ ( nat2 @ Z ) )
        = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).

% Suc_nat_eq_nat_zadd1
thf(fact_5634_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_5635_diff__nat__eq__if,axiom,
    ! [Z6: int,Z: int] :
      ( ( ( ord_less_int @ Z6 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
          = ( nat2 @ Z ) ) )
      & ( ~ ( ord_less_int @ Z6 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
          = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z6 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z6 ) ) ) ) ) ) ).

% diff_nat_eq_if
thf(fact_5636_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv2 @ Uw ) @ X )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
thf(fact_5637_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
thf(fact_5638_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
thf(fact_5639_insersimp_H,axiom,
    ! [T: vEBT_VEBT,N3: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ T @ Y ) @ one_one_nat ) ) ) ).

% insersimp'
thf(fact_5640_insertsimp_H,axiom,
    ! [T: vEBT_VEBT,N3: nat,L2: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ( vEBT_VEBT_minNull @ T )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ T @ L2 ) @ one_one_nat ) ) ) ).

% insertsimp'
thf(fact_5641_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
thf(fact_5642_insert_H__bound__height,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ T @ X ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% insert'_bound_height
thf(fact_5643_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
thf(fact_5644_member__bound__height_H,axiom,
    ! [T: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( ord_less_eq_nat @ ( vEBT_T_m_e_m_b_e_r2 @ T @ X ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% member_bound_height'
thf(fact_5645_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
      = ( if_nat
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
          & ~ ( ( X = Mi )
              | ( X = Ma ) ) )
        @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
        @ one_one_nat ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
thf(fact_5646_TBOUND__vebt__memberi,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_TBOUND_o @ ( vEBT_V854960066525838166emberi @ T @ Ti @ X ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% TBOUND_vebt_memberi
thf(fact_5647_httI,axiom,
    ! [P: assn,C2: heap_Time_Heap_o,Q: $o > assn,T: nat] :
      ( ( hoare_hoare_triple_o @ P @ C2 @ Q )
     => ( ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H4 @ As ) )
           => ( ord_less_eq_nat @ ( time_time_o @ C2 @ H4 ) @ T ) )
       => ( time_htt_o @ P @ C2 @ Q @ T ) ) ) ).

% httI
thf(fact_5648_httI,axiom,
    ! [P: assn,C2: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,T: nat] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C2 @ Q )
     => ( ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H4 @ As ) )
           => ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ C2 @ H4 ) @ T ) )
       => ( time_htt_VEBT_VEBTi @ P @ C2 @ Q @ T ) ) ) ).

% httI
thf(fact_5649_httI,axiom,
    ! [P: assn,C2: heap_T2636463487746394924on_nat,Q: option_nat > assn,T: nat] :
      ( ( hoare_7629718768684598413on_nat @ P @ C2 @ Q )
     => ( ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H4 @ As ) )
           => ( ord_less_eq_nat @ ( time_time_option_nat @ C2 @ H4 ) @ T ) )
       => ( time_htt_option_nat @ P @ C2 @ Q @ T ) ) ) ).

% httI
thf(fact_5650_httI,axiom,
    ! [P: assn,C2: heap_Time_Heap_nat,Q: nat > assn,T: nat] :
      ( ( hoare_3067605981109127869le_nat @ P @ C2 @ Q )
     => ( ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H4 @ As ) )
           => ( ord_less_eq_nat @ ( time_time_nat @ C2 @ H4 ) @ T ) )
       => ( time_htt_nat @ P @ C2 @ Q @ T ) ) ) ).

% httI
thf(fact_5651_htt__def,axiom,
    ( time_htt_o
    = ( ^ [P3: assn,C5: heap_Time_Heap_o,Q6: $o > assn,T2: nat] :
          ( ( hoare_hoare_triple_o @ P3 @ C5 @ Q6 )
          & ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( rep_assn @ P3 @ ( produc7507926704131184380et_nat @ H @ As2 ) )
             => ( ord_less_eq_nat @ ( time_time_o @ C5 @ H ) @ T2 ) ) ) ) ) ).

% htt_def
thf(fact_5652_htt__def,axiom,
    ( time_htt_VEBT_VEBTi
    = ( ^ [P3: assn,C5: heap_T8145700208782473153_VEBTi,Q6: vEBT_VEBTi > assn,T2: nat] :
          ( ( hoare_1429296392585015714_VEBTi @ P3 @ C5 @ Q6 )
          & ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( rep_assn @ P3 @ ( produc7507926704131184380et_nat @ H @ As2 ) )
             => ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ C5 @ H ) @ T2 ) ) ) ) ) ).

% htt_def
thf(fact_5653_htt__def,axiom,
    ( time_htt_option_nat
    = ( ^ [P3: assn,C5: heap_T2636463487746394924on_nat,Q6: option_nat > assn,T2: nat] :
          ( ( hoare_7629718768684598413on_nat @ P3 @ C5 @ Q6 )
          & ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( rep_assn @ P3 @ ( produc7507926704131184380et_nat @ H @ As2 ) )
             => ( ord_less_eq_nat @ ( time_time_option_nat @ C5 @ H ) @ T2 ) ) ) ) ) ).

% htt_def
thf(fact_5654_htt__def,axiom,
    ( time_htt_nat
    = ( ^ [P3: assn,C5: heap_Time_Heap_nat,Q6: nat > assn,T2: nat] :
          ( ( hoare_3067605981109127869le_nat @ P3 @ C5 @ Q6 )
          & ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
              ( ( rep_assn @ P3 @ ( produc7507926704131184380et_nat @ H @ As2 ) )
             => ( ord_less_eq_nat @ ( time_time_nat @ C5 @ H ) @ T2 ) ) ) ) ) ).

% htt_def
thf(fact_5655_TBOUND__vebt__succi,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_T8353473612707095248on_nat @ ( vEBT_VEBT_vebt_succi @ T @ Ti @ X ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% TBOUND_vebt_succi
thf(fact_5656_TBOUND__vebt__predi,axiom,
    ! [T: vEBT_VEBT,Ti: vEBT_VEBTi,X: nat] : ( time_T8353473612707095248on_nat @ ( vEBT_VEBT_vebt_predi @ T @ Ti @ X ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T ) ) ) ) ).

% TBOUND_vebt_predi
thf(fact_5657_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_d_e_l_e_t_e @ X @ Xa )
        = Y )
     => ( ( ? [A3: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A3: $o,B3: $o] :
                ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( ( Xa
                = ( suc @ zero_zero_nat ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [A3: $o,B3: $o] :
                  ( X
                  = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ? [N: nat] :
                    ( Xa
                    = ( suc @ ( suc @ N ) ) )
               => ( Y != one_one_nat ) ) )
           => ( ( ? [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList3 @ Summary2 ) )
                 => ( Y != one_one_nat ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList3 @ Summary2 ) )
                   => ( Y != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                       => ( Y
                         != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                            @ ( if_nat
                              @ ( ( ord_less_nat @ Xa @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa ) )
                              @ one_one_nat
                              @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                @ ( if_nat
                                  @ ( ( Xa = Mi2 )
                                    & ( Xa = Ma2 ) )
                                  @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_m_i_n_t @ Summary2 ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) ) @ one_one_nat ) ) @ one_one_nat )
                                    @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                      @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                          @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_d_e_l_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                              @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                                                @ ( if_nat
                                                  @ ( ( ( Xa = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa != Mi2 )
                                                     => ( Xa = Ma2 ) ) )
                                                  @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                                    @ ( plus_plus_nat @ one_one_nat
                                                      @ ( if_nat
                                                        @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                          = none_nat )
                                                        @ one_one_nat
                                                        @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                                                  @ one_one_nat ) ) )
                                            @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                                              @ ( if_nat
                                                @ ( ( ( Xa = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa != Mi2 )
                                                   => ( Xa = Ma2 ) ) )
                                                @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                                @ one_one_nat ) ) ) ) )
                                      @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.elims
thf(fact_5658_TBOUND__vebt__maxti,axiom,
    ! [T: vEBT_VEBTi] : ( time_T8353473612707095248on_nat @ ( vEBT_vebt_maxti @ T ) @ one_one_nat ) ).

% TBOUND_vebt_maxti
thf(fact_5659_TBOUND__vebt__minti,axiom,
    ! [T: vEBT_VEBTi] : ( time_T8353473612707095248on_nat @ ( vEBT_vebt_minti @ T ) @ one_one_nat ) ).

% TBOUND_vebt_minti
thf(fact_5660_TBOUND__minNulli,axiom,
    ! [T: vEBT_VEBTi] : ( time_TBOUND_o @ ( vEBT_VEBT_minNulli @ T ) @ one_one_nat ) ).

% TBOUND_minNulli
thf(fact_5661_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      ( ( vEBT_T_m_a_x_t @ ( vEBT_Leaf @ A2 @ B2 ) )
      = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B2 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(1)
thf(fact_5662_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv2 @ Uw ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
thf(fact_5663_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv2 @ Uw ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
thf(fact_5664_maxt__bound,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_eq_nat @ ( vEBT_T_m_a_x_t @ T ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% maxt_bound
thf(fact_5665_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
      ( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
thf(fact_5666_mint__bound,axiom,
    ! [T: vEBT_VEBT] : ( ord_less_eq_nat @ ( vEBT_T_m_i_n_t @ T ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% mint_bound
thf(fact_5667_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      ( ( vEBT_T_m_i_n_t @ ( vEBT_Leaf @ A2 @ B2 ) )
      = ( plus_plus_nat @ one_one_nat @ ( if_nat @ A2 @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
thf(fact_5668_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
thf(fact_5669_TBOUND__mono,axiom,
    ! [C2: heap_T8145700208782473153_VEBTi,T: nat,T3: nat] :
      ( ( time_T5737551269749752165_VEBTi @ C2 @ T )
     => ( ( ord_less_eq_nat @ T @ T3 )
       => ( time_T5737551269749752165_VEBTi @ C2 @ T3 ) ) ) ).

% TBOUND_mono
thf(fact_5670_TBOUND__mono,axiom,
    ! [C2: heap_Time_Heap_o,T: nat,T3: nat] :
      ( ( time_TBOUND_o @ C2 @ T )
     => ( ( ord_less_eq_nat @ T @ T3 )
       => ( time_TBOUND_o @ C2 @ T3 ) ) ) ).

% TBOUND_mono
thf(fact_5671_TBOUND__mono,axiom,
    ! [C2: heap_T2636463487746394924on_nat,T: nat,T3: nat] :
      ( ( time_T8353473612707095248on_nat @ C2 @ T )
     => ( ( ord_less_eq_nat @ T @ T3 )
       => ( time_T8353473612707095248on_nat @ C2 @ T3 ) ) ) ).

% TBOUND_mono
thf(fact_5672_TBOUND__mono,axiom,
    ! [C2: heap_Time_Heap_nat,T: nat,T3: nat] :
      ( ( time_TBOUND_nat @ C2 @ T )
     => ( ( ord_less_eq_nat @ T @ T3 )
       => ( time_TBOUND_nat @ C2 @ T3 ) ) ) ).

% TBOUND_mono
thf(fact_5673_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_a_x_t @ X )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( Y
             != ( plus_plus_nat @ one_one_nat @ ( if_nat @ B3 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) )
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
thf(fact_5674_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_i_n_t @ X )
        = Y )
     => ( ! [A3: $o] :
            ( ? [B3: $o] :
                ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( Y
             != ( plus_plus_nat @ one_one_nat @ ( if_nat @ A3 @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) )
           => ( Y != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
             => ( Y != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
thf(fact_5675_TBOUNDD,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,T: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_T5737551269749752165_VEBTi @ M @ T )
     => ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ M @ H2 ) @ T ) ) ).

% TBOUNDD
thf(fact_5676_TBOUNDD,axiom,
    ! [M: heap_Time_Heap_o,T: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_TBOUND_o @ M @ T )
     => ( ord_less_eq_nat @ ( time_time_o @ M @ H2 ) @ T ) ) ).

% TBOUNDD
thf(fact_5677_TBOUNDD,axiom,
    ! [M: heap_T2636463487746394924on_nat,T: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_T8353473612707095248on_nat @ M @ T )
     => ( ord_less_eq_nat @ ( time_time_option_nat @ M @ H2 ) @ T ) ) ).

% TBOUNDD
thf(fact_5678_TBOUNDD,axiom,
    ! [M: heap_Time_Heap_nat,T: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_TBOUND_nat @ M @ T )
     => ( ord_less_eq_nat @ ( time_time_nat @ M @ H2 ) @ T ) ) ).

% TBOUNDD
thf(fact_5679_TBOUNDI,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,T: nat] :
      ( ! [H4: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ M @ H4 ) @ T )
     => ( time_T5737551269749752165_VEBTi @ M @ T ) ) ).

% TBOUNDI
thf(fact_5680_TBOUNDI,axiom,
    ! [M: heap_Time_Heap_o,T: nat] :
      ( ! [H4: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_o @ M @ H4 ) @ T )
     => ( time_TBOUND_o @ M @ T ) ) ).

% TBOUNDI
thf(fact_5681_TBOUNDI,axiom,
    ! [M: heap_T2636463487746394924on_nat,T: nat] :
      ( ! [H4: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_option_nat @ M @ H4 ) @ T )
     => ( time_T8353473612707095248on_nat @ M @ T ) ) ).

% TBOUNDI
thf(fact_5682_TBOUNDI,axiom,
    ! [M: heap_Time_Heap_nat,T: nat] :
      ( ! [H4: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_nat @ M @ H4 ) @ T )
     => ( time_TBOUND_nat @ M @ T ) ) ).

% TBOUNDI
thf(fact_5683_TBOUND__def,axiom,
    ( time_T5737551269749752165_VEBTi
    = ( ^ [M5: heap_T8145700208782473153_VEBTi,T2: nat] :
        ! [H: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ M5 @ H ) @ T2 ) ) ) ).

% TBOUND_def
thf(fact_5684_TBOUND__def,axiom,
    ( time_TBOUND_o
    = ( ^ [M5: heap_Time_Heap_o,T2: nat] :
        ! [H: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_o @ M5 @ H ) @ T2 ) ) ) ).

% TBOUND_def
thf(fact_5685_TBOUND__def,axiom,
    ( time_T8353473612707095248on_nat
    = ( ^ [M5: heap_T2636463487746394924on_nat,T2: nat] :
        ! [H: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_option_nat @ M5 @ H ) @ T2 ) ) ) ).

% TBOUND_def
thf(fact_5686_TBOUND__def,axiom,
    ( time_TBOUND_nat
    = ( ^ [M5: heap_Time_Heap_nat,T2: nat] :
        ! [H: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_nat @ M5 @ H ) @ T2 ) ) ) ).

% TBOUND_def
thf(fact_5687_norm__pre__pure__iff__htt_H,axiom,
    ! [B2: $o,P: assn,F2: heap_Time_Heap_o,Q: $o > assn,T: nat] :
      ( ( time_htt_o @ ( times_times_assn @ ( pure_assn @ B2 ) @ P ) @ F2 @ Q @ T )
      = ( B2
       => ( time_htt_o @ P @ F2 @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt'
thf(fact_5688_norm__pre__pure__iff__htt_H,axiom,
    ! [B2: $o,P: assn,F2: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,T: nat] :
      ( ( time_htt_VEBT_VEBTi @ ( times_times_assn @ ( pure_assn @ B2 ) @ P ) @ F2 @ Q @ T )
      = ( B2
       => ( time_htt_VEBT_VEBTi @ P @ F2 @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt'
thf(fact_5689_norm__pre__pure__iff__htt_H,axiom,
    ! [B2: $o,P: assn,F2: heap_T2636463487746394924on_nat,Q: option_nat > assn,T: nat] :
      ( ( time_htt_option_nat @ ( times_times_assn @ ( pure_assn @ B2 ) @ P ) @ F2 @ Q @ T )
      = ( B2
       => ( time_htt_option_nat @ P @ F2 @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt'
thf(fact_5690_norm__pre__pure__iff__htt_H,axiom,
    ! [B2: $o,P: assn,F2: heap_Time_Heap_nat,Q: nat > assn,T: nat] :
      ( ( time_htt_nat @ ( times_times_assn @ ( pure_assn @ B2 ) @ P ) @ F2 @ Q @ T )
      = ( B2
       => ( time_htt_nat @ P @ F2 @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt'
thf(fact_5691_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B2: $o,F2: heap_Time_Heap_o,Q: $o > assn,T: nat] :
      ( ( time_htt_o @ ( times_times_assn @ P @ ( pure_assn @ B2 ) ) @ F2 @ Q @ T )
      = ( B2
       => ( time_htt_o @ P @ F2 @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt
thf(fact_5692_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B2: $o,F2: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,T: nat] :
      ( ( time_htt_VEBT_VEBTi @ ( times_times_assn @ P @ ( pure_assn @ B2 ) ) @ F2 @ Q @ T )
      = ( B2
       => ( time_htt_VEBT_VEBTi @ P @ F2 @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt
thf(fact_5693_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B2: $o,F2: heap_T2636463487746394924on_nat,Q: option_nat > assn,T: nat] :
      ( ( time_htt_option_nat @ ( times_times_assn @ P @ ( pure_assn @ B2 ) ) @ F2 @ Q @ T )
      = ( B2
       => ( time_htt_option_nat @ P @ F2 @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt
thf(fact_5694_norm__pre__pure__iff__htt,axiom,
    ! [P: assn,B2: $o,F2: heap_Time_Heap_nat,Q: nat > assn,T: nat] :
      ( ( time_htt_nat @ ( times_times_assn @ P @ ( pure_assn @ B2 ) ) @ F2 @ Q @ T )
      = ( B2
       => ( time_htt_nat @ P @ F2 @ Q @ T ) ) ) ).

% norm_pre_pure_iff_htt
thf(fact_5695_htt__cons__rule,axiom,
    ! [P2: assn,C2: heap_Time_Heap_o,Q2: $o > assn,T3: nat,P: assn,Q: $o > assn,T: nat] :
      ( ( time_htt_o @ P2 @ C2 @ Q2 @ T3 )
     => ( ( entails @ P @ P2 )
       => ( ! [X4: $o] : ( entails @ ( Q2 @ X4 ) @ ( Q @ X4 ) )
         => ( ( ord_less_eq_nat @ T3 @ T )
           => ( time_htt_o @ P @ C2 @ Q @ T ) ) ) ) ) ).

% htt_cons_rule
thf(fact_5696_htt__cons__rule,axiom,
    ! [P2: assn,C2: heap_T8145700208782473153_VEBTi,Q2: vEBT_VEBTi > assn,T3: nat,P: assn,Q: vEBT_VEBTi > assn,T: nat] :
      ( ( time_htt_VEBT_VEBTi @ P2 @ C2 @ Q2 @ T3 )
     => ( ( entails @ P @ P2 )
       => ( ! [X4: vEBT_VEBTi] : ( entails @ ( Q2 @ X4 ) @ ( Q @ X4 ) )
         => ( ( ord_less_eq_nat @ T3 @ T )
           => ( time_htt_VEBT_VEBTi @ P @ C2 @ Q @ T ) ) ) ) ) ).

% htt_cons_rule
thf(fact_5697_htt__cons__rule,axiom,
    ! [P2: assn,C2: heap_T2636463487746394924on_nat,Q2: option_nat > assn,T3: nat,P: assn,Q: option_nat > assn,T: nat] :
      ( ( time_htt_option_nat @ P2 @ C2 @ Q2 @ T3 )
     => ( ( entails @ P @ P2 )
       => ( ! [X4: option_nat] : ( entails @ ( Q2 @ X4 ) @ ( Q @ X4 ) )
         => ( ( ord_less_eq_nat @ T3 @ T )
           => ( time_htt_option_nat @ P @ C2 @ Q @ T ) ) ) ) ) ).

% htt_cons_rule
thf(fact_5698_htt__cons__rule,axiom,
    ! [P2: assn,C2: heap_Time_Heap_nat,Q2: nat > assn,T3: nat,P: assn,Q: nat > assn,T: nat] :
      ( ( time_htt_nat @ P2 @ C2 @ Q2 @ T3 )
     => ( ( entails @ P @ P2 )
       => ( ! [X4: nat] : ( entails @ ( Q2 @ X4 ) @ ( Q @ X4 ) )
         => ( ( ord_less_eq_nat @ T3 @ T )
           => ( time_htt_nat @ P @ C2 @ Q @ T ) ) ) ) ) ).

% htt_cons_rule
thf(fact_5699_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ one_one_nat
        @ ( if_nat @ ( ord_less_nat @ Ma @ X ) @ one_one_nat
          @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ one_one_nat )
            @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
              @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
                @ ( if_nat
                  @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                     != none_nat )
                    & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                  @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                    @ ( if_nat
                      @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                        = none_nat )
                      @ ( plus_plus_nat @ one_one_nat @ one_one_nat )
                      @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
              @ one_one_nat ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
thf(fact_5700_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ one_one_nat
        @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ one_one_nat
          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) )
            @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
              @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                  @ ( if_nat
                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                       != none_nat )
                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                    @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                      @ ( if_nat
                        @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                          = none_nat )
                        @ one_one_nat
                        @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) )
              @ one_one_nat ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
thf(fact_5701_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_p_r_e_d @ X @ Xa )
        = Y )
     => ( ( ? [Uu2: $o,Uv: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y != one_one_nat ) ) )
       => ( ( ? [A3: $o,Uw2: $o] :
                ( X
                = ( vEBT_Leaf @ A3 @ Uw2 ) )
           => ( ( Xa
                = ( suc @ zero_zero_nat ) )
             => ( Y
               != ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ B3 ) )
               => ( ? [Va3: nat] :
                      ( Xa
                      = ( suc @ ( suc @ Va3 ) ) )
                 => ( Y
                   != ( plus_plus_nat @ one_one_nat @ ( if_nat @ B3 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ) )
           => ( ( ? [Uy: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va2 ) )
               => ( Y != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd @ Ve ) )
                 => ( Y != one_one_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 != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                       => ( Y
                         != ( plus_plus_nat @ one_one_nat
                            @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa ) @ one_one_nat
                              @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ one_one_nat )
                                @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
                                    @ ( if_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                        @ ( if_nat
                                          @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                            = none_nat )
                                          @ ( plus_plus_nat @ one_one_nat @ one_one_nat )
                                          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                                  @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
thf(fact_5702_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_s_u_c_c @ X @ Xa )
        = Y )
     => ( ( ? [Uu2: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ B3 ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y
             != ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
       => ( ( ? [Uv: $o,Uw2: $o] :
                ( X
                = ( vEBT_Leaf @ Uv @ Uw2 ) )
           => ( ? [N: nat] :
                  ( Xa
                  = ( suc @ N ) )
             => ( Y != one_one_nat ) ) )
         => ( ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) )
             => ( Y != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc @ Vd ) )
               => ( Y != one_one_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 != one_one_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                     => ( Y
                       != ( plus_plus_nat @ one_one_nat
                          @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ one_one_nat
                            @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) )
                              @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                  @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                    @ ( if_nat
                                      @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                        @ ( if_nat
                                          @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                            = none_nat )
                                          @ one_one_nat
                                          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
thf(fact_5703_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
        @ ( if_nat
          @ ( ( ord_less_nat @ X @ Mi )
            | ( ord_less_nat @ Ma @ X ) )
          @ one_one_nat
          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
            @ ( if_nat
              @ ( ( X = Mi )
                & ( X = Ma ) )
              @ ( numeral_numeral_nat @ ( bit1 @ one ) )
              @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_m_i_n_t @ Summary ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) ) @ one_one_nat ) ) @ one_one_nat )
                @ ( if_nat @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_i_n_N_u_l_l @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                      @ ( if_nat @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_d_e_l_e_t_e @ 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                            @ ( 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 @ Va ) ) @ ( 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 @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                @ ( plus_plus_nat @ one_one_nat
                                  @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      = none_nat )
                                    @ one_one_nat
                                    @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                              @ one_one_nat ) ) )
                        @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                          @ ( 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 @ Va ) ) @ ( 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 @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                            @ one_one_nat ) ) ) ) )
                  @ one_one_nat ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(7)
thf(fact_5704_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_d_e_l_e_t_e @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ B3 ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A3 @ B3 ) )
                 => ! [N: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ N ) ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ N ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList3 @ Summary2 ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList3 @ Summary2 ) @ Xa ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList3 @ Summary2 ) )
                       => ( ( Y = one_one_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                         => ( ( Y
                              = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                @ ( if_nat
                                  @ ( ( ord_less_nat @ Xa @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa ) )
                                  @ one_one_nat
                                  @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                    @ ( if_nat
                                      @ ( ( Xa = Mi2 )
                                        & ( Xa = Ma2 ) )
                                      @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                      @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_m_i_n_t @ Summary2 ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) ) @ one_one_nat ) ) @ one_one_nat )
                                        @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                          @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                              @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_d_e_l_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                  @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                                                    @ ( if_nat
                                                      @ ( ( ( Xa = Mi2 )
                                                         => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                            = Ma2 ) )
                                                        & ( ( Xa != Mi2 )
                                                         => ( Xa = Ma2 ) ) )
                                                      @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                                        @ ( plus_plus_nat @ one_one_nat
                                                          @ ( if_nat
                                                            @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                              = none_nat )
                                                            @ one_one_nat
                                                            @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                                                      @ one_one_nat ) ) )
                                                @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                                                  @ ( if_nat
                                                    @ ( ( ( Xa = Mi2 )
                                                       => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                          = Ma2 ) )
                                                      & ( ( Xa != Mi2 )
                                                       => ( Xa = Ma2 ) ) )
                                                    @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                                    @ one_one_nat ) ) ) ) )
                                          @ one_one_nat ) ) ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
thf(fact_5705_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X @ Xa )
     => ( ! [Uu2: $o,Uv: $o] :
            ( X
           != ( vEBT_Leaf @ Uu2 @ Uv ) )
       => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) )
               => ( ( Xa = Mi2 )
                  | ( Xa = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                  ( ? [Vc: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc ) )
                 => ( ( Xa = Mi2 )
                    | ( Xa = Ma2 )
                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
thf(fact_5706_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X @ Xa )
        = Y )
     => ( ( ? [Uu2: $o,Uv: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv ) )
         => Y )
       => ( ( ? [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) )
           => Y )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) )
               => ( Y
                  = ( ~ ( ( Xa = Mi2 )
                        | ( Xa = Ma2 ) ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                  ( ? [Vc: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc ) )
                 => ( Y
                    = ( ~ ( ( Xa = Mi2 )
                          | ( Xa = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
                   => ( Y
                      = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
thf(fact_5707_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( ( ( Xa = zero_zero_nat )
               => A3 )
              & ( ( Xa != zero_zero_nat )
               => ( ( ( Xa = one_one_nat )
                   => B3 )
                  & ( Xa = one_one_nat ) ) ) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw2 ) )
         => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
thf(fact_5708_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_V5719532721284313246member @ X @ Xa )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => A3 )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => B3 )
                    & ( Xa = one_one_nat ) ) ) ) )
       => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [S3: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
             => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
thf(fact_5709_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X @ Xa )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( Y
              = ( ~ ( ( ( Xa = zero_zero_nat )
                     => A3 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B3 )
                        & ( Xa = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw2 ) )
           => Y )
         => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
               => ( Y
                  = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
thf(fact_5710_buildup__nothing__in__min__max,axiom,
    ! [N3: nat,X: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N3 ) @ X ) ).

% buildup_nothing_in_min_max
thf(fact_5711_buildup__nothing__in__leaf,axiom,
    ! [N3: nat,X: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N3 ) @ X ) ).

% buildup_nothing_in_leaf
thf(fact_5712_both__member__options__def,axiom,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [T2: vEBT_VEBT,X3: nat] :
          ( ( vEBT_V5719532721284313246member @ T2 @ X3 )
          | ( vEBT_VEBT_membermima @ T2 @ X3 ) ) ) ) ).

% both_member_options_def
thf(fact_5713_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N3: nat,X: nat] :
      ( ( vEBT_invar_vebt @ Tree @ N3 )
     => ( ( vEBT_vebt_member @ Tree @ X )
       => ( ( vEBT_V5719532721284313246member @ Tree @ X )
          | ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).

% member_valid_both_member_options
thf(fact_5714_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv2: $o,Uw: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv2 ) @ Uw ) ).

% VEBT_internal.membermima.simps(1)
thf(fact_5715_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux2: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv2 @ Uw ) @ Ux2 ) ).

% VEBT_internal.naive_member.simps(2)
thf(fact_5716_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) ).

% VEBT_internal.membermima.simps(2)
thf(fact_5717_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,X: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
      = ( ( ( X = zero_zero_nat )
         => A2 )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => B2 )
            & ( X = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
thf(fact_5718_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb2: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb2 ) @ X )
      = ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
thf(fact_5719_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd2 ) @ 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_5720_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy2: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy2 @ ( suc @ V ) @ TreeList @ S2 ) @ 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_5721_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList @ Vc2 ) @ 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_5722_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_VEBT_membermima @ X @ Xa )
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) )
           => ~ ( ( Xa = Mi2 )
                | ( Xa = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [Vc: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc ) )
             => ~ ( ( Xa = Mi2 )
                  | ( Xa = Ma2 )
                  | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
         => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [Vd: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
thf(fact_5723_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_s_u_c_c @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [Uu2: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ B3 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y
                    = ( plus_plus_nat @ one_one_nat @ one_one_nat ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B3 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv: $o,Uw2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uv @ Uw2 ) )
               => ! [N: nat] :
                    ( ( Xa
                      = ( suc @ N ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw2 ) @ ( suc @ N ) ) ) ) ) )
           => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc @ Vd ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc @ Vd ) @ Xa ) ) ) )
               => ( ! [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 = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                       => ( ( Y
                            = ( plus_plus_nat @ one_one_nat
                              @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ one_one_nat
                                @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) )
                                  @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                    @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                        @ ( if_nat
                                          @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                             != none_nat )
                                            & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                            @ ( if_nat
                                              @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                                = none_nat )
                                              @ one_one_nat
                                              @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                    @ one_one_nat ) ) ) ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
thf(fact_5724_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_p_r_e_d @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,Uw2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ Uw2 ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y
                      = ( plus_plus_nat @ one_one_nat @ one_one_nat ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A3 @ B3 ) )
                 => ! [Va3: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ Va3 ) ) )
                     => ( ( Y
                          = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B3 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
             => ( ! [Uy: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va2 ) @ Xa ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd @ Ve ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd @ Ve ) @ Xa ) ) ) )
                 => ( ! [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 = one_one_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                         => ( ( Y
                              = ( plus_plus_nat @ one_one_nat
                                @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa ) @ one_one_nat
                                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ one_one_nat )
                                    @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                      @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
                                        @ ( if_nat
                                          @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                             != none_nat )
                                            & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                            @ ( if_nat
                                              @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                                = none_nat )
                                              @ ( plus_plus_nat @ one_one_nat @ one_one_nat )
                                              @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                                      @ one_one_nat ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
thf(fact_5725_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_V1232361888498592333_e_t_e @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ B3 ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A3 @ B3 ) )
                 => ! [N: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ N ) ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ N ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList3 @ Summary2 ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList3 @ Summary2 ) @ Xa ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList3 @ Summary2 ) )
                       => ( ( Y = one_one_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                         => ( ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa ) )
                               => ( Y = one_one_nat ) )
                              & ( ~ ( ( ord_less_nat @ Xa @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa ) )
                               => ( ( ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                   => ( Y = one_one_nat ) )
                                  & ( ~ ( ( Xa = Mi2 )
                                        & ( Xa = Ma2 ) )
                                   => ( Y
                                      = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) @ ( plus_plus_nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) ) @ one_one_nat ) ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
thf(fact_5726_vebt__delete_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y
                    = ( vEBT_Leaf @ $false @ B3 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ B3 ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y
                      = ( vEBT_Leaf @ A3 @ $false ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A3 @ B3 ) )
                 => ! [N: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ N ) ) )
                     => ( ( Y
                          = ( vEBT_Leaf @ A3 @ B3 ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ N ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
                   => ( ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                     => ( ( Y
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ Xa ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                       => ( ( Y
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                         => ( ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa ) )
                               => ( Y
                                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) ) )
                              & ( ~ ( ( ord_less_nat @ Xa @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa ) )
                               => ( ( ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                   => ( Y
                                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) ) )
                                  & ( ~ ( ( Xa = Mi2 )
                                        & ( Xa = Ma2 ) )
                                   => ( Y
                                      = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                        @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some_P7363390416028606310at_nat
                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa = Mi2 ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa != Mi2 )
                                                     => ( Xa = Ma2 ) ) )
                                                  @ ( if_nat
                                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                      = none_nat )
                                                    @ ( if_nat @ ( Xa = Mi2 ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va3 ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some_P7363390416028606310at_nat
                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa = Mi2 ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa != Mi2 )
                                                     => ( Xa = Ma2 ) ) )
                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va3 ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ Summary2 ) )
                                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
thf(fact_5727_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_s_u_c_c2 @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [Uu2: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ B3 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B3 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv: $o,Uw2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uv @ Uw2 ) )
               => ! [N: nat] :
                    ( ( Xa
                      = ( suc @ N ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw2 ) @ ( suc @ N ) ) ) ) ) )
           => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc @ Vd ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc @ Vd ) @ Xa ) ) ) )
               => ( ! [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 = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                       => ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                             => ( Y = one_one_nat ) )
                            & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                             => ( Y
                                = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                  @ ( if_nat
                                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c2 @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( plus_plus_nat @ ( vEBT_T_s_u_c_c2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                                  @ one_one_nat ) ) ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
thf(fact_5728_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y
                  = ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) @ Xa ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) @ Xa ) ) ) )
             => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
                   => ( ( Y
                        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( Y
                          = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
                            @ ( if_nat
                              @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                & ~ ( ( Xa = Mi2 )
                                    | ( Xa = Ma2 ) ) )
                              @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                              @ one_one_nat ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
thf(fact_5729_vebt__insert_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( ( ( Xa = zero_zero_nat )
                   => ( Y
                      = ( vEBT_Leaf @ $true @ B3 ) ) )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A3 @ $true ) ) )
                      & ( ( Xa != one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) )
               => ( ( Y
                    = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) @ Xa ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
                 => ( ( Y
                      = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) @ Xa ) ) ) )
             => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
                   => ( ( Y
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( Y
                          = ( if_VEBT_VEBT
                            @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                              & ~ ( ( Xa = Mi2 )
                                  | ( Xa = Ma2 ) ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
thf(fact_5730_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_p_r_e_d2 @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,Uw2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ Uw2 ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A3 @ B3 ) )
                 => ! [Va3: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ Va3 ) ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
             => ( ! [Uy: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va2 ) @ Xa ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd @ Ve ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd @ Ve ) @ Xa ) ) ) )
                 => ( ! [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 = one_one_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                         => ( ( ( ( ord_less_nat @ Ma2 @ Xa )
                               => ( Y = one_one_nat ) )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                               => ( Y
                                  = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                    @ ( if_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                      @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d2 @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( vEBT_T_p_r_e_d2 @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                                    @ one_one_nat ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
thf(fact_5731_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t2 @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y = one_one_nat )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S3 ) @ Xa ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) @ Xa ) ) ) )
             => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( Y
                          = ( if_nat
                            @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                              & ~ ( ( Xa = Mi2 )
                                  | ( Xa = Ma2 ) ) )
                            @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                            @ one_one_nat ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
thf(fact_5732_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y
                  = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa ) ) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) )
               => ( ( Y
                    = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) @ Xa ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) )
                 => ( ( Y
                      = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) )
                   => ( ( Y
                        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( Y
                          = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa = Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa = Ma2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
thf(fact_5733_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r2 @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y = one_one_nat )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa ) ) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) @ Xa ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( Y
                          = ( plus_plus_nat @ one_one_nat
                            @ ( if_nat @ ( Xa = Mi2 ) @ zero_zero_nat
                              @ ( if_nat @ ( Xa = Ma2 ) @ zero_zero_nat
                                @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ zero_zero_nat
                                  @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa ) @ zero_zero_nat
                                    @ ( if_nat
                                      @ ( ( ord_less_nat @ Mi2 @ Xa )
                                        & ( ord_less_nat @ Xa @ Ma2 ) )
                                      @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ zero_zero_nat )
                                      @ zero_zero_nat ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
thf(fact_5734_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_vebt_member @ X @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa ) )
               => ( ( ( Xa = zero_zero_nat )
                   => A3 )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => B3 )
                      & ( Xa = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) @ Xa ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) @ Xa ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ Xa ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) )
                       => ( ( Xa != Mi2 )
                         => ( ( Xa != Ma2 )
                           => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                              & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                               => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                  & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                       => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
thf(fact_5735_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_vebt_member @ X @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa ) )
               => ~ ( ( ( Xa = zero_zero_nat )
                     => A3 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B3 )
                        & ( Xa = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) )
                 => ~ ( ( Xa != Mi2 )
                     => ( ( Xa != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
thf(fact_5736_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y
                  = ( ( ( Xa = zero_zero_nat )
                     => A3 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B3 )
                        & ( Xa = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa ) ) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) @ Xa ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) )
                 => ( ~ Y
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) )
                   => ( ~ Y
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( Y
                          = ( ( Xa != Mi2 )
                           => ( ( Xa != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
thf(fact_5737_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y
                  = ( ( ( Xa = zero_zero_nat )
                     => A3 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B3 )
                        & ( Xa = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw2 ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw2 ) @ Xa ) ) ) )
           => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
                 => ( ( Y
                      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
thf(fact_5738_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_V5719532721284313246member @ X @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa ) )
               => ~ ( ( ( Xa = zero_zero_nat )
                     => A3 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B3 )
                        & ( Xa = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa ) )
                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
thf(fact_5739_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa ) )
               => ( ( ( Xa = zero_zero_nat )
                   => A3 )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => B3 )
                      & ( Xa = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv @ Uw2 ) @ Xa ) ) )
           => ~ ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
thf(fact_5740_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv ) )
             => ( ~ Y
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ Xa ) ) ) )
         => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Xa ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) )
                 => ( ( Y
                      = ( ( Xa = Mi2 )
                        | ( Xa = Ma2 ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) @ Xa ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc ) )
                   => ( ( Y
                        = ( ( Xa = Mi2 )
                          | ( Xa = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc ) @ Xa ) ) ) )
               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
                     => ( ( Y
                          = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
thf(fact_5741_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv ) )
             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ Xa ) ) )
         => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Xa ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) @ Xa ) )
                   => ( ( Xa = Mi2 )
                      | ( Xa = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc ) )
                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc ) @ Xa ) )
                     => ( ( Xa = Mi2 )
                        | ( Xa = Ma2 )
                        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ Xa ) )
                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
thf(fact_5742_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_VEBT_membermima @ X @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb ) @ Xa ) )
               => ~ ( ( Xa = Mi2 )
                    | ( Xa = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc ) @ Xa ) )
                 => ~ ( ( Xa = Mi2 )
                      | ( Xa = Ma2 )
                      | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
           => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ Xa ) )
                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
thf(fact_5743_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_5744_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_5745_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_5746_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_5747_foldr__zero,axiom,
    ! [Xs2: list_nat,D2: nat] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs2 ) )
         => ( ord_less_nat @ zero_zero_nat @ ( nth_nat @ Xs2 @ I5 ) ) )
     => ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ ( minus_minus_nat @ ( foldr_nat_nat @ plus_plus_nat @ Xs2 @ D2 ) @ D2 ) ) ) ).

% foldr_zero
thf(fact_5748_add__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = Z )
      = ( ( vEBT_VEBT_add @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z ) ) ) ).

% add_shift
thf(fact_5749_mul__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = Z )
      = ( ( vEBT_VEBT_mul @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z ) ) ) ).

% mul_shift
thf(fact_5750_foldr__one,axiom,
    ! [D2: nat,Ys: list_nat] : ( ord_less_eq_nat @ D2 @ ( foldr_nat_nat @ plus_plus_nat @ Ys @ D2 ) ) ).

% foldr_one
thf(fact_5751_add__def,axiom,
    ( vEBT_VEBT_add
    = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).

% add_def
thf(fact_5752_mul__def,axiom,
    ( vEBT_VEBT_mul
    = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).

% mul_def
thf(fact_5753_foldr__same__int,axiom,
    ! [Xs2: list_nat,Y: nat] :
      ( ! [X4: nat,Y3: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
         => ( ( member_nat @ Y3 @ ( set_nat2 @ Xs2 ) )
           => ( X4 = Y3 ) ) )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
           => ( X4 = Y ) )
       => ( ( foldr_nat_nat @ plus_plus_nat @ Xs2 @ zero_zero_nat )
          = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ Y ) ) ) ) ).

% foldr_same_int
thf(fact_5754_foldr__mono,axiom,
    ! [Xs2: list_nat,Ys: list_nat,C2: nat,D2: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs2 ) )
           => ( ord_less_nat @ ( nth_nat @ Xs2 @ I5 ) @ ( nth_nat @ Ys @ I5 ) ) )
       => ( ( ord_less_eq_nat @ C2 @ D2 )
         => ( ord_less_eq_nat @ ( plus_plus_nat @ ( foldr_nat_nat @ plus_plus_nat @ Xs2 @ C2 ) @ ( size_size_list_nat @ Ys ) ) @ ( foldr_nat_nat @ plus_plus_nat @ Ys @ D2 ) ) ) ) ) ).

% foldr_mono
thf(fact_5755_foldr__length,axiom,
    ! [L2: list_VEBT_VEBT] :
      ( ( foldr_VEBT_VEBT_nat
        @ ^ [X3: vEBT_VEBT] : suc
        @ L2
        @ zero_zero_nat )
      = ( size_s6755466524823107622T_VEBT @ L2 ) ) ).

% foldr_length
thf(fact_5756_foldr__length,axiom,
    ! [L2: list_real] :
      ( ( foldr_real_nat
        @ ^ [X3: real] : suc
        @ L2
        @ zero_zero_nat )
      = ( size_size_list_real @ L2 ) ) ).

% foldr_length
thf(fact_5757_foldr__length,axiom,
    ! [L2: list_o] :
      ( ( foldr_o_nat
        @ ^ [X3: $o] : suc
        @ L2
        @ zero_zero_nat )
      = ( size_size_list_o @ L2 ) ) ).

% foldr_length
thf(fact_5758_foldr__length,axiom,
    ! [L2: list_nat] :
      ( ( foldr_nat_nat
        @ ^ [X3: nat] : suc
        @ L2
        @ zero_zero_nat )
      = ( size_size_list_nat @ L2 ) ) ).

% foldr_length
thf(fact_5759_foldr__length,axiom,
    ! [L2: list_VEBT_VEBTi] :
      ( ( foldr_VEBT_VEBTi_nat
        @ ^ [X3: vEBT_VEBTi] : suc
        @ L2
        @ zero_zero_nat )
      = ( size_s7982070591426661849_VEBTi @ L2 ) ) ).

% foldr_length
thf(fact_5760_foldr__cong,axiom,
    ! [A2: nat,B2: nat,L2: list_nat,K: list_nat,F2: nat > nat > nat,G: nat > nat > nat] :
      ( ( A2 = B2 )
     => ( ( L2 = K )
       => ( ! [A3: nat,X4: nat] :
              ( ( member_nat @ X4 @ ( set_nat2 @ L2 ) )
             => ( ( F2 @ X4 @ A3 )
                = ( G @ X4 @ A3 ) ) )
         => ( ( foldr_nat_nat @ F2 @ L2 @ A2 )
            = ( foldr_nat_nat @ G @ K @ B2 ) ) ) ) ) ).

% foldr_cong
thf(fact_5761_foldr__cong,axiom,
    ! [A2: real,B2: real,L2: list_real,K: list_real,F2: real > real > real,G: real > real > real] :
      ( ( A2 = B2 )
     => ( ( L2 = K )
       => ( ! [A3: real,X4: real] :
              ( ( member_real @ X4 @ ( set_real2 @ L2 ) )
             => ( ( F2 @ X4 @ A3 )
                = ( G @ X4 @ A3 ) ) )
         => ( ( foldr_real_real @ F2 @ L2 @ A2 )
            = ( foldr_real_real @ G @ K @ B2 ) ) ) ) ) ).

% foldr_cong
thf(fact_5762_foldr__length__aux,axiom,
    ! [L2: list_VEBT_VEBT,A2: nat] :
      ( ( foldr_VEBT_VEBT_nat
        @ ^ [X3: vEBT_VEBT] : suc
        @ L2
        @ A2 )
      = ( plus_plus_nat @ A2 @ ( size_s6755466524823107622T_VEBT @ L2 ) ) ) ).

% foldr_length_aux
thf(fact_5763_foldr__length__aux,axiom,
    ! [L2: list_real,A2: nat] :
      ( ( foldr_real_nat
        @ ^ [X3: real] : suc
        @ L2
        @ A2 )
      = ( plus_plus_nat @ A2 @ ( size_size_list_real @ L2 ) ) ) ).

% foldr_length_aux
thf(fact_5764_foldr__length__aux,axiom,
    ! [L2: list_o,A2: nat] :
      ( ( foldr_o_nat
        @ ^ [X3: $o] : suc
        @ L2
        @ A2 )
      = ( plus_plus_nat @ A2 @ ( size_size_list_o @ L2 ) ) ) ).

% foldr_length_aux
thf(fact_5765_foldr__length__aux,axiom,
    ! [L2: list_nat,A2: nat] :
      ( ( foldr_nat_nat
        @ ^ [X3: nat] : suc
        @ L2
        @ A2 )
      = ( plus_plus_nat @ A2 @ ( size_size_list_nat @ L2 ) ) ) ).

% foldr_length_aux
thf(fact_5766_foldr__length__aux,axiom,
    ! [L2: list_VEBT_VEBTi,A2: nat] :
      ( ( foldr_VEBT_VEBTi_nat
        @ ^ [X3: vEBT_VEBTi] : suc
        @ L2
        @ A2 )
      = ( plus_plus_nat @ A2 @ ( size_s7982070591426661849_VEBTi @ L2 ) ) ) ).

% foldr_length_aux
thf(fact_5767_vebt__succ_Osimps_I6_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va: 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 @ Va ) ) @ 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 @ Va ) ) @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( 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 @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_succ.simps(6)
thf(fact_5768_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X: nat,Mi: nat,Va: 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 @ Va ) ) @ 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 @ Va ) ) @ TreeList @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_pred.simps(7)
thf(fact_5769_vebt__succ_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X @ Xa )
        = Y )
     => ( ! [Uu2: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ Uu2 @ B3 ) )
           => ( ( Xa = zero_zero_nat )
             => ~ ( ( B3
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B3
                   => ( Y = none_nat ) ) ) ) )
       => ( ( ? [Uv: $o,Uw2: $o] :
                ( X
                = ( vEBT_Leaf @ Uv @ Uw2 ) )
           => ( ? [N: nat] :
                  ( Xa
                  = ( suc @ N ) )
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) )
             => ( Y != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc @ Vd ) )
               => ( 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,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                     => ~ ( ( ( ord_less_nat @ Xa @ Mi2 )
                           => ( Y
                              = ( some_nat @ Mi2 ) ) )
                          & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                           => ( Y
                              = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                @ ( if_option_nat
                                  @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                     != none_nat )
                                    & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  @ ( if_option_nat
                                    @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
thf(fact_5770_vebt__pred_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_pred @ X @ Xa )
        = Y )
     => ( ( ? [Uu2: $o,Uv: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A3: $o] :
              ( ? [Uw2: $o] :
                  ( X
                  = ( vEBT_Leaf @ A3 @ Uw2 ) )
             => ( ( Xa
                  = ( suc @ zero_zero_nat ) )
               => ~ ( ( A3
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A3
                     => ( Y = none_nat ) ) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ B3 ) )
               => ( ? [Va3: nat] :
                      ( Xa
                      = ( suc @ ( suc @ Va3 ) ) )
                 => ~ ( ( B3
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B3
                       => ( ( A3
                           => ( Y
                              = ( some_nat @ zero_zero_nat ) ) )
                          & ( ~ A3
                           => ( Y = none_nat ) ) ) ) ) ) )
           => ( ( ? [Uy: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va2 ) )
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd @ Ve ) )
                 => ( 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,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                       => ~ ( ( ( ord_less_nat @ Ma2 @ Xa )
                             => ( Y
                                = ( some_nat @ Ma2 ) ) )
                            & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        = none_nat )
                                      @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
thf(fact_5771_vebt__succ_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [Uu2: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ B3 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( ( B3
                     => ( Y
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B3
                     => ( Y = none_nat ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B3 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv: $o,Uw2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uv @ Uw2 ) )
               => ! [N: nat] :
                    ( ( Xa
                      = ( suc @ N ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw2 ) @ ( suc @ N ) ) ) ) ) )
           => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) )
                 => ( ( Y = none_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc @ Vd ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc @ Vd ) @ Xa ) ) ) )
               => ( ! [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 ) @ Xa ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                       => ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                             => ( Y
                                = ( some_nat @ Mi2 ) ) )
                            & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
thf(fact_5772_vebt__pred_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_pred @ X @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,Uw2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ Uw2 ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( ( A3
                       => ( Y
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A3
                       => ( Y = none_nat ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A3 @ B3 ) )
                 => ! [Va3: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ Va3 ) ) )
                     => ( ( ( B3
                           => ( Y
                              = ( some_nat @ one_one_nat ) ) )
                          & ( ~ B3
                           => ( ( A3
                               => ( Y
                                  = ( some_nat @ zero_zero_nat ) ) )
                              & ( ~ A3
                               => ( Y = none_nat ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
             => ( ! [Uy: nat,Uz: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va2 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va2 ) @ Xa ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd @ Ve ) )
                     => ( ( Y = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd @ Ve ) @ Xa ) ) ) )
                 => ( ! [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 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
                         => ( ( ( ( ord_less_nat @ Ma2 @ Xa )
                               => ( Y
                                  = ( some_nat @ Ma2 ) ) )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                               => ( Y
                                  = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                    @ ( if_option_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( if_option_nat
                                        @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                          = none_nat )
                                        @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
thf(fact_5773_lowi__hT,axiom,
    ! [X: nat,N3: nat] :
      ( time_htt_nat @ one_one_assn @ ( vEBT_VEBT_lowi @ X @ N3 )
      @ ^ [R5: nat] :
          ( pure_assn
          @ ( R5
            = ( vEBT_VEBT_low @ X @ N3 ) ) )
      @ one_one_nat ) ).

% lowi_hT
thf(fact_5774_highi__hT,axiom,
    ! [X: nat,N3: nat] :
      ( time_htt_nat @ one_one_assn @ ( vEBT_VEBT_highi @ X @ N3 )
      @ ^ [R5: nat] :
          ( pure_assn
          @ ( R5
            = ( vEBT_VEBT_high @ X @ N3 ) ) )
      @ one_one_nat ) ).

% highi_hT
thf(fact_5775_foldr__same,axiom,
    ! [Xs2: list_real,Y: real] :
      ( ! [X4: real,Y3: real] :
          ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
         => ( ( member_real @ Y3 @ ( set_real2 @ Xs2 ) )
           => ( X4 = Y3 ) ) )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
           => ( X4 = Y ) )
       => ( ( foldr_real_real @ plus_plus_real @ Xs2 @ zero_zero_real )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_real @ Xs2 ) ) @ Y ) ) ) ) ).

% foldr_same
thf(fact_5776_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs2: list_VEBT_VEBT,F2: vEBT_VEBT > nat,Bound: nat,I: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F2 @ X4 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ F2 @ Xs2 ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_5777_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs2: list_real,F2: real > nat,Bound: nat,I: nat] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F2 @ X4 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_real_nat @ F2 @ Xs2 ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_size_list_real @ Xs2 ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_5778_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs2: list_o,F2: $o > nat,Bound: nat,I: nat] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F2 @ X4 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_o_nat @ F2 @ Xs2 ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_5779_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs2: list_nat,F2: nat > nat,Bound: nat,I: nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F2 @ X4 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_nat_nat @ F2 @ Xs2 ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_5780_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs2: list_VEBT_VEBTi,F2: vEBT_VEBTi > nat,Bound: nat,I: nat] :
      ( ! [X4: vEBT_VEBTi] :
          ( ( member_VEBT_VEBTi @ X4 @ ( set_VEBT_VEBTi2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F2 @ X4 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBTi_nat @ F2 @ Xs2 ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_5781_TBOUND__highi,axiom,
    ! [X: nat,N3: nat] : ( time_TBOUND_nat @ ( vEBT_VEBT_highi @ X @ N3 ) @ one_one_nat ) ).

% TBOUND_highi
thf(fact_5782_TBOUND__lowi,axiom,
    ! [X: nat,N3: nat] : ( time_TBOUND_nat @ ( vEBT_VEBT_lowi @ X @ N3 ) @ one_one_nat ) ).

% TBOUND_lowi
thf(fact_5783_foldr0,axiom,
    ! [Xs2: list_real,C2: real,D2: real] :
      ( ( foldr_real_real @ plus_plus_real @ Xs2 @ ( plus_plus_real @ C2 @ D2 ) )
      = ( plus_plus_real @ ( foldr_real_real @ plus_plus_real @ Xs2 @ D2 ) @ C2 ) ) ).

% foldr0
thf(fact_5784_highi__h,axiom,
    ! [X: nat,N3: nat] :
      ( hoare_3067605981109127869le_nat @ one_one_assn @ ( vEBT_VEBT_highi @ X @ N3 )
      @ ^ [R5: nat] :
          ( pure_assn
          @ ( R5
            = ( vEBT_VEBT_high @ X @ N3 ) ) ) ) ).

% highi_h
thf(fact_5785_lowi__h,axiom,
    ! [X: nat,N3: nat] :
      ( hoare_3067605981109127869le_nat @ one_one_assn @ ( vEBT_VEBT_lowi @ X @ N3 )
      @ ^ [R5: nat] :
          ( pure_assn
          @ ( R5
            = ( vEBT_VEBT_low @ X @ N3 ) ) ) ) ).

% lowi_h
thf(fact_5786_length__map,axiom,
    ! [F2: vEBT_VEBT > vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( map_VE8901447254227204932T_VEBT @ F2 @ Xs2 ) )
      = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).

% length_map
thf(fact_5787_length__map,axiom,
    ! [F2: real > vEBT_VEBT,Xs2: list_real] :
      ( ( size_s6755466524823107622T_VEBT @ ( map_real_VEBT_VEBT @ F2 @ Xs2 ) )
      = ( size_size_list_real @ Xs2 ) ) ).

% length_map
thf(fact_5788_length__map,axiom,
    ! [F2: $o > vEBT_VEBT,Xs2: list_o] :
      ( ( size_s6755466524823107622T_VEBT @ ( map_o_VEBT_VEBT @ F2 @ Xs2 ) )
      = ( size_size_list_o @ Xs2 ) ) ).

% length_map
thf(fact_5789_length__map,axiom,
    ! [F2: nat > vEBT_VEBT,Xs2: list_nat] :
      ( ( size_s6755466524823107622T_VEBT @ ( map_nat_VEBT_VEBT @ F2 @ Xs2 ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_map
thf(fact_5790_length__map,axiom,
    ! [F2: vEBT_VEBTi > vEBT_VEBT,Xs2: list_VEBT_VEBTi] :
      ( ( size_s6755466524823107622T_VEBT @ ( map_VE7998069337340375161T_VEBT @ F2 @ Xs2 ) )
      = ( size_s7982070591426661849_VEBTi @ Xs2 ) ) ).

% length_map
thf(fact_5791_length__map,axiom,
    ! [F2: vEBT_VEBT > real,Xs2: list_VEBT_VEBT] :
      ( ( size_size_list_real @ ( map_VEBT_VEBT_real @ F2 @ Xs2 ) )
      = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).

% length_map
thf(fact_5792_length__map,axiom,
    ! [F2: real > real,Xs2: list_real] :
      ( ( size_size_list_real @ ( map_real_real @ F2 @ Xs2 ) )
      = ( size_size_list_real @ Xs2 ) ) ).

% length_map
thf(fact_5793_length__map,axiom,
    ! [F2: $o > real,Xs2: list_o] :
      ( ( size_size_list_real @ ( map_o_real @ F2 @ Xs2 ) )
      = ( size_size_list_o @ Xs2 ) ) ).

% length_map
thf(fact_5794_length__map,axiom,
    ! [F2: nat > real,Xs2: list_nat] :
      ( ( size_size_list_real @ ( map_nat_real @ F2 @ Xs2 ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_map
thf(fact_5795_length__map,axiom,
    ! [F2: vEBT_VEBTi > real,Xs2: list_VEBT_VEBTi] :
      ( ( size_size_list_real @ ( map_VEBT_VEBTi_real @ F2 @ Xs2 ) )
      = ( size_s7982070591426661849_VEBTi @ Xs2 ) ) ).

% length_map
thf(fact_5796_map__eq__conv,axiom,
    ! [F2: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,G: vEBT_VEBT > nat] :
      ( ( ( map_VEBT_VEBT_nat @ F2 @ Xs2 )
        = ( map_VEBT_VEBT_nat @ G @ Xs2 ) )
      = ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( ( F2 @ X3 )
              = ( G @ X3 ) ) ) ) ) ).

% map_eq_conv
thf(fact_5797_map__eq__conv,axiom,
    ! [F2: vEBT_VEBT > real,Xs2: list_VEBT_VEBT,G: vEBT_VEBT > real] :
      ( ( ( map_VEBT_VEBT_real @ F2 @ Xs2 )
        = ( map_VEBT_VEBT_real @ G @ Xs2 ) )
      = ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( ( F2 @ X3 )
              = ( G @ X3 ) ) ) ) ) ).

% map_eq_conv
thf(fact_5798_nth__map,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT,F2: vEBT_VEBT > vEBT_VEBT] :
      ( ( ord_less_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_VEBT_VEBT @ ( map_VE8901447254227204932T_VEBT @ F2 @ Xs2 ) @ N3 )
        = ( F2 @ ( nth_VEBT_VEBT @ Xs2 @ N3 ) ) ) ) ).

% nth_map
thf(fact_5799_nth__map,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT,F2: vEBT_VEBT > vEBT_VEBTi] :
      ( ( ord_less_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_VEBT_VEBTi @ ( map_VE7029150624388687525_VEBTi @ F2 @ Xs2 ) @ N3 )
        = ( F2 @ ( nth_VEBT_VEBT @ Xs2 @ N3 ) ) ) ) ).

% nth_map
thf(fact_5800_nth__map,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT,F2: vEBT_VEBT > nat] :
      ( ( ord_less_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_nat @ ( map_VEBT_VEBT_nat @ F2 @ Xs2 ) @ N3 )
        = ( F2 @ ( nth_VEBT_VEBT @ Xs2 @ N3 ) ) ) ) ).

% nth_map
thf(fact_5801_nth__map,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT,F2: vEBT_VEBT > real] :
      ( ( ord_less_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_real @ ( map_VEBT_VEBT_real @ F2 @ Xs2 ) @ N3 )
        = ( F2 @ ( nth_VEBT_VEBT @ Xs2 @ N3 ) ) ) ) ).

% nth_map
thf(fact_5802_nth__map,axiom,
    ! [N3: nat,Xs2: list_real,F2: real > vEBT_VEBT] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_real @ Xs2 ) )
     => ( ( nth_VEBT_VEBT @ ( map_real_VEBT_VEBT @ F2 @ Xs2 ) @ N3 )
        = ( F2 @ ( nth_real @ Xs2 @ N3 ) ) ) ) ).

% nth_map
thf(fact_5803_nth__map,axiom,
    ! [N3: nat,Xs2: list_real,F2: real > vEBT_VEBTi] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_real @ Xs2 ) )
     => ( ( nth_VEBT_VEBTi @ ( map_real_VEBT_VEBTi @ F2 @ Xs2 ) @ N3 )
        = ( F2 @ ( nth_real @ Xs2 @ N3 ) ) ) ) ).

% nth_map
thf(fact_5804_nth__map,axiom,
    ! [N3: nat,Xs2: list_real,F2: real > nat] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_real @ Xs2 ) )
     => ( ( nth_nat @ ( map_real_nat @ F2 @ Xs2 ) @ N3 )
        = ( F2 @ ( nth_real @ Xs2 @ N3 ) ) ) ) ).

% nth_map
thf(fact_5805_nth__map,axiom,
    ! [N3: nat,Xs2: list_o,F2: $o > vEBT_VEBT] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_VEBT_VEBT @ ( map_o_VEBT_VEBT @ F2 @ Xs2 ) @ N3 )
        = ( F2 @ ( nth_o @ Xs2 @ N3 ) ) ) ) ).

% nth_map
thf(fact_5806_nth__map,axiom,
    ! [N3: nat,Xs2: list_o,F2: $o > vEBT_VEBTi] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_VEBT_VEBTi @ ( map_o_VEBT_VEBTi @ F2 @ Xs2 ) @ N3 )
        = ( F2 @ ( nth_o @ Xs2 @ N3 ) ) ) ) ).

% nth_map
thf(fact_5807_nth__map,axiom,
    ! [N3: nat,Xs2: list_o,F2: $o > nat] :
      ( ( ord_less_nat @ N3 @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_nat @ ( map_o_nat @ F2 @ Xs2 ) @ N3 )
        = ( F2 @ ( nth_o @ Xs2 @ N3 ) ) ) ) ).

% nth_map
thf(fact_5808_map__eq__imp__length__eq,axiom,
    ! [F2: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,G: vEBT_VEBT > nat,Ys: list_VEBT_VEBT] :
      ( ( ( map_VEBT_VEBT_nat @ F2 @ Xs2 )
        = ( map_VEBT_VEBT_nat @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5809_map__eq__imp__length__eq,axiom,
    ! [F2: vEBT_VEBT > real,Xs2: list_VEBT_VEBT,G: vEBT_VEBT > real,Ys: list_VEBT_VEBT] :
      ( ( ( map_VEBT_VEBT_real @ F2 @ Xs2 )
        = ( map_VEBT_VEBT_real @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5810_map__eq__imp__length__eq,axiom,
    ! [F2: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,G: real > nat,Ys: list_real] :
      ( ( ( map_VEBT_VEBT_nat @ F2 @ Xs2 )
        = ( map_real_nat @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_real @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5811_map__eq__imp__length__eq,axiom,
    ! [F2: vEBT_VEBT > real,Xs2: list_VEBT_VEBT,G: real > real,Ys: list_real] :
      ( ( ( map_VEBT_VEBT_real @ F2 @ Xs2 )
        = ( map_real_real @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_real @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5812_map__eq__imp__length__eq,axiom,
    ! [F2: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,G: $o > nat,Ys: list_o] :
      ( ( ( map_VEBT_VEBT_nat @ F2 @ Xs2 )
        = ( map_o_nat @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_o @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5813_map__eq__imp__length__eq,axiom,
    ! [F2: vEBT_VEBT > real,Xs2: list_VEBT_VEBT,G: $o > real,Ys: list_o] :
      ( ( ( map_VEBT_VEBT_real @ F2 @ Xs2 )
        = ( map_o_real @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_o @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5814_map__eq__imp__length__eq,axiom,
    ! [F2: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,G: nat > nat,Ys: list_nat] :
      ( ( ( map_VEBT_VEBT_nat @ F2 @ Xs2 )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5815_map__eq__imp__length__eq,axiom,
    ! [F2: vEBT_VEBT > real,Xs2: list_VEBT_VEBT,G: nat > real,Ys: list_nat] :
      ( ( ( map_VEBT_VEBT_real @ F2 @ Xs2 )
        = ( map_nat_real @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5816_map__eq__imp__length__eq,axiom,
    ! [F2: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,G: vEBT_VEBTi > nat,Ys: list_VEBT_VEBTi] :
      ( ( ( map_VEBT_VEBT_nat @ F2 @ Xs2 )
        = ( map_VEBT_VEBTi_nat @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s7982070591426661849_VEBTi @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5817_map__eq__imp__length__eq,axiom,
    ! [F2: vEBT_VEBT > real,Xs2: list_VEBT_VEBT,G: vEBT_VEBTi > real,Ys: list_VEBT_VEBTi] :
      ( ( ( map_VEBT_VEBT_real @ F2 @ Xs2 )
        = ( map_VEBT_VEBTi_real @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s7982070591426661849_VEBTi @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_5818_list_Omap__cong,axiom,
    ! [X: list_VEBT_VEBT,Ya: list_VEBT_VEBT,F2: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ( X = Ya )
     => ( ! [Z2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ Z2 @ ( set_VEBT_VEBT2 @ Ya ) )
           => ( ( F2 @ Z2 )
              = ( G @ Z2 ) ) )
       => ( ( map_VEBT_VEBT_nat @ F2 @ X )
          = ( map_VEBT_VEBT_nat @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_5819_list_Omap__cong,axiom,
    ! [X: list_VEBT_VEBT,Ya: list_VEBT_VEBT,F2: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ( X = Ya )
     => ( ! [Z2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ Z2 @ ( set_VEBT_VEBT2 @ Ya ) )
           => ( ( F2 @ Z2 )
              = ( G @ Z2 ) ) )
       => ( ( map_VEBT_VEBT_real @ F2 @ X )
          = ( map_VEBT_VEBT_real @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_5820_list_Omap__cong0,axiom,
    ! [X: list_VEBT_VEBT,F2: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ! [Z2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ Z2 @ ( set_VEBT_VEBT2 @ X ) )
         => ( ( F2 @ Z2 )
            = ( G @ Z2 ) ) )
     => ( ( map_VEBT_VEBT_nat @ F2 @ X )
        = ( map_VEBT_VEBT_nat @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_5821_list_Omap__cong0,axiom,
    ! [X: list_VEBT_VEBT,F2: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ! [Z2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ Z2 @ ( set_VEBT_VEBT2 @ X ) )
         => ( ( F2 @ Z2 )
            = ( G @ Z2 ) ) )
     => ( ( map_VEBT_VEBT_real @ F2 @ X )
        = ( map_VEBT_VEBT_real @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_5822_list_Oinj__map__strong,axiom,
    ! [X: list_VEBT_VEBT,Xa: list_VEBT_VEBT,F2: vEBT_VEBT > nat,Fa: vEBT_VEBT > nat] :
      ( ! [Z2: vEBT_VEBT,Za: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ Z2 @ ( set_VEBT_VEBT2 @ X ) )
         => ( ( member_VEBT_VEBT @ Za @ ( set_VEBT_VEBT2 @ Xa ) )
           => ( ( ( F2 @ Z2 )
                = ( Fa @ Za ) )
             => ( Z2 = Za ) ) ) )
     => ( ( ( map_VEBT_VEBT_nat @ F2 @ X )
          = ( map_VEBT_VEBT_nat @ Fa @ Xa ) )
       => ( X = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_5823_list_Oinj__map__strong,axiom,
    ! [X: list_VEBT_VEBT,Xa: list_VEBT_VEBT,F2: vEBT_VEBT > real,Fa: vEBT_VEBT > real] :
      ( ! [Z2: vEBT_VEBT,Za: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ Z2 @ ( set_VEBT_VEBT2 @ X ) )
         => ( ( member_VEBT_VEBT @ Za @ ( set_VEBT_VEBT2 @ Xa ) )
           => ( ( ( F2 @ Z2 )
                = ( Fa @ Za ) )
             => ( Z2 = Za ) ) ) )
     => ( ( ( map_VEBT_VEBT_real @ F2 @ X )
          = ( map_VEBT_VEBT_real @ Fa @ Xa ) )
       => ( X = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_5824_map__ext,axiom,
    ! [Xs2: list_VEBT_VEBT,F2: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ( F2 @ X4 )
            = ( G @ X4 ) ) )
     => ( ( map_VEBT_VEBT_nat @ F2 @ Xs2 )
        = ( map_VEBT_VEBT_nat @ G @ Xs2 ) ) ) ).

% map_ext
thf(fact_5825_map__ext,axiom,
    ! [Xs2: list_VEBT_VEBT,F2: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ( F2 @ X4 )
            = ( G @ X4 ) ) )
     => ( ( map_VEBT_VEBT_real @ F2 @ Xs2 )
        = ( map_VEBT_VEBT_real @ G @ Xs2 ) ) ) ).

% map_ext
thf(fact_5826_map__idI,axiom,
    ! [Xs2: list_int,F2: int > int] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
         => ( ( F2 @ X4 )
            = X4 ) )
     => ( ( map_int_int @ F2 @ Xs2 )
        = Xs2 ) ) ).

% map_idI
thf(fact_5827_map__idI,axiom,
    ! [Xs2: list_complex,F2: complex > complex] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ ( set_complex2 @ Xs2 ) )
         => ( ( F2 @ X4 )
            = X4 ) )
     => ( ( map_complex_complex @ F2 @ Xs2 )
        = Xs2 ) ) ).

% map_idI
thf(fact_5828_map__idI,axiom,
    ! [Xs2: list_VEBT_VEBT,F2: vEBT_VEBT > vEBT_VEBT] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ( F2 @ X4 )
            = X4 ) )
     => ( ( map_VE8901447254227204932T_VEBT @ F2 @ Xs2 )
        = Xs2 ) ) ).

% map_idI
thf(fact_5829_map__idI,axiom,
    ! [Xs2: list_nat,F2: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
         => ( ( F2 @ X4 )
            = X4 ) )
     => ( ( map_nat_nat @ F2 @ Xs2 )
        = Xs2 ) ) ).

% map_idI
thf(fact_5830_map__idI,axiom,
    ! [Xs2: list_real,F2: real > real] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
         => ( ( F2 @ X4 )
            = X4 ) )
     => ( ( map_real_real @ F2 @ Xs2 )
        = Xs2 ) ) ).

% map_idI
thf(fact_5831_map__cong,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT,F2: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ( Xs2 = Ys )
     => ( ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Ys ) )
           => ( ( F2 @ X4 )
              = ( G @ X4 ) ) )
       => ( ( map_VEBT_VEBT_nat @ F2 @ Xs2 )
          = ( map_VEBT_VEBT_nat @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_5832_map__cong,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT,F2: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ( Xs2 = Ys )
     => ( ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Ys ) )
           => ( ( F2 @ X4 )
              = ( G @ X4 ) ) )
       => ( ( map_VEBT_VEBT_real @ F2 @ Xs2 )
          = ( map_VEBT_VEBT_real @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_5833_ex__map__conv,axiom,
    ! [Ys: list_nat,F2: vEBT_VEBT > nat] :
      ( ( ? [Xs: list_VEBT_VEBT] :
            ( Ys
            = ( map_VEBT_VEBT_nat @ F2 @ Xs ) ) )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
           => ? [Y2: vEBT_VEBT] :
                ( X3
                = ( F2 @ Y2 ) ) ) ) ) ).

% ex_map_conv
thf(fact_5834_ex__map__conv,axiom,
    ! [Ys: list_real,F2: vEBT_VEBT > real] :
      ( ( ? [Xs: list_VEBT_VEBT] :
            ( Ys
            = ( map_VEBT_VEBT_real @ F2 @ Xs ) ) )
      = ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_real2 @ Ys ) )
           => ? [Y2: vEBT_VEBT] :
                ( X3
                = ( F2 @ Y2 ) ) ) ) ) ).

% ex_map_conv
thf(fact_5835_map__eq__nth__eq,axiom,
    ! [F2: vEBT_VEBT > nat,L2: list_VEBT_VEBT,L3: list_VEBT_VEBT,I: nat] :
      ( ( ( map_VEBT_VEBT_nat @ F2 @ L2 )
        = ( map_VEBT_VEBT_nat @ F2 @ L3 ) )
     => ( ( F2 @ ( nth_VEBT_VEBT @ L2 @ I ) )
        = ( F2 @ ( nth_VEBT_VEBT @ L3 @ I ) ) ) ) ).

% map_eq_nth_eq
thf(fact_5836_map__eq__nth__eq,axiom,
    ! [F2: vEBT_VEBT > real,L2: list_VEBT_VEBT,L3: list_VEBT_VEBT,I: nat] :
      ( ( ( map_VEBT_VEBT_real @ F2 @ L2 )
        = ( map_VEBT_VEBT_real @ F2 @ L3 ) )
     => ( ( F2 @ ( nth_VEBT_VEBT @ L2 @ I ) )
        = ( F2 @ ( nth_VEBT_VEBT @ L3 @ I ) ) ) ) ).

% map_eq_nth_eq
thf(fact_5837_map__update,axiom,
    ! [F2: vEBT_VEBTi > vEBT_VEBTi,Xs2: list_VEBT_VEBTi,K: nat,Y: vEBT_VEBTi] :
      ( ( map_VE483055756984248624_VEBTi @ F2 @ ( list_u6098035379799741383_VEBTi @ Xs2 @ K @ Y ) )
      = ( list_u6098035379799741383_VEBTi @ ( map_VE483055756984248624_VEBTi @ F2 @ Xs2 ) @ K @ ( F2 @ Y ) ) ) ).

% map_update
thf(fact_5838_map__update,axiom,
    ! [F2: vEBT_VEBTi > vEBT_VEBT,Xs2: list_VEBT_VEBTi,K: nat,Y: vEBT_VEBTi] :
      ( ( map_VE7998069337340375161T_VEBT @ F2 @ ( list_u6098035379799741383_VEBTi @ Xs2 @ K @ Y ) )
      = ( list_u1324408373059187874T_VEBT @ ( map_VE7998069337340375161T_VEBT @ F2 @ Xs2 ) @ K @ ( F2 @ Y ) ) ) ).

% map_update
thf(fact_5839_map__update,axiom,
    ! [F2: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,K: nat,Y: vEBT_VEBT] :
      ( ( map_VEBT_VEBT_nat @ F2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ K @ Y ) )
      = ( list_update_nat @ ( map_VEBT_VEBT_nat @ F2 @ Xs2 ) @ K @ ( F2 @ Y ) ) ) ).

% map_update
thf(fact_5840_map__update,axiom,
    ! [F2: vEBT_VEBT > real,Xs2: list_VEBT_VEBT,K: nat,Y: vEBT_VEBT] :
      ( ( map_VEBT_VEBT_real @ F2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ K @ Y ) )
      = ( list_update_real @ ( map_VEBT_VEBT_real @ F2 @ Xs2 ) @ K @ ( F2 @ Y ) ) ) ).

% map_update
thf(fact_5841_map__update,axiom,
    ! [F2: vEBT_VEBT > vEBT_VEBTi,Xs2: list_VEBT_VEBT,K: nat,Y: vEBT_VEBT] :
      ( ( map_VE7029150624388687525_VEBTi @ F2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ K @ Y ) )
      = ( list_u6098035379799741383_VEBTi @ ( map_VE7029150624388687525_VEBTi @ F2 @ Xs2 ) @ K @ ( F2 @ Y ) ) ) ).

% map_update
thf(fact_5842_map__update,axiom,
    ! [F2: vEBT_VEBT > vEBT_VEBT,Xs2: list_VEBT_VEBT,K: nat,Y: vEBT_VEBT] :
      ( ( map_VE8901447254227204932T_VEBT @ F2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ K @ Y ) )
      = ( list_u1324408373059187874T_VEBT @ ( map_VE8901447254227204932T_VEBT @ F2 @ Xs2 ) @ K @ ( F2 @ Y ) ) ) ).

% map_update
thf(fact_5843_map__upd__eq,axiom,
    ! [I: nat,L2: list_VEBT_VEBT,F2: vEBT_VEBT > nat,X: vEBT_VEBT] :
      ( ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) )
       => ( ( F2 @ ( nth_VEBT_VEBT @ L2 @ I ) )
          = ( F2 @ X ) ) )
     => ( ( map_VEBT_VEBT_nat @ F2 @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ X ) )
        = ( map_VEBT_VEBT_nat @ F2 @ L2 ) ) ) ).

% map_upd_eq
thf(fact_5844_map__upd__eq,axiom,
    ! [I: nat,L2: list_VEBT_VEBT,F2: vEBT_VEBT > real,X: vEBT_VEBT] :
      ( ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) )
       => ( ( F2 @ ( nth_VEBT_VEBT @ L2 @ I ) )
          = ( F2 @ X ) ) )
     => ( ( map_VEBT_VEBT_real @ F2 @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ X ) )
        = ( map_VEBT_VEBT_real @ F2 @ L2 ) ) ) ).

% map_upd_eq
thf(fact_5845_VEBT__internal_Ocnt_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_cnt2 @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_cnt2 @ Summary ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_cnt2 @ TreeList ) @ zero_zero_nat ) ) ) ).

% VEBT_internal.cnt'.simps(2)
thf(fact_5846_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_cnt2 @ X )
        = Y )
     => ( ( ? [A3: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( Y != one_one_nat ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( Y
               != ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_cnt2 @ Summary2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_cnt2 @ TreeList3 ) @ zero_zero_nat ) ) ) ) ) ) ).

% VEBT_internal.cnt'.elims
thf(fact_5847_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_space2 @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_space2 @ Summary ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space2 @ TreeList ) @ zero_zero_nat ) ) ) ).

% VEBT_internal.space'.simps(2)
thf(fact_5848_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_space2 @ X )
        = Y )
     => ( ( ? [A3: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( Y
           != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( Y
               != ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_space2 @ Summary2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space2 @ TreeList3 ) @ zero_zero_nat ) ) ) ) ) ) ).

% VEBT_internal.space'.elims
thf(fact_5849_VEBT__internal_Ospace_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_space @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_space @ Summary ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space @ TreeList ) @ zero_zero_nat ) ) ) ).

% VEBT_internal.space.simps(2)
thf(fact_5850_VEBT__internal_Ospace_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_space @ X )
        = Y )
     => ( ( ? [A3: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( Y
           != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( Y
               != ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_space @ Summary2 ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space @ TreeList3 ) @ zero_zero_nat ) ) ) ) ) ) ).

% VEBT_internal.space.elims
thf(fact_5851_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs2: list_VEBT_VEBT,F2: vEBT_VEBT > real,Bound: real,I: real] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F2 @ X4 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ F2 @ Xs2 ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_5852_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs2: list_real,F2: real > real,Bound: real,I: real] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F2 @ X4 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_real_real @ F2 @ Xs2 ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_real @ Xs2 ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_5853_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs2: list_o,F2: $o > real,Bound: real,I: real] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F2 @ X4 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_o_real @ F2 @ Xs2 ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_o @ Xs2 ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_5854_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs2: list_nat,F2: nat > real,Bound: real,I: real] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F2 @ X4 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_nat_real @ F2 @ Xs2 ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_nat @ Xs2 ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_5855_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs2: list_VEBT_VEBTi,F2: vEBT_VEBTi > real,Bound: real,I: real] :
      ( ! [X4: vEBT_VEBTi] :
          ( ( member_VEBT_VEBTi @ X4 @ ( set_VEBT_VEBTi2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F2 @ X4 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBTi_real @ F2 @ Xs2 ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_s7982070591426661849_VEBTi @ Xs2 ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_5856_real__nat__list,axiom,
    ! [F2: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,C2: nat] :
      ( ( semiri5074537144036343181t_real @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ F2 @ Xs2 ) @ C2 ) )
      = ( foldr_real_real @ plus_plus_real
        @ ( map_VEBT_VEBT_real
          @ ^ [X3: vEBT_VEBT] : ( semiri5074537144036343181t_real @ ( F2 @ X3 ) )
          @ Xs2 )
        @ ( semiri5074537144036343181t_real @ C2 ) ) ) ).

% real_nat_list
thf(fact_5857_f__g__map__foldr__bound,axiom,
    ! [Xs2: list_VEBT_VEBT,F2: vEBT_VEBT > real,C2: real,G: vEBT_VEBT > real,D2: real] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F2 @ X4 ) @ ( times_times_real @ C2 @ ( G @ X4 ) ) ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ F2 @ Xs2 ) @ D2 ) @ ( plus_plus_real @ ( times_times_real @ C2 @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ G @ Xs2 ) @ zero_zero_real ) ) @ D2 ) ) ) ).

% f_g_map_foldr_bound
thf(fact_5858_f__g__map__foldr__bound,axiom,
    ! [Xs2: list_nat,F2: nat > real,C2: real,G: nat > real,D2: real] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F2 @ X4 ) @ ( times_times_real @ C2 @ ( G @ X4 ) ) ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_nat_real @ F2 @ Xs2 ) @ D2 ) @ ( plus_plus_real @ ( times_times_real @ C2 @ ( foldr_real_real @ plus_plus_real @ ( map_nat_real @ G @ Xs2 ) @ zero_zero_real ) ) @ D2 ) ) ) ).

% f_g_map_foldr_bound
thf(fact_5859_f__g__map__foldr__bound,axiom,
    ! [Xs2: list_real,F2: real > real,C2: real,G: real > real,D2: real] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F2 @ X4 ) @ ( times_times_real @ C2 @ ( G @ X4 ) ) ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_real_real @ F2 @ Xs2 ) @ D2 ) @ ( plus_plus_real @ ( times_times_real @ C2 @ ( foldr_real_real @ plus_plus_real @ ( map_real_real @ G @ Xs2 ) @ zero_zero_real ) ) @ D2 ) ) ) ).

% f_g_map_foldr_bound
thf(fact_5860_VEBTi_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] :
      ( ( size_size_VEBT_VEBTi @ ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( size_a6397454172108246045_VEBTi @ size_size_VEBT_VEBTi @ X13 ) @ ( size_size_VEBT_VEBTi @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% VEBTi.size(3)
thf(fact_5861_vebt__memberi__refines,axiom,
    ! [Ti: vEBT_VEBTi,X: nat,T: vEBT_VEBT] : ( refine_Imp_refines_o @ ( vEBT_vebt_memberi @ Ti @ X ) @ ( vEBT_V854960066525838166emberi @ T @ Ti @ X ) ) ).

% vebt_memberi_refines
thf(fact_5862_refines__replicate,axiom,
    ! [F2: heap_Time_Heap_o,F5: heap_Time_Heap_o,N3: nat] :
      ( ( refine_Imp_refines_o @ F2 @ F5 )
     => ( refine5896690332125372649list_o @ ( vEBT_V2326993469660664182atei_o @ N3 @ F2 ) @ ( vEBT_V2326993469660664182atei_o @ N3 @ F5 ) ) ) ).

% refines_replicate
thf(fact_5863_refines__replicate,axiom,
    ! [F2: heap_T2636463487746394924on_nat,F5: heap_T2636463487746394924on_nat,N3: nat] :
      ( ( refine7594492741263601813on_nat @ F2 @ F5 )
     => ( refine1935026298455697829on_nat @ ( vEBT_V792416675989592002on_nat @ N3 @ F2 ) @ ( vEBT_V792416675989592002on_nat @ N3 @ F5 ) ) ) ).

% refines_replicate
thf(fact_5864_refines__replicate,axiom,
    ! [F2: heap_T8145700208782473153_VEBTi,F5: heap_T8145700208782473153_VEBTi,N3: nat] :
      ( ( refine5565527176597971370_VEBTi @ F2 @ F5 )
     => ( refine3700189196150522554_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ N3 @ F2 ) @ ( vEBT_V1859673955506687831_VEBTi @ N3 @ F5 ) ) ) ).

% refines_replicate
thf(fact_5865_listsum__bound,axiom,
    ! [Xs2: list_int,F2: int > real,Y: real] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X4 ) ) )
     => ( ord_less_eq_real @ Y @ ( foldr_real_real @ plus_plus_real @ ( map_int_real @ F2 @ Xs2 ) @ Y ) ) ) ).

% listsum_bound
thf(fact_5866_listsum__bound,axiom,
    ! [Xs2: list_complex,F2: complex > real,Y: real] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ ( set_complex2 @ Xs2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X4 ) ) )
     => ( ord_less_eq_real @ Y @ ( foldr_real_real @ plus_plus_real @ ( map_complex_real @ F2 @ Xs2 ) @ Y ) ) ) ).

% listsum_bound
thf(fact_5867_listsum__bound,axiom,
    ! [Xs2: list_VEBT_VEBT,F2: vEBT_VEBT > real,Y: real] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X4 ) ) )
     => ( ord_less_eq_real @ Y @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ F2 @ Xs2 ) @ Y ) ) ) ).

% listsum_bound
thf(fact_5868_listsum__bound,axiom,
    ! [Xs2: list_nat,F2: nat > real,Y: real] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X4 ) ) )
     => ( ord_less_eq_real @ Y @ ( foldr_real_real @ plus_plus_real @ ( map_nat_real @ F2 @ Xs2 ) @ Y ) ) ) ).

% listsum_bound
thf(fact_5869_listsum__bound,axiom,
    ! [Xs2: list_real,F2: real > real,Y: real] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X4 ) ) )
     => ( ord_less_eq_real @ Y @ ( foldr_real_real @ plus_plus_real @ ( map_real_real @ F2 @ Xs2 ) @ Y ) ) ) ).

% listsum_bound
thf(fact_5870_refines__case__VEBTi,axiom,
    ! [Ti: vEBT_VEBTi,Ti2: vEBT_VEBTi,F1: $o > $o > heap_Time_Heap_o,F12: $o > $o > heap_Time_Heap_o,F22: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_o,F23: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_o] :
      ( ( Ti = Ti2 )
     => ( ! [A3: $o,B3: $o] : ( refine_Imp_refines_o @ ( F1 @ A3 @ B3 ) @ ( F12 @ A3 @ B3 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] : ( refine_Imp_refines_o @ ( F22 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) @ ( F23 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) )
         => ( refine_Imp_refines_o @ ( vEBT_c6104975476656191286Heap_o @ F22 @ F1 @ Ti ) @ ( vEBT_c6104975476656191286Heap_o @ F23 @ F12 @ Ti2 ) ) ) ) ) ).

% refines_case_VEBTi
thf(fact_5871_refines__case__VEBTi,axiom,
    ! [Ti: vEBT_VEBTi,Ti2: vEBT_VEBTi,F1: $o > $o > heap_T2636463487746394924on_nat,F12: $o > $o > heap_T2636463487746394924on_nat,F22: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat,F23: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat] :
      ( ( Ti = Ti2 )
     => ( ! [A3: $o,B3: $o] : ( refine7594492741263601813on_nat @ ( F1 @ A3 @ B3 ) @ ( F12 @ A3 @ B3 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] : ( refine7594492741263601813on_nat @ ( F22 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) @ ( F23 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) )
         => ( refine7594492741263601813on_nat @ ( vEBT_c6250501799366334488on_nat @ F22 @ F1 @ Ti ) @ ( vEBT_c6250501799366334488on_nat @ F23 @ F12 @ Ti2 ) ) ) ) ) ).

% refines_case_VEBTi
thf(fact_5872_refines__case__VEBTi,axiom,
    ! [Ti: vEBT_VEBTi,Ti2: vEBT_VEBTi,F1: $o > $o > heap_T8145700208782473153_VEBTi,F12: $o > $o > heap_T8145700208782473153_VEBTi,F22: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi,F23: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi] :
      ( ( Ti = Ti2 )
     => ( ! [A3: $o,B3: $o] : ( refine5565527176597971370_VEBTi @ ( F1 @ A3 @ B3 ) @ ( F12 @ A3 @ B3 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] : ( refine5565527176597971370_VEBTi @ ( F22 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) @ ( F23 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) )
         => ( refine5565527176597971370_VEBTi @ ( vEBT_c6028912655521741485_VEBTi @ F22 @ F1 @ Ti ) @ ( vEBT_c6028912655521741485_VEBTi @ F23 @ F12 @ Ti2 ) ) ) ) ) ).

% refines_case_VEBTi
thf(fact_5873_VEBT__internal_Ocnt_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_cnt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) )
      = ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( vEBT_VEBT_cnt @ Summary ) ) @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ vEBT_VEBT_cnt @ TreeList ) @ zero_zero_real ) ) ) ).

% VEBT_internal.cnt.simps(2)
thf(fact_5874_VEBT__internal_Ocnt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: real] :
      ( ( ( vEBT_VEBT_cnt @ X )
        = Y )
     => ( ( ? [A3: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( Y != one_one_real ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( Y
               != ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( vEBT_VEBT_cnt @ Summary2 ) ) @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ vEBT_VEBT_cnt @ TreeList3 ) @ zero_zero_real ) ) ) ) ) ) ).

% VEBT_internal.cnt.elims
thf(fact_5875_VEBTi_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] :
      ( ( vEBT_size_VEBTi @ ( vEBT_Nodei @ X11 @ X12 @ X13 @ X14 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( size_a6397454172108246045_VEBTi @ vEBT_size_VEBTi @ X13 ) @ ( vEBT_size_VEBTi @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% VEBTi.size_gen(1)
thf(fact_5876_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_5877_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_5878_mult__le__cancel__iff2,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_5879_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_5880_mult__le__cancel__iff2,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_5881_mult__le__cancel__iff1,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_5882_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_5883_mult__le__cancel__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_5884_vebt__predi__refines,axiom,
    ! [Ti: vEBT_VEBTi,X: nat,T: vEBT_VEBT] : ( refine7594492741263601813on_nat @ ( vEBT_vebt_predi @ Ti @ X ) @ ( vEBT_VEBT_vebt_predi @ T @ Ti @ X ) ) ).

% vebt_predi_refines
thf(fact_5885_vebt__succi__refines,axiom,
    ! [Ti: vEBT_VEBTi,X: nat,T: vEBT_VEBT] : ( refine7594492741263601813on_nat @ ( vEBT_vebt_succi @ Ti @ X ) @ ( vEBT_VEBT_vebt_succi @ T @ Ti @ X ) ) ).

% vebt_succi_refines
thf(fact_5886_vebt__buildupi__refines,axiom,
    ! [N3: nat] : ( refine5565527176597971370_VEBTi @ ( vEBT_vebt_buildupi @ N3 ) @ ( vEBT_V739175172307565963ildupi @ N3 ) ) ).

% vebt_buildupi_refines
thf(fact_5887_vebt__inserti__refines,axiom,
    ! [Ti: vEBT_VEBTi,X: nat,T: vEBT_VEBT] : ( refine5565527176597971370_VEBTi @ ( vEBT_vebt_inserti @ Ti @ X ) @ ( vEBT_V3964819847710782039nserti @ T @ Ti @ X ) ) ).

% vebt_inserti_refines
thf(fact_5888_round__0,axiom,
    ( ( archim8280529875227126926d_real @ zero_zero_real )
    = zero_zero_int ) ).

% round_0
thf(fact_5889_round__0,axiom,
    ( ( archim7778729529865785530nd_rat @ zero_zero_rat )
    = zero_zero_int ) ).

% round_0
thf(fact_5890_round__numeral,axiom,
    ! [N3: num] :
      ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N3 ) )
      = ( numeral_numeral_int @ N3 ) ) ).

% round_numeral
thf(fact_5891_round__numeral,axiom,
    ! [N3: num] :
      ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N3 ) )
      = ( numeral_numeral_int @ N3 ) ) ).

% round_numeral
thf(fact_5892_TBOUND__VEBT__case,axiom,
    ! [Ti: vEBT_VEBTi,F2: $o > $o > heap_T8145700208782473153_VEBTi,Bnd: $o > $o > nat,F5: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi,Bnd2: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > nat] :
      ( ! [A3: $o,B3: $o] :
          ( ( Ti
            = ( vEBT_Leafi @ A3 @ B3 ) )
         => ( time_T5737551269749752165_VEBTi @ ( F2 @ A3 @ B3 ) @ ( Bnd @ A3 @ B3 ) ) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
            ( ( Ti
              = ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray @ Summary2 ) )
           => ( time_T5737551269749752165_VEBTi @ ( F5 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) @ ( Bnd2 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) ) )
       => ( time_T5737551269749752165_VEBTi @ ( vEBT_c6028912655521741485_VEBTi @ F5 @ F2 @ Ti ) @ ( vEBT_case_VEBTi_nat @ Bnd2 @ Bnd @ Ti ) ) ) ) ).

% TBOUND_VEBT_case
thf(fact_5893_TBOUND__VEBT__case,axiom,
    ! [Ti: vEBT_VEBTi,F2: $o > $o > heap_Time_Heap_o,Bnd: $o > $o > nat,F5: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_o,Bnd2: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > nat] :
      ( ! [A3: $o,B3: $o] :
          ( ( Ti
            = ( vEBT_Leafi @ A3 @ B3 ) )
         => ( time_TBOUND_o @ ( F2 @ A3 @ B3 ) @ ( Bnd @ A3 @ B3 ) ) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
            ( ( Ti
              = ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray @ Summary2 ) )
           => ( time_TBOUND_o @ ( F5 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) @ ( Bnd2 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) ) )
       => ( time_TBOUND_o @ ( vEBT_c6104975476656191286Heap_o @ F5 @ F2 @ Ti ) @ ( vEBT_case_VEBTi_nat @ Bnd2 @ Bnd @ Ti ) ) ) ) ).

% TBOUND_VEBT_case
thf(fact_5894_TBOUND__VEBT__case,axiom,
    ! [Ti: vEBT_VEBTi,F2: $o > $o > heap_T2636463487746394924on_nat,Bnd: $o > $o > nat,F5: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat,Bnd2: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > nat] :
      ( ! [A3: $o,B3: $o] :
          ( ( Ti
            = ( vEBT_Leafi @ A3 @ B3 ) )
         => ( time_T8353473612707095248on_nat @ ( F2 @ A3 @ B3 ) @ ( Bnd @ A3 @ B3 ) ) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
            ( ( Ti
              = ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray @ Summary2 ) )
           => ( time_T8353473612707095248on_nat @ ( F5 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) @ ( Bnd2 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) ) )
       => ( time_T8353473612707095248on_nat @ ( vEBT_c6250501799366334488on_nat @ F5 @ F2 @ Ti ) @ ( vEBT_case_VEBTi_nat @ Bnd2 @ Bnd @ Ti ) ) ) ) ).

% TBOUND_VEBT_case
thf(fact_5895_TBOUND__VEBT__case,axiom,
    ! [Ti: vEBT_VEBTi,F2: $o > $o > heap_Time_Heap_nat,Bnd: $o > $o > nat,F5: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_nat,Bnd2: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > nat] :
      ( ! [A3: $o,B3: $o] :
          ( ( Ti
            = ( vEBT_Leafi @ A3 @ B3 ) )
         => ( time_TBOUND_nat @ ( F2 @ A3 @ B3 ) @ ( Bnd @ A3 @ B3 ) ) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
            ( ( Ti
              = ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray @ Summary2 ) )
           => ( time_TBOUND_nat @ ( F5 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) @ ( Bnd2 @ Info2 @ Deg2 @ TreeArray @ Summary2 ) ) )
       => ( time_TBOUND_nat @ ( vEBT_c1335663792808957512ap_nat @ F5 @ F2 @ Ti ) @ ( vEBT_case_VEBTi_nat @ Bnd2 @ Bnd @ Ti ) ) ) ) ).

% TBOUND_VEBT_case
thf(fact_5896_round__mono,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y ) ) ) ).

% round_mono
thf(fact_5897_VEBTi_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] :
      ( ( vEBT_size_VEBTi @ ( vEBT_Leafi @ X21 @ X222 ) )
      = zero_zero_nat ) ).

% VEBTi.size_gen(2)
thf(fact_5898_mult__less__iff1,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
        = ( ord_less_real @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_5899_mult__less__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
        = ( ord_less_rat @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_5900_mult__less__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_int @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_5901_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_5902_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_5903_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_5904_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_5905_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_5906_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_5907_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N3: num] :
      ( ( ( ord_less_eq_num @ M @ N3 )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N3 )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N3 ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N3 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_5908_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N3: num] :
      ( ( ( ord_less_eq_num @ M @ N3 )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N3 )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N3 ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N3 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_5909_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N3: num] :
      ( ( ( ord_less_eq_num @ M @ N3 )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N3 )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N3 ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N3 ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N3 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_5910_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N3: num] :
      ( ( ( ord_less_num @ M @ N3 )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N3 )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N3 ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N3 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_5911_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N3: num] :
      ( ( ( ord_less_num @ M @ N3 )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N3 )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N3 ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N3 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_5912_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N3: num] :
      ( ( ( ord_less_num @ M @ N3 )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N3 )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N3 ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N3 ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N3 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_5913_divides__aux__eq,axiom,
    ! [Q3: code_integer,R3: code_integer] :
      ( ( unique5706413561485394159nteger @ ( produc1086072967326762835nteger @ Q3 @ R3 ) )
      = ( R3 = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
thf(fact_5914_divides__aux__eq,axiom,
    ! [Q3: nat,R3: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q3 @ R3 ) )
      = ( R3 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_5915_divides__aux__eq,axiom,
    ! [Q3: int,R3: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q3 @ R3 ) )
      = ( R3 = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_5916_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
    ! [X: nat,Y: int] :
      ( ( ( vEBT_V9176841429113362141ildupi @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != one_one_int ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y != one_one_int ) )
         => ~ ! [N: nat] :
                ( ( X
                  = ( suc @ ( suc @ N ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                     => ( Y
                        = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                     => ( Y
                        = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.elims
thf(fact_5917_arsinh__0,axiom,
    ( ( arsinh_real @ zero_zero_real )
    = zero_zero_real ) ).

% arsinh_0
thf(fact_5918_artanh__0,axiom,
    ( ( artanh_real @ zero_zero_real )
    = zero_zero_real ) ).

% artanh_0
thf(fact_5919_dvd__0__left__iff,axiom,
    ! [A2: complex] :
      ( ( dvd_dvd_complex @ zero_zero_complex @ A2 )
      = ( A2 = zero_zero_complex ) ) ).

% dvd_0_left_iff
thf(fact_5920_dvd__0__left__iff,axiom,
    ! [A2: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A2 )
      = ( A2 = zero_zero_real ) ) ).

% dvd_0_left_iff
thf(fact_5921_dvd__0__left__iff,axiom,
    ! [A2: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A2 )
      = ( A2 = zero_zero_rat ) ) ).

% dvd_0_left_iff
thf(fact_5922_dvd__0__left__iff,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
      = ( A2 = zero_zero_nat ) ) ).

% dvd_0_left_iff
thf(fact_5923_dvd__0__left__iff,axiom,
    ! [A2: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A2 )
      = ( A2 = zero_zero_int ) ) ).

% dvd_0_left_iff
thf(fact_5924_dvd__0__right,axiom,
    ! [A2: complex] : ( dvd_dvd_complex @ A2 @ zero_zero_complex ) ).

% dvd_0_right
thf(fact_5925_dvd__0__right,axiom,
    ! [A2: real] : ( dvd_dvd_real @ A2 @ zero_zero_real ) ).

% dvd_0_right
thf(fact_5926_dvd__0__right,axiom,
    ! [A2: rat] : ( dvd_dvd_rat @ A2 @ zero_zero_rat ) ).

% dvd_0_right
thf(fact_5927_dvd__0__right,axiom,
    ! [A2: nat] : ( dvd_dvd_nat @ A2 @ zero_zero_nat ) ).

% dvd_0_right
thf(fact_5928_dvd__0__right,axiom,
    ! [A2: int] : ( dvd_dvd_int @ A2 @ zero_zero_int ) ).

% dvd_0_right
thf(fact_5929_dvd__add__triv__right__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( dvd_dvd_real @ A2 @ ( plus_plus_real @ B2 @ A2 ) )
      = ( dvd_dvd_real @ A2 @ B2 ) ) ).

% dvd_add_triv_right_iff
thf(fact_5930_dvd__add__triv__right__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( dvd_dvd_rat @ A2 @ ( plus_plus_rat @ B2 @ A2 ) )
      = ( dvd_dvd_rat @ A2 @ B2 ) ) ).

% dvd_add_triv_right_iff
thf(fact_5931_dvd__add__triv__right__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
      = ( dvd_dvd_nat @ A2 @ B2 ) ) ).

% dvd_add_triv_right_iff
thf(fact_5932_dvd__add__triv__right__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
      = ( dvd_dvd_int @ A2 @ B2 ) ) ).

% dvd_add_triv_right_iff
thf(fact_5933_dvd__add__triv__left__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( dvd_dvd_real @ A2 @ ( plus_plus_real @ A2 @ B2 ) )
      = ( dvd_dvd_real @ A2 @ B2 ) ) ).

% dvd_add_triv_left_iff
thf(fact_5934_dvd__add__triv__left__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( dvd_dvd_rat @ A2 @ ( plus_plus_rat @ A2 @ B2 ) )
      = ( dvd_dvd_rat @ A2 @ B2 ) ) ).

% dvd_add_triv_left_iff
thf(fact_5935_dvd__add__triv__left__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
      = ( dvd_dvd_nat @ A2 @ B2 ) ) ).

% dvd_add_triv_left_iff
thf(fact_5936_dvd__add__triv__left__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
      = ( dvd_dvd_int @ A2 @ B2 ) ) ).

% dvd_add_triv_left_iff
thf(fact_5937_div__dvd__div,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A2 @ B2 )
     => ( ( dvd_dvd_nat @ A2 @ C2 )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B2 @ A2 ) @ ( divide_divide_nat @ C2 @ A2 ) )
          = ( dvd_dvd_nat @ B2 @ C2 ) ) ) ) ).

% div_dvd_div
thf(fact_5938_div__dvd__div,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( dvd_dvd_int @ A2 @ B2 )
     => ( ( dvd_dvd_int @ A2 @ C2 )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ B2 @ A2 ) @ ( divide_divide_int @ C2 @ A2 ) )
          = ( dvd_dvd_int @ B2 @ C2 ) ) ) ) ).

% div_dvd_div
thf(fact_5939_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_5940_dvd__times__right__cancel__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ A2 ) @ ( times_times_nat @ C2 @ A2 ) )
        = ( dvd_dvd_nat @ B2 @ C2 ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_5941_dvd__times__right__cancel__iff,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ B2 @ A2 ) @ ( times_times_int @ C2 @ A2 ) )
        = ( dvd_dvd_int @ B2 @ C2 ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_5942_dvd__times__left__cancel__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C2 ) )
        = ( dvd_dvd_nat @ B2 @ C2 ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_5943_dvd__times__left__cancel__iff,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C2 ) )
        = ( dvd_dvd_int @ B2 @ C2 ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_5944_dvd__mult__cancel__right,axiom,
    ! [A2: complex,C2: complex,B2: complex] :
      ( ( dvd_dvd_complex @ ( times_times_complex @ A2 @ C2 ) @ ( times_times_complex @ B2 @ C2 ) )
      = ( ( C2 = zero_zero_complex )
        | ( dvd_dvd_complex @ A2 @ B2 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_5945_dvd__mult__cancel__right,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ C2 ) )
      = ( ( C2 = zero_zero_real )
        | ( dvd_dvd_real @ A2 @ B2 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_5946_dvd__mult__cancel__right,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ C2 ) )
      = ( ( C2 = zero_zero_rat )
        | ( dvd_dvd_rat @ A2 @ B2 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_5947_dvd__mult__cancel__right,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) )
      = ( ( C2 = zero_zero_int )
        | ( dvd_dvd_int @ A2 @ B2 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_5948_dvd__mult__cancel__left,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( dvd_dvd_complex @ ( times_times_complex @ C2 @ A2 ) @ ( times_times_complex @ C2 @ B2 ) )
      = ( ( C2 = zero_zero_complex )
        | ( dvd_dvd_complex @ A2 @ B2 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_5949_dvd__mult__cancel__left,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ C2 @ A2 ) @ ( times_times_real @ C2 @ B2 ) )
      = ( ( C2 = zero_zero_real )
        | ( dvd_dvd_real @ A2 @ B2 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_5950_dvd__mult__cancel__left,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ C2 @ A2 ) @ ( times_times_rat @ C2 @ B2 ) )
      = ( ( C2 = zero_zero_rat )
        | ( dvd_dvd_rat @ A2 @ B2 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_5951_dvd__mult__cancel__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) )
      = ( ( C2 = zero_zero_int )
        | ( dvd_dvd_int @ A2 @ B2 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_5952_dvd__add__times__triv__right__iff,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( dvd_dvd_real @ A2 @ ( plus_plus_real @ B2 @ ( times_times_real @ C2 @ A2 ) ) )
      = ( dvd_dvd_real @ A2 @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5953_dvd__add__times__triv__right__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( dvd_dvd_rat @ A2 @ ( plus_plus_rat @ B2 @ ( times_times_rat @ C2 @ A2 ) ) )
      = ( dvd_dvd_rat @ A2 @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5954_dvd__add__times__triv__right__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ ( times_times_nat @ C2 @ A2 ) ) )
      = ( dvd_dvd_nat @ A2 @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5955_dvd__add__times__triv__right__iff,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B2 @ ( times_times_int @ C2 @ A2 ) ) )
      = ( dvd_dvd_int @ A2 @ B2 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5956_dvd__add__times__triv__left__iff,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( dvd_dvd_real @ A2 @ ( plus_plus_real @ ( times_times_real @ C2 @ A2 ) @ B2 ) )
      = ( dvd_dvd_real @ A2 @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5957_dvd__add__times__triv__left__iff,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( dvd_dvd_rat @ A2 @ ( plus_plus_rat @ ( times_times_rat @ C2 @ A2 ) @ B2 ) )
      = ( dvd_dvd_rat @ A2 @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5958_dvd__add__times__triv__left__iff,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ ( times_times_nat @ C2 @ A2 ) @ B2 ) )
      = ( dvd_dvd_nat @ A2 @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5959_dvd__add__times__triv__left__iff,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ ( times_times_int @ C2 @ A2 ) @ B2 ) )
      = ( dvd_dvd_int @ A2 @ B2 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5960_unit__prod,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ).

% unit_prod
thf(fact_5961_unit__prod,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ one_one_int ) ) ) ).

% unit_prod
thf(fact_5962_div__add,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C2 @ A2 )
     => ( ( dvd_dvd_nat @ C2 @ B2 )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 )
          = ( plus_plus_nat @ ( divide_divide_nat @ A2 @ C2 ) @ ( divide_divide_nat @ B2 @ C2 ) ) ) ) ) ).

% div_add
thf(fact_5963_div__add,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( dvd_dvd_int @ C2 @ A2 )
     => ( ( dvd_dvd_int @ C2 @ B2 )
       => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 )
          = ( plus_plus_int @ ( divide_divide_int @ A2 @ C2 ) @ ( divide_divide_int @ B2 @ C2 ) ) ) ) ) ).

% div_add
thf(fact_5964_unit__div__1__div__1,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A2 ) )
        = A2 ) ) ).

% unit_div_1_div_1
thf(fact_5965_unit__div__1__div__1,axiom,
    ! [A2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A2 ) )
        = A2 ) ) ).

% unit_div_1_div_1
thf(fact_5966_unit__div__1__unit,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A2 ) @ one_one_nat ) ) ).

% unit_div_1_unit
thf(fact_5967_unit__div__1__unit,axiom,
    ! [A2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A2 ) @ one_one_int ) ) ).

% unit_div_1_unit
thf(fact_5968_unit__div,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( dvd_dvd_nat @ ( divide_divide_nat @ A2 @ B2 ) @ one_one_nat ) ) ) ).

% unit_div
thf(fact_5969_unit__div,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( dvd_dvd_int @ ( divide_divide_int @ A2 @ B2 ) @ one_one_int ) ) ) ).

% unit_div
thf(fact_5970_dvd__mult__div__cancel,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ B2 )
     => ( ( times_times_nat @ A2 @ ( divide_divide_nat @ B2 @ A2 ) )
        = B2 ) ) ).

% dvd_mult_div_cancel
thf(fact_5971_dvd__mult__div__cancel,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ B2 )
     => ( ( times_times_int @ A2 @ ( divide_divide_int @ B2 @ A2 ) )
        = B2 ) ) ).

% dvd_mult_div_cancel
thf(fact_5972_dvd__div__mult__self,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ B2 )
     => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A2 ) @ A2 )
        = B2 ) ) ).

% dvd_div_mult_self
thf(fact_5973_dvd__div__mult__self,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ B2 )
     => ( ( times_times_int @ ( divide_divide_int @ B2 @ A2 ) @ A2 )
        = B2 ) ) ).

% dvd_div_mult_self
thf(fact_5974_div__diff,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( dvd_dvd_int @ C2 @ A2 )
     => ( ( dvd_dvd_int @ C2 @ B2 )
       => ( ( divide_divide_int @ ( minus_minus_int @ A2 @ B2 ) @ C2 )
          = ( minus_minus_int @ ( divide_divide_int @ A2 @ C2 ) @ ( divide_divide_int @ B2 @ C2 ) ) ) ) ) ).

% div_diff
thf(fact_5975_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_5976_dvd__1__left,axiom,
    ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).

% dvd_1_left
thf(fact_5977_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N3 ) )
      = ( ( K = zero_zero_nat )
        | ( dvd_dvd_nat @ M @ N3 ) ) ) ).

% nat_mult_dvd_cancel_disj
thf(fact_5978_unit__div__mult__self,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ A2 ) @ A2 )
        = B2 ) ) ).

% unit_div_mult_self
thf(fact_5979_unit__div__mult__self,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ B2 @ A2 ) @ A2 )
        = B2 ) ) ).

% unit_div_mult_self
thf(fact_5980_unit__mult__div__div,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( times_times_nat @ B2 @ ( divide_divide_nat @ one_one_nat @ A2 ) )
        = ( divide_divide_nat @ B2 @ A2 ) ) ) ).

% unit_mult_div_div
thf(fact_5981_unit__mult__div__div,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( times_times_int @ B2 @ ( divide_divide_int @ one_one_int @ A2 ) )
        = ( divide_divide_int @ B2 @ A2 ) ) ) ).

% unit_mult_div_div
thf(fact_5982_even__add,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A2 @ B2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_add
thf(fact_5983_even__add,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ B2 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_add
thf(fact_5984_odd__add,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A2 @ B2 ) ) )
      = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) )
       != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).

% odd_add
thf(fact_5985_odd__add,axiom,
    ! [A2: int,B2: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ B2 ) ) )
      = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
       != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ) ).

% odd_add
thf(fact_5986_even__mult__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A2 @ B2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_mult_iff
thf(fact_5987_even__mult__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A2 @ B2 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).

% even_mult_iff
thf(fact_5988_even__Suc__Suc__iff,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N3 ) ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% even_Suc_Suc_iff
thf(fact_5989_even__Suc,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% even_Suc
thf(fact_5990_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_5991_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_5992_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique3479559517661332726nteger @ M @ one )
      = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).

% divmod_algorithm_code(2)
thf(fact_5993_dvd__numeral__simp,axiom,
    ! [M: num,N3: num] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N3 ) )
      = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N3 @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_5994_dvd__numeral__simp,axiom,
    ! [M: num,N3: num] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N3 ) )
      = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N3 @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_5995_dvd__numeral__simp,axiom,
    ! [M: num,N3: num] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N3 ) )
      = ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N3 @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_5996_even__plus__one__iff,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A2 @ one_one_nat ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) ) ) ).

% even_plus_one_iff
thf(fact_5997_even__plus__one__iff,axiom,
    ! [A2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ one_one_int ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) ) ).

% even_plus_one_iff
thf(fact_5998_even__diff,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A2 @ B2 ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ B2 ) ) ) ).

% even_diff
thf(fact_5999_even__Suc__div__two,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( divide_divide_nat @ ( suc @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_Suc_div_two
thf(fact_6000_odd__Suc__div__two,axiom,
    ! [N3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( divide_divide_nat @ ( suc @ N3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% odd_Suc_div_two
thf(fact_6001_divmod__algorithm__code_I3_J,axiom,
    ! [N3: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N3 ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_6002_divmod__algorithm__code_I3_J,axiom,
    ! [N3: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N3 ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_6003_divmod__algorithm__code_I3_J,axiom,
    ! [N3: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N3 ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_6004_divmod__algorithm__code_I4_J,axiom,
    ! [N3: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N3 ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_6005_divmod__algorithm__code_I4_J,axiom,
    ! [N3: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N3 ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_6006_divmod__algorithm__code_I4_J,axiom,
    ! [N3: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N3 ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_6007_even__succ__div__two,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_6008_even__succ__div__two,axiom,
    ! [A2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_6009_odd__succ__div__two,axiom,
    ! [A2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( plus_plus_nat @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).

% odd_succ_div_two
thf(fact_6010_odd__succ__div__two,axiom,
    ! [A2: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( plus_plus_int @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% odd_succ_div_two
thf(fact_6011_even__succ__div__2,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_6012_even__succ__div__2,axiom,
    ! [A2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_6013_even__power,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A2 @ N3 ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
        & ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% even_power
thf(fact_6014_even__power,axiom,
    ! [A2: nat,N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A2 @ N3 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
        & ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% even_power
thf(fact_6015_even__power,axiom,
    ! [A2: int,N3: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A2 @ N3 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
        & ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% even_power
thf(fact_6016_zero__le__power__eq__numeral,axiom,
    ! [A2: real,W: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ ( 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 @ A2 ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_6017_zero__le__power__eq__numeral,axiom,
    ! [A2: code_integer,W: num] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A2 @ ( 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_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_6018_zero__le__power__eq__numeral,axiom,
    ! [A2: rat,W: num] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ ( 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 @ A2 ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_6019_zero__le__power__eq__numeral,axiom,
    ! [A2: int,W: num] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ ( 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 @ A2 ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_6020_power__less__zero__eq__numeral,axiom,
    ! [A2: code_integer,W: num] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ W ) ) @ zero_z3403309356797280102nteger )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_6021_power__less__zero__eq__numeral,axiom,
    ! [A2: real,W: num] :
      ( ( ord_less_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_real @ A2 @ zero_zero_real ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_6022_power__less__zero__eq__numeral,axiom,
    ! [A2: rat,W: num] :
      ( ( ord_less_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_rat @ A2 @ zero_zero_rat ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_6023_power__less__zero__eq__numeral,axiom,
    ! [A2: int,W: num] :
      ( ( ord_less_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_6024_power__less__zero__eq,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ zero_z3403309356797280102nteger )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
        & ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ) ).

% power_less_zero_eq
thf(fact_6025_power__less__zero__eq,axiom,
    ! [A2: real,N3: nat] :
      ( ( ord_less_real @ ( power_power_real @ A2 @ N3 ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
        & ( ord_less_real @ A2 @ zero_zero_real ) ) ) ).

% power_less_zero_eq
thf(fact_6026_power__less__zero__eq,axiom,
    ! [A2: rat,N3: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ A2 @ N3 ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
        & ( ord_less_rat @ A2 @ zero_zero_rat ) ) ) ).

% power_less_zero_eq
thf(fact_6027_power__less__zero__eq,axiom,
    ! [A2: int,N3: nat] :
      ( ( ord_less_int @ ( power_power_int @ A2 @ N3 ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
        & ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).

% power_less_zero_eq
thf(fact_6028_even__of__nat,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N3 ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% even_of_nat
thf(fact_6029_even__of__nat,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N3 ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% even_of_nat
thf(fact_6030_even__of__nat,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N3 ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% even_of_nat
thf(fact_6031_odd__Suc__minus__one,axiom,
    ! [N3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( suc @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) )
        = N3 ) ) ).

% odd_Suc_minus_one
thf(fact_6032_even__diff__nat,axiom,
    ! [M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N3 ) )
      = ( ( ord_less_nat @ M @ N3 )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N3 ) ) ) ) ).

% even_diff_nat
thf(fact_6033_odd__two__times__div__two__succ,axiom,
    ! [A2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
        = A2 ) ) ).

% odd_two_times_div_two_succ
thf(fact_6034_odd__two__times__div__two__succ,axiom,
    ! [A2: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
        = A2 ) ) ).

% odd_two_times_div_two_succ
thf(fact_6035_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) @ one_one_Code_integer ) )
      = ( N3 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_6036_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) )
      = ( N3 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_6037_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) )
      = ( N3 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_6038_zero__less__power__eq__numeral,axiom,
    ! [A2: code_integer,W: num] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A2 != zero_z3403309356797280102nteger ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_6039_zero__less__power__eq__numeral,axiom,
    ! [A2: real,W: num] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A2 != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_real @ zero_zero_real @ A2 ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_6040_zero__less__power__eq__numeral,axiom,
    ! [A2: rat,W: num] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A2 != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_rat @ zero_zero_rat @ A2 ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_6041_zero__less__power__eq__numeral,axiom,
    ! [A2: int,W: num] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A2 != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_int @ zero_zero_int @ A2 ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_6042_odd__two__times__div__two__nat,axiom,
    ! [N3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ).

% odd_two_times_div_two_nat
thf(fact_6043_power__le__zero__eq__numeral,axiom,
    ! [A2: real,W: num] :
      ( ( ord_less_eq_real @ ( power_power_real @ A2 @ ( 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 @ A2 @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A2 = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_6044_power__le__zero__eq__numeral,axiom,
    ! [A2: code_integer,W: num] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ W ) ) @ zero_z3403309356797280102nteger )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A2 = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_6045_power__le__zero__eq__numeral,axiom,
    ! [A2: rat,W: num] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ ( 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 @ A2 @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A2 = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_6046_power__le__zero__eq__numeral,axiom,
    ! [A2: int,W: num] :
      ( ( ord_less_eq_int @ ( power_power_int @ A2 @ ( 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 @ A2 @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A2 = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_6047_even__succ__div__exp,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) )
          = ( divide6298287555418463151nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_6048_even__succ__div__exp,axiom,
    ! [A2: nat,N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
          = ( divide_divide_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_6049_even__succ__div__exp,axiom,
    ! [A2: int,N3: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
          = ( divide_divide_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_6050_dvd__antisym,axiom,
    ! [M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ M @ N3 )
     => ( ( dvd_dvd_nat @ N3 @ M )
       => ( M = N3 ) ) ) ).

% dvd_antisym
thf(fact_6051_dvd__trans,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A2 @ B2 )
     => ( ( dvd_dvd_nat @ B2 @ C2 )
       => ( dvd_dvd_nat @ A2 @ C2 ) ) ) ).

% dvd_trans
thf(fact_6052_dvd__trans,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( dvd_dvd_int @ A2 @ B2 )
     => ( ( dvd_dvd_int @ B2 @ C2 )
       => ( dvd_dvd_int @ A2 @ C2 ) ) ) ).

% dvd_trans
thf(fact_6053_dvd__refl,axiom,
    ! [A2: nat] : ( dvd_dvd_nat @ A2 @ A2 ) ).

% dvd_refl
thf(fact_6054_dvd__refl,axiom,
    ! [A2: int] : ( dvd_dvd_int @ A2 @ A2 ) ).

% dvd_refl
thf(fact_6055_of__nat__dvd__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) )
      = ( dvd_dvd_nat @ M @ N3 ) ) ).

% of_nat_dvd_iff
thf(fact_6056_of__nat__dvd__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N3 ) )
      = ( dvd_dvd_nat @ M @ N3 ) ) ).

% of_nat_dvd_iff
thf(fact_6057_of__nat__dvd__iff,axiom,
    ! [M: nat,N3: nat] :
      ( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N3 ) )
      = ( dvd_dvd_nat @ M @ N3 ) ) ).

% of_nat_dvd_iff
thf(fact_6058_dvd__0__left,axiom,
    ! [A2: complex] :
      ( ( dvd_dvd_complex @ zero_zero_complex @ A2 )
     => ( A2 = zero_zero_complex ) ) ).

% dvd_0_left
thf(fact_6059_dvd__0__left,axiom,
    ! [A2: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A2 )
     => ( A2 = zero_zero_real ) ) ).

% dvd_0_left
thf(fact_6060_dvd__0__left,axiom,
    ! [A2: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A2 )
     => ( A2 = zero_zero_rat ) ) ).

% dvd_0_left
thf(fact_6061_dvd__0__left,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
     => ( A2 = zero_zero_nat ) ) ).

% dvd_0_left
thf(fact_6062_dvd__0__left,axiom,
    ! [A2: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A2 )
     => ( A2 = zero_zero_int ) ) ).

% dvd_0_left
thf(fact_6063_dvd__field__iff,axiom,
    ( dvd_dvd_complex
    = ( ^ [A7: complex,B7: complex] :
          ( ( A7 = zero_zero_complex )
         => ( B7 = zero_zero_complex ) ) ) ) ).

% dvd_field_iff
thf(fact_6064_dvd__field__iff,axiom,
    ( dvd_dvd_real
    = ( ^ [A7: real,B7: real] :
          ( ( A7 = zero_zero_real )
         => ( B7 = zero_zero_real ) ) ) ) ).

% dvd_field_iff
thf(fact_6065_dvd__field__iff,axiom,
    ( dvd_dvd_rat
    = ( ^ [A7: rat,B7: rat] :
          ( ( A7 = zero_zero_rat )
         => ( B7 = zero_zero_rat ) ) ) ) ).

% dvd_field_iff
thf(fact_6066_dvd__add__right__iff,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( dvd_dvd_real @ A2 @ B2 )
     => ( ( dvd_dvd_real @ A2 @ ( plus_plus_real @ B2 @ C2 ) )
        = ( dvd_dvd_real @ A2 @ C2 ) ) ) ).

% dvd_add_right_iff
thf(fact_6067_dvd__add__right__iff,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( dvd_dvd_rat @ A2 @ B2 )
     => ( ( dvd_dvd_rat @ A2 @ ( plus_plus_rat @ B2 @ C2 ) )
        = ( dvd_dvd_rat @ A2 @ C2 ) ) ) ).

% dvd_add_right_iff
thf(fact_6068_dvd__add__right__iff,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A2 @ B2 )
     => ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ C2 ) )
        = ( dvd_dvd_nat @ A2 @ C2 ) ) ) ).

% dvd_add_right_iff
thf(fact_6069_dvd__add__right__iff,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( dvd_dvd_int @ A2 @ B2 )
     => ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B2 @ C2 ) )
        = ( dvd_dvd_int @ A2 @ C2 ) ) ) ).

% dvd_add_right_iff
thf(fact_6070_dvd__add__left__iff,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( dvd_dvd_real @ A2 @ C2 )
     => ( ( dvd_dvd_real @ A2 @ ( plus_plus_real @ B2 @ C2 ) )
        = ( dvd_dvd_real @ A2 @ B2 ) ) ) ).

% dvd_add_left_iff
thf(fact_6071_dvd__add__left__iff,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( dvd_dvd_rat @ A2 @ C2 )
     => ( ( dvd_dvd_rat @ A2 @ ( plus_plus_rat @ B2 @ C2 ) )
        = ( dvd_dvd_rat @ A2 @ B2 ) ) ) ).

% dvd_add_left_iff
thf(fact_6072_dvd__add__left__iff,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ C2 )
     => ( ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ C2 ) )
        = ( dvd_dvd_nat @ A2 @ B2 ) ) ) ).

% dvd_add_left_iff
thf(fact_6073_dvd__add__left__iff,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ C2 )
     => ( ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B2 @ C2 ) )
        = ( dvd_dvd_int @ A2 @ B2 ) ) ) ).

% dvd_add_left_iff
thf(fact_6074_dvd__add,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( dvd_dvd_real @ A2 @ B2 )
     => ( ( dvd_dvd_real @ A2 @ C2 )
       => ( dvd_dvd_real @ A2 @ ( plus_plus_real @ B2 @ C2 ) ) ) ) ).

% dvd_add
thf(fact_6075_dvd__add,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( dvd_dvd_rat @ A2 @ B2 )
     => ( ( dvd_dvd_rat @ A2 @ C2 )
       => ( dvd_dvd_rat @ A2 @ ( plus_plus_rat @ B2 @ C2 ) ) ) ) ).

% dvd_add
thf(fact_6076_dvd__add,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A2 @ B2 )
     => ( ( dvd_dvd_nat @ A2 @ C2 )
       => ( dvd_dvd_nat @ A2 @ ( plus_plus_nat @ B2 @ C2 ) ) ) ) ).

% dvd_add
thf(fact_6077_dvd__add,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( dvd_dvd_int @ A2 @ B2 )
     => ( ( dvd_dvd_int @ A2 @ C2 )
       => ( dvd_dvd_int @ A2 @ ( plus_plus_int @ B2 @ C2 ) ) ) ) ).

% dvd_add
thf(fact_6078_dvd__unit__imp__unit,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ B2 )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( dvd_dvd_nat @ A2 @ one_one_nat ) ) ) ).

% dvd_unit_imp_unit
thf(fact_6079_dvd__unit__imp__unit,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ B2 )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( dvd_dvd_int @ A2 @ one_one_int ) ) ) ).

% dvd_unit_imp_unit
thf(fact_6080_unit__imp__dvd,axiom,
    ! [B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( dvd_dvd_nat @ B2 @ A2 ) ) ).

% unit_imp_dvd
thf(fact_6081_unit__imp__dvd,axiom,
    ! [B2: int,A2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( dvd_dvd_int @ B2 @ A2 ) ) ).

% unit_imp_dvd
thf(fact_6082_one__dvd,axiom,
    ! [A2: assn] : ( dvd_dvd_assn @ one_one_assn @ A2 ) ).

% one_dvd
thf(fact_6083_one__dvd,axiom,
    ! [A2: real] : ( dvd_dvd_real @ one_one_real @ A2 ) ).

% one_dvd
thf(fact_6084_one__dvd,axiom,
    ! [A2: rat] : ( dvd_dvd_rat @ one_one_rat @ A2 ) ).

% one_dvd
thf(fact_6085_one__dvd,axiom,
    ! [A2: nat] : ( dvd_dvd_nat @ one_one_nat @ A2 ) ).

% one_dvd
thf(fact_6086_one__dvd,axiom,
    ! [A2: int] : ( dvd_dvd_int @ one_one_int @ A2 ) ).

% one_dvd
thf(fact_6087_dvd__triv__right,axiom,
    ! [A2: real,B2: real] : ( dvd_dvd_real @ A2 @ ( times_times_real @ B2 @ A2 ) ) ).

% dvd_triv_right
thf(fact_6088_dvd__triv__right,axiom,
    ! [A2: rat,B2: rat] : ( dvd_dvd_rat @ A2 @ ( times_times_rat @ B2 @ A2 ) ) ).

% dvd_triv_right
thf(fact_6089_dvd__triv__right,axiom,
    ! [A2: nat,B2: nat] : ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ A2 ) ) ).

% dvd_triv_right
thf(fact_6090_dvd__triv__right,axiom,
    ! [A2: int,B2: int] : ( dvd_dvd_int @ A2 @ ( times_times_int @ B2 @ A2 ) ) ).

% dvd_triv_right
thf(fact_6091_dvd__triv__right,axiom,
    ! [A2: assn,B2: assn] : ( dvd_dvd_assn @ A2 @ ( times_times_assn @ B2 @ A2 ) ) ).

% dvd_triv_right
thf(fact_6092_dvd__mult__right,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A2 @ B2 ) @ C2 )
     => ( dvd_dvd_real @ B2 @ C2 ) ) ).

% dvd_mult_right
thf(fact_6093_dvd__mult__right,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A2 @ B2 ) @ C2 )
     => ( dvd_dvd_rat @ B2 @ C2 ) ) ).

% dvd_mult_right
thf(fact_6094_dvd__mult__right,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 )
     => ( dvd_dvd_nat @ B2 @ C2 ) ) ).

% dvd_mult_right
thf(fact_6095_dvd__mult__right,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ C2 )
     => ( dvd_dvd_int @ B2 @ C2 ) ) ).

% dvd_mult_right
thf(fact_6096_dvd__mult__right,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( dvd_dvd_assn @ ( times_times_assn @ A2 @ B2 ) @ C2 )
     => ( dvd_dvd_assn @ B2 @ C2 ) ) ).

% dvd_mult_right
thf(fact_6097_mult__dvd__mono,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( dvd_dvd_real @ A2 @ B2 )
     => ( ( dvd_dvd_real @ C2 @ D2 )
       => ( dvd_dvd_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ D2 ) ) ) ) ).

% mult_dvd_mono
thf(fact_6098_mult__dvd__mono,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] :
      ( ( dvd_dvd_rat @ A2 @ B2 )
     => ( ( dvd_dvd_rat @ C2 @ D2 )
       => ( dvd_dvd_rat @ ( times_times_rat @ A2 @ C2 ) @ ( times_times_rat @ B2 @ D2 ) ) ) ) ).

% mult_dvd_mono
thf(fact_6099_mult__dvd__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat,D2: nat] :
      ( ( dvd_dvd_nat @ A2 @ B2 )
     => ( ( dvd_dvd_nat @ C2 @ D2 )
       => ( dvd_dvd_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ).

% mult_dvd_mono
thf(fact_6100_mult__dvd__mono,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] :
      ( ( dvd_dvd_int @ A2 @ B2 )
     => ( ( dvd_dvd_int @ C2 @ D2 )
       => ( dvd_dvd_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ).

% mult_dvd_mono
thf(fact_6101_mult__dvd__mono,axiom,
    ! [A2: assn,B2: assn,C2: assn,D2: assn] :
      ( ( dvd_dvd_assn @ A2 @ B2 )
     => ( ( dvd_dvd_assn @ C2 @ D2 )
       => ( dvd_dvd_assn @ ( times_times_assn @ A2 @ C2 ) @ ( times_times_assn @ B2 @ D2 ) ) ) ) ).

% mult_dvd_mono
thf(fact_6102_dvd__triv__left,axiom,
    ! [A2: real,B2: real] : ( dvd_dvd_real @ A2 @ ( times_times_real @ A2 @ B2 ) ) ).

% dvd_triv_left
thf(fact_6103_dvd__triv__left,axiom,
    ! [A2: rat,B2: rat] : ( dvd_dvd_rat @ A2 @ ( times_times_rat @ A2 @ B2 ) ) ).

% dvd_triv_left
thf(fact_6104_dvd__triv__left,axiom,
    ! [A2: nat,B2: nat] : ( dvd_dvd_nat @ A2 @ ( times_times_nat @ A2 @ B2 ) ) ).

% dvd_triv_left
thf(fact_6105_dvd__triv__left,axiom,
    ! [A2: int,B2: int] : ( dvd_dvd_int @ A2 @ ( times_times_int @ A2 @ B2 ) ) ).

% dvd_triv_left
thf(fact_6106_dvd__triv__left,axiom,
    ! [A2: assn,B2: assn] : ( dvd_dvd_assn @ A2 @ ( times_times_assn @ A2 @ B2 ) ) ).

% dvd_triv_left
thf(fact_6107_dvd__mult__left,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A2 @ B2 ) @ C2 )
     => ( dvd_dvd_real @ A2 @ C2 ) ) ).

% dvd_mult_left
thf(fact_6108_dvd__mult__left,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A2 @ B2 ) @ C2 )
     => ( dvd_dvd_rat @ A2 @ C2 ) ) ).

% dvd_mult_left
thf(fact_6109_dvd__mult__left,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 )
     => ( dvd_dvd_nat @ A2 @ C2 ) ) ).

% dvd_mult_left
thf(fact_6110_dvd__mult__left,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ C2 )
     => ( dvd_dvd_int @ A2 @ C2 ) ) ).

% dvd_mult_left
thf(fact_6111_dvd__mult__left,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( dvd_dvd_assn @ ( times_times_assn @ A2 @ B2 ) @ C2 )
     => ( dvd_dvd_assn @ A2 @ C2 ) ) ).

% dvd_mult_left
thf(fact_6112_dvd__mult2,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( dvd_dvd_real @ A2 @ B2 )
     => ( dvd_dvd_real @ A2 @ ( times_times_real @ B2 @ C2 ) ) ) ).

% dvd_mult2
thf(fact_6113_dvd__mult2,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( dvd_dvd_rat @ A2 @ B2 )
     => ( dvd_dvd_rat @ A2 @ ( times_times_rat @ B2 @ C2 ) ) ) ).

% dvd_mult2
thf(fact_6114_dvd__mult2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A2 @ B2 )
     => ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) ) ) ).

% dvd_mult2
thf(fact_6115_dvd__mult2,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( dvd_dvd_int @ A2 @ B2 )
     => ( dvd_dvd_int @ A2 @ ( times_times_int @ B2 @ C2 ) ) ) ).

% dvd_mult2
thf(fact_6116_dvd__mult2,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( dvd_dvd_assn @ A2 @ B2 )
     => ( dvd_dvd_assn @ A2 @ ( times_times_assn @ B2 @ C2 ) ) ) ).

% dvd_mult2
thf(fact_6117_dvd__mult,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( dvd_dvd_real @ A2 @ C2 )
     => ( dvd_dvd_real @ A2 @ ( times_times_real @ B2 @ C2 ) ) ) ).

% dvd_mult
thf(fact_6118_dvd__mult,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( dvd_dvd_rat @ A2 @ C2 )
     => ( dvd_dvd_rat @ A2 @ ( times_times_rat @ B2 @ C2 ) ) ) ).

% dvd_mult
thf(fact_6119_dvd__mult,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ C2 )
     => ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) ) ) ).

% dvd_mult
thf(fact_6120_dvd__mult,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ C2 )
     => ( dvd_dvd_int @ A2 @ ( times_times_int @ B2 @ C2 ) ) ) ).

% dvd_mult
thf(fact_6121_dvd__mult,axiom,
    ! [A2: assn,C2: assn,B2: assn] :
      ( ( dvd_dvd_assn @ A2 @ C2 )
     => ( dvd_dvd_assn @ A2 @ ( times_times_assn @ B2 @ C2 ) ) ) ).

% dvd_mult
thf(fact_6122_dvd__def,axiom,
    ( dvd_dvd_real
    = ( ^ [B7: real,A7: real] :
        ? [K3: real] :
          ( A7
          = ( times_times_real @ B7 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_6123_dvd__def,axiom,
    ( dvd_dvd_rat
    = ( ^ [B7: rat,A7: rat] :
        ? [K3: rat] :
          ( A7
          = ( times_times_rat @ B7 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_6124_dvd__def,axiom,
    ( dvd_dvd_nat
    = ( ^ [B7: nat,A7: nat] :
        ? [K3: nat] :
          ( A7
          = ( times_times_nat @ B7 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_6125_dvd__def,axiom,
    ( dvd_dvd_int
    = ( ^ [B7: int,A7: int] :
        ? [K3: int] :
          ( A7
          = ( times_times_int @ B7 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_6126_dvd__def,axiom,
    ( dvd_dvd_assn
    = ( ^ [B7: assn,A7: assn] :
        ? [K3: assn] :
          ( A7
          = ( times_times_assn @ B7 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_6127_dvdI,axiom,
    ! [A2: real,B2: real,K: real] :
      ( ( A2
        = ( times_times_real @ B2 @ K ) )
     => ( dvd_dvd_real @ B2 @ A2 ) ) ).

% dvdI
thf(fact_6128_dvdI,axiom,
    ! [A2: rat,B2: rat,K: rat] :
      ( ( A2
        = ( times_times_rat @ B2 @ K ) )
     => ( dvd_dvd_rat @ B2 @ A2 ) ) ).

% dvdI
thf(fact_6129_dvdI,axiom,
    ! [A2: nat,B2: nat,K: nat] :
      ( ( A2
        = ( times_times_nat @ B2 @ K ) )
     => ( dvd_dvd_nat @ B2 @ A2 ) ) ).

% dvdI
thf(fact_6130_dvdI,axiom,
    ! [A2: int,B2: int,K: int] :
      ( ( A2
        = ( times_times_int @ B2 @ K ) )
     => ( dvd_dvd_int @ B2 @ A2 ) ) ).

% dvdI
thf(fact_6131_dvdI,axiom,
    ! [A2: assn,B2: assn,K: assn] :
      ( ( A2
        = ( times_times_assn @ B2 @ K ) )
     => ( dvd_dvd_assn @ B2 @ A2 ) ) ).

% dvdI
thf(fact_6132_dvdE,axiom,
    ! [B2: real,A2: real] :
      ( ( dvd_dvd_real @ B2 @ A2 )
     => ~ ! [K2: real] :
            ( A2
           != ( times_times_real @ B2 @ K2 ) ) ) ).

% dvdE
thf(fact_6133_dvdE,axiom,
    ! [B2: rat,A2: rat] :
      ( ( dvd_dvd_rat @ B2 @ A2 )
     => ~ ! [K2: rat] :
            ( A2
           != ( times_times_rat @ B2 @ K2 ) ) ) ).

% dvdE
thf(fact_6134_dvdE,axiom,
    ! [B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ B2 @ A2 )
     => ~ ! [K2: nat] :
            ( A2
           != ( times_times_nat @ B2 @ K2 ) ) ) ).

% dvdE
thf(fact_6135_dvdE,axiom,
    ! [B2: int,A2: int] :
      ( ( dvd_dvd_int @ B2 @ A2 )
     => ~ ! [K2: int] :
            ( A2
           != ( times_times_int @ B2 @ K2 ) ) ) ).

% dvdE
thf(fact_6136_dvdE,axiom,
    ! [B2: assn,A2: assn] :
      ( ( dvd_dvd_assn @ B2 @ A2 )
     => ~ ! [K2: assn] :
            ( A2
           != ( times_times_assn @ B2 @ K2 ) ) ) ).

% dvdE
thf(fact_6137_dvd__diff,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( ( dvd_dvd_real @ X @ Z )
       => ( dvd_dvd_real @ X @ ( minus_minus_real @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_6138_dvd__diff,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( dvd_dvd_rat @ X @ Y )
     => ( ( dvd_dvd_rat @ X @ Z )
       => ( dvd_dvd_rat @ X @ ( minus_minus_rat @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_6139_dvd__diff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( ( dvd_dvd_int @ X @ Z )
       => ( dvd_dvd_int @ X @ ( minus_minus_int @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_6140_dvd__diff__commute,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ ( minus_minus_int @ C2 @ B2 ) )
      = ( dvd_dvd_int @ A2 @ ( minus_minus_int @ B2 @ C2 ) ) ) ).

% dvd_diff_commute
thf(fact_6141_div__div__div__same,axiom,
    ! [D2: nat,B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ D2 @ B2 )
     => ( ( dvd_dvd_nat @ B2 @ A2 )
       => ( ( divide_divide_nat @ ( divide_divide_nat @ A2 @ D2 ) @ ( divide_divide_nat @ B2 @ D2 ) )
          = ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).

% div_div_div_same
thf(fact_6142_div__div__div__same,axiom,
    ! [D2: int,B2: int,A2: int] :
      ( ( dvd_dvd_int @ D2 @ B2 )
     => ( ( dvd_dvd_int @ B2 @ A2 )
       => ( ( divide_divide_int @ ( divide_divide_int @ A2 @ D2 ) @ ( divide_divide_int @ B2 @ D2 ) )
          = ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% div_div_div_same
thf(fact_6143_dvd__div__eq__cancel,axiom,
    ! [A2: complex,C2: complex,B2: complex] :
      ( ( ( divide1717551699836669952omplex @ A2 @ C2 )
        = ( divide1717551699836669952omplex @ B2 @ C2 ) )
     => ( ( dvd_dvd_complex @ C2 @ A2 )
       => ( ( dvd_dvd_complex @ C2 @ B2 )
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_6144_dvd__div__eq__cancel,axiom,
    ! [A2: real,C2: real,B2: real] :
      ( ( ( divide_divide_real @ A2 @ C2 )
        = ( divide_divide_real @ B2 @ C2 ) )
     => ( ( dvd_dvd_real @ C2 @ A2 )
       => ( ( dvd_dvd_real @ C2 @ B2 )
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_6145_dvd__div__eq__cancel,axiom,
    ! [A2: rat,C2: rat,B2: rat] :
      ( ( ( divide_divide_rat @ A2 @ C2 )
        = ( divide_divide_rat @ B2 @ C2 ) )
     => ( ( dvd_dvd_rat @ C2 @ A2 )
       => ( ( dvd_dvd_rat @ C2 @ B2 )
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_6146_dvd__div__eq__cancel,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( ( divide_divide_nat @ A2 @ C2 )
        = ( divide_divide_nat @ B2 @ C2 ) )
     => ( ( dvd_dvd_nat @ C2 @ A2 )
       => ( ( dvd_dvd_nat @ C2 @ B2 )
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_6147_dvd__div__eq__cancel,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ( divide_divide_int @ A2 @ C2 )
        = ( divide_divide_int @ B2 @ C2 ) )
     => ( ( dvd_dvd_int @ C2 @ A2 )
       => ( ( dvd_dvd_int @ C2 @ B2 )
         => ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_6148_dvd__div__eq__iff,axiom,
    ! [C2: complex,A2: complex,B2: complex] :
      ( ( dvd_dvd_complex @ C2 @ A2 )
     => ( ( dvd_dvd_complex @ C2 @ B2 )
       => ( ( ( divide1717551699836669952omplex @ A2 @ C2 )
            = ( divide1717551699836669952omplex @ B2 @ C2 ) )
          = ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_6149_dvd__div__eq__iff,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( dvd_dvd_real @ C2 @ A2 )
     => ( ( dvd_dvd_real @ C2 @ B2 )
       => ( ( ( divide_divide_real @ A2 @ C2 )
            = ( divide_divide_real @ B2 @ C2 ) )
          = ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_6150_dvd__div__eq__iff,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( dvd_dvd_rat @ C2 @ A2 )
     => ( ( dvd_dvd_rat @ C2 @ B2 )
       => ( ( ( divide_divide_rat @ A2 @ C2 )
            = ( divide_divide_rat @ B2 @ C2 ) )
          = ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_6151_dvd__div__eq__iff,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C2 @ A2 )
     => ( ( dvd_dvd_nat @ C2 @ B2 )
       => ( ( ( divide_divide_nat @ A2 @ C2 )
            = ( divide_divide_nat @ B2 @ C2 ) )
          = ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_6152_dvd__div__eq__iff,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( dvd_dvd_int @ C2 @ A2 )
     => ( ( dvd_dvd_int @ C2 @ B2 )
       => ( ( ( divide_divide_int @ A2 @ C2 )
            = ( divide_divide_int @ B2 @ C2 ) )
          = ( A2 = B2 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_6153_dvd__power__same,axiom,
    ! [X: nat,Y: nat,N3: nat] :
      ( ( dvd_dvd_nat @ X @ Y )
     => ( dvd_dvd_nat @ ( power_power_nat @ X @ N3 ) @ ( power_power_nat @ Y @ N3 ) ) ) ).

% dvd_power_same
thf(fact_6154_dvd__power__same,axiom,
    ! [X: real,Y: real,N3: nat] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( dvd_dvd_real @ ( power_power_real @ X @ N3 ) @ ( power_power_real @ Y @ N3 ) ) ) ).

% dvd_power_same
thf(fact_6155_dvd__power__same,axiom,
    ! [X: int,Y: int,N3: nat] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( dvd_dvd_int @ ( power_power_int @ X @ N3 ) @ ( power_power_int @ Y @ N3 ) ) ) ).

% dvd_power_same
thf(fact_6156_dvd__power__same,axiom,
    ! [X: complex,Y: complex,N3: nat] :
      ( ( dvd_dvd_complex @ X @ Y )
     => ( dvd_dvd_complex @ ( power_power_complex @ X @ N3 ) @ ( power_power_complex @ Y @ N3 ) ) ) ).

% dvd_power_same
thf(fact_6157_dvd__power__same,axiom,
    ! [X: code_integer,Y: code_integer,N3: nat] :
      ( ( dvd_dvd_Code_integer @ X @ Y )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N3 ) @ ( power_8256067586552552935nteger @ Y @ N3 ) ) ) ).

% dvd_power_same
thf(fact_6158_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N3 )
       => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N3 ) ) ) ) ).

% dvd_diff_nat
thf(fact_6159_subset__divisors__dvd,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ord_le211207098394363844omplex
        @ ( collect_complex
          @ ^ [C5: complex] : ( dvd_dvd_complex @ C5 @ A2 ) )
        @ ( collect_complex
          @ ^ [C5: complex] : ( dvd_dvd_complex @ C5 @ B2 ) ) )
      = ( dvd_dvd_complex @ A2 @ B2 ) ) ).

% subset_divisors_dvd
thf(fact_6160_subset__divisors__dvd,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_set_int
        @ ( collect_int
          @ ^ [C5: int] : ( dvd_dvd_int @ C5 @ A2 ) )
        @ ( collect_int
          @ ^ [C5: int] : ( dvd_dvd_int @ C5 @ B2 ) ) )
      = ( dvd_dvd_int @ A2 @ B2 ) ) ).

% subset_divisors_dvd
thf(fact_6161_subset__divisors__dvd,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_set_nat
        @ ( collect_nat
          @ ^ [C5: nat] : ( dvd_dvd_nat @ C5 @ A2 ) )
        @ ( collect_nat
          @ ^ [C5: nat] : ( dvd_dvd_nat @ C5 @ B2 ) ) )
      = ( dvd_dvd_nat @ A2 @ B2 ) ) ).

% subset_divisors_dvd
thf(fact_6162_strict__subset__divisors__dvd,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ord_less_set_complex
        @ ( collect_complex
          @ ^ [C5: complex] : ( dvd_dvd_complex @ C5 @ A2 ) )
        @ ( collect_complex
          @ ^ [C5: complex] : ( dvd_dvd_complex @ C5 @ B2 ) ) )
      = ( ( dvd_dvd_complex @ A2 @ B2 )
        & ~ ( dvd_dvd_complex @ B2 @ A2 ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_6163_strict__subset__divisors__dvd,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_set_nat
        @ ( collect_nat
          @ ^ [C5: nat] : ( dvd_dvd_nat @ C5 @ A2 ) )
        @ ( collect_nat
          @ ^ [C5: nat] : ( dvd_dvd_nat @ C5 @ B2 ) ) )
      = ( ( dvd_dvd_nat @ A2 @ B2 )
        & ~ ( dvd_dvd_nat @ B2 @ A2 ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_6164_strict__subset__divisors__dvd,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_set_int
        @ ( collect_int
          @ ^ [C5: int] : ( dvd_dvd_int @ C5 @ A2 ) )
        @ ( collect_int
          @ ^ [C5: int] : ( dvd_dvd_int @ C5 @ B2 ) ) )
      = ( ( dvd_dvd_int @ A2 @ B2 )
        & ~ ( dvd_dvd_int @ B2 @ A2 ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_6165_not__is__unit__0,axiom,
    ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).

% not_is_unit_0
thf(fact_6166_not__is__unit__0,axiom,
    ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).

% not_is_unit_0
thf(fact_6167_pinf_I9_J,axiom,
    ! [D2: real,S2: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ( ( dvd_dvd_real @ D2 @ ( plus_plus_real @ X5 @ S2 ) )
        = ( dvd_dvd_real @ D2 @ ( plus_plus_real @ X5 @ S2 ) ) ) ) ).

% pinf(9)
thf(fact_6168_pinf_I9_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ( ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X5 @ S2 ) )
        = ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X5 @ S2 ) ) ) ) ).

% pinf(9)
thf(fact_6169_pinf_I9_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X5 @ S2 ) )
        = ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X5 @ S2 ) ) ) ) ).

% pinf(9)
thf(fact_6170_pinf_I9_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X5 @ S2 ) )
        = ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X5 @ S2 ) ) ) ) ).

% pinf(9)
thf(fact_6171_pinf_I10_J,axiom,
    ! [D2: real,S2: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z2 @ X5 )
     => ( ( ~ ( dvd_dvd_real @ D2 @ ( plus_plus_real @ X5 @ S2 ) ) )
        = ( ~ ( dvd_dvd_real @ D2 @ ( plus_plus_real @ X5 @ S2 ) ) ) ) ) ).

% pinf(10)
thf(fact_6172_pinf_I10_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z2 @ X5 )
     => ( ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X5 @ S2 ) ) )
        = ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X5 @ S2 ) ) ) ) ) ).

% pinf(10)
thf(fact_6173_pinf_I10_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z2 @ X5 )
     => ( ( ~ ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X5 @ S2 ) ) )
        = ( ~ ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X5 @ S2 ) ) ) ) ) ).

% pinf(10)
thf(fact_6174_pinf_I10_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z2 @ X5 )
     => ( ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X5 @ S2 ) ) )
        = ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X5 @ S2 ) ) ) ) ) ).

% pinf(10)
thf(fact_6175_minf_I9_J,axiom,
    ! [D2: real,S2: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ( ( dvd_dvd_real @ D2 @ ( plus_plus_real @ X5 @ S2 ) )
        = ( dvd_dvd_real @ D2 @ ( plus_plus_real @ X5 @ S2 ) ) ) ) ).

% minf(9)
thf(fact_6176_minf_I9_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ( ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X5 @ S2 ) )
        = ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X5 @ S2 ) ) ) ) ).

% minf(9)
thf(fact_6177_minf_I9_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X5 @ S2 ) )
        = ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X5 @ S2 ) ) ) ) ).

% minf(9)
thf(fact_6178_minf_I9_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X5 @ S2 ) )
        = ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X5 @ S2 ) ) ) ) ).

% minf(9)
thf(fact_6179_minf_I10_J,axiom,
    ! [D2: real,S2: real] :
    ? [Z2: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z2 )
     => ( ( ~ ( dvd_dvd_real @ D2 @ ( plus_plus_real @ X5 @ S2 ) ) )
        = ( ~ ( dvd_dvd_real @ D2 @ ( plus_plus_real @ X5 @ S2 ) ) ) ) ) ).

% minf(10)
thf(fact_6180_minf_I10_J,axiom,
    ! [D2: rat,S2: rat] :
    ? [Z2: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z2 )
     => ( ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X5 @ S2 ) ) )
        = ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X5 @ S2 ) ) ) ) ) ).

% minf(10)
thf(fact_6181_minf_I10_J,axiom,
    ! [D2: nat,S2: nat] :
    ? [Z2: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z2 )
     => ( ( ~ ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X5 @ S2 ) ) )
        = ( ~ ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ X5 @ S2 ) ) ) ) ) ).

% minf(10)
thf(fact_6182_minf_I10_J,axiom,
    ! [D2: int,S2: int] :
    ? [Z2: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z2 )
     => ( ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X5 @ S2 ) ) )
        = ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X5 @ S2 ) ) ) ) ) ).

% minf(10)
thf(fact_6183_dvd__div__eq__0__iff,axiom,
    ! [B2: complex,A2: complex] :
      ( ( dvd_dvd_complex @ B2 @ A2 )
     => ( ( ( divide1717551699836669952omplex @ A2 @ B2 )
          = zero_zero_complex )
        = ( A2 = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_6184_dvd__div__eq__0__iff,axiom,
    ! [B2: real,A2: real] :
      ( ( dvd_dvd_real @ B2 @ A2 )
     => ( ( ( divide_divide_real @ A2 @ B2 )
          = zero_zero_real )
        = ( A2 = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_6185_dvd__div__eq__0__iff,axiom,
    ! [B2: rat,A2: rat] :
      ( ( dvd_dvd_rat @ B2 @ A2 )
     => ( ( ( divide_divide_rat @ A2 @ B2 )
          = zero_zero_rat )
        = ( A2 = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_6186_dvd__div__eq__0__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ B2 @ A2 )
     => ( ( ( divide_divide_nat @ A2 @ B2 )
          = zero_zero_nat )
        = ( A2 = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_6187_dvd__div__eq__0__iff,axiom,
    ! [B2: int,A2: int] :
      ( ( dvd_dvd_int @ B2 @ A2 )
     => ( ( ( divide_divide_int @ A2 @ B2 )
          = zero_zero_int )
        = ( A2 = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_6188_unit__mult__right__cancel,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( ( times_times_nat @ B2 @ A2 )
          = ( times_times_nat @ C2 @ A2 ) )
        = ( B2 = C2 ) ) ) ).

% unit_mult_right_cancel
thf(fact_6189_unit__mult__right__cancel,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( ( times_times_int @ B2 @ A2 )
          = ( times_times_int @ C2 @ A2 ) )
        = ( B2 = C2 ) ) ) ).

% unit_mult_right_cancel
thf(fact_6190_unit__mult__left__cancel,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( ( times_times_nat @ A2 @ B2 )
          = ( times_times_nat @ A2 @ C2 ) )
        = ( B2 = C2 ) ) ) ).

% unit_mult_left_cancel
thf(fact_6191_unit__mult__left__cancel,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( ( times_times_int @ A2 @ B2 )
          = ( times_times_int @ A2 @ C2 ) )
        = ( B2 = C2 ) ) ) ).

% unit_mult_left_cancel
thf(fact_6192_mult__unit__dvd__iff_H,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 )
        = ( dvd_dvd_nat @ B2 @ C2 ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_6193_mult__unit__dvd__iff_H,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ C2 )
        = ( dvd_dvd_int @ B2 @ C2 ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_6194_dvd__mult__unit__iff_H,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) )
        = ( dvd_dvd_nat @ A2 @ C2 ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_6195_dvd__mult__unit__iff_H,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ A2 @ ( times_times_int @ B2 @ C2 ) )
        = ( dvd_dvd_int @ A2 @ C2 ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_6196_mult__unit__dvd__iff,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 )
        = ( dvd_dvd_nat @ A2 @ C2 ) ) ) ).

% mult_unit_dvd_iff
thf(fact_6197_mult__unit__dvd__iff,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ C2 )
        = ( dvd_dvd_int @ A2 @ C2 ) ) ) ).

% mult_unit_dvd_iff
thf(fact_6198_dvd__mult__unit__iff,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ C2 @ B2 ) )
        = ( dvd_dvd_nat @ A2 @ C2 ) ) ) ).

% dvd_mult_unit_iff
thf(fact_6199_dvd__mult__unit__iff,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ A2 @ ( times_times_int @ C2 @ B2 ) )
        = ( dvd_dvd_int @ A2 @ C2 ) ) ) ).

% dvd_mult_unit_iff
thf(fact_6200_is__unit__mult__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ B2 ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A2 @ one_one_nat )
        & ( dvd_dvd_nat @ B2 @ one_one_nat ) ) ) ).

% is_unit_mult_iff
thf(fact_6201_is__unit__mult__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A2 @ B2 ) @ one_one_int )
      = ( ( dvd_dvd_int @ A2 @ one_one_int )
        & ( dvd_dvd_int @ B2 @ one_one_int ) ) ) ).

% is_unit_mult_iff
thf(fact_6202_div__plus__div__distrib__dvd__right,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ C2 @ B2 )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 )
        = ( plus_plus_nat @ ( divide_divide_nat @ A2 @ C2 ) @ ( divide_divide_nat @ B2 @ C2 ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_6203_div__plus__div__distrib__dvd__right,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( dvd_dvd_int @ C2 @ B2 )
     => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 )
        = ( plus_plus_int @ ( divide_divide_int @ A2 @ C2 ) @ ( divide_divide_int @ B2 @ C2 ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_6204_div__plus__div__distrib__dvd__left,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C2 @ A2 )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 )
        = ( plus_plus_nat @ ( divide_divide_nat @ A2 @ C2 ) @ ( divide_divide_nat @ B2 @ C2 ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_6205_div__plus__div__distrib__dvd__left,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( dvd_dvd_int @ C2 @ A2 )
     => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 )
        = ( plus_plus_int @ ( divide_divide_int @ A2 @ C2 ) @ ( divide_divide_int @ B2 @ C2 ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_6206_unit__div__cancel,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ( ( ( divide_divide_nat @ B2 @ A2 )
          = ( divide_divide_nat @ C2 @ A2 ) )
        = ( B2 = C2 ) ) ) ).

% unit_div_cancel
thf(fact_6207_unit__div__cancel,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ( ( ( divide_divide_int @ B2 @ A2 )
          = ( divide_divide_int @ C2 @ A2 ) )
        = ( B2 = C2 ) ) ) ).

% unit_div_cancel
thf(fact_6208_div__unit__dvd__iff,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C2 )
        = ( dvd_dvd_nat @ A2 @ C2 ) ) ) ).

% div_unit_dvd_iff
thf(fact_6209_div__unit__dvd__iff,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ ( divide_divide_int @ A2 @ B2 ) @ C2 )
        = ( dvd_dvd_int @ A2 @ C2 ) ) ) ).

% div_unit_dvd_iff
thf(fact_6210_dvd__div__unit__iff,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ A2 @ ( divide_divide_nat @ C2 @ B2 ) )
        = ( dvd_dvd_nat @ A2 @ C2 ) ) ) ).

% dvd_div_unit_iff
thf(fact_6211_dvd__div__unit__iff,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ A2 @ ( divide_divide_int @ C2 @ B2 ) )
        = ( dvd_dvd_int @ A2 @ C2 ) ) ) ).

% dvd_div_unit_iff
thf(fact_6212_div__mult__div__if__dvd,axiom,
    ! [B2: nat,A2: nat,D2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B2 @ A2 )
     => ( ( dvd_dvd_nat @ D2 @ C2 )
       => ( ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ ( divide_divide_nat @ C2 @ D2 ) )
          = ( divide_divide_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_6213_div__mult__div__if__dvd,axiom,
    ! [B2: int,A2: int,D2: int,C2: int] :
      ( ( dvd_dvd_int @ B2 @ A2 )
     => ( ( dvd_dvd_int @ D2 @ C2 )
       => ( ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ ( divide_divide_int @ C2 @ D2 ) )
          = ( divide_divide_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_6214_dvd__mult__imp__div,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A2 @ C2 ) @ B2 )
     => ( dvd_dvd_nat @ A2 @ ( divide_divide_nat @ B2 @ C2 ) ) ) ).

% dvd_mult_imp_div
thf(fact_6215_dvd__mult__imp__div,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A2 @ C2 ) @ B2 )
     => ( dvd_dvd_int @ A2 @ ( divide_divide_int @ B2 @ C2 ) ) ) ).

% dvd_mult_imp_div
thf(fact_6216_dvd__div__mult2__eq,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ B2 @ C2 ) @ A2 )
     => ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C2 ) ) ) ).

% dvd_div_mult2_eq
thf(fact_6217_dvd__div__mult2__eq,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ B2 @ C2 ) @ A2 )
     => ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ C2 ) )
        = ( divide_divide_int @ ( divide_divide_int @ A2 @ B2 ) @ C2 ) ) ) ).

% dvd_div_mult2_eq
thf(fact_6218_div__div__eq__right,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ C2 @ B2 )
     => ( ( dvd_dvd_nat @ B2 @ A2 )
       => ( ( divide_divide_nat @ A2 @ ( divide_divide_nat @ B2 @ C2 ) )
          = ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C2 ) ) ) ) ).

% div_div_eq_right
thf(fact_6219_div__div__eq__right,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( dvd_dvd_int @ C2 @ B2 )
     => ( ( dvd_dvd_int @ B2 @ A2 )
       => ( ( divide_divide_int @ A2 @ ( divide_divide_int @ B2 @ C2 ) )
          = ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ C2 ) ) ) ) ).

% div_div_eq_right
thf(fact_6220_div__mult__swap,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ C2 @ B2 )
     => ( ( times_times_nat @ A2 @ ( divide_divide_nat @ B2 @ C2 ) )
        = ( divide_divide_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 ) ) ) ).

% div_mult_swap
thf(fact_6221_div__mult__swap,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( dvd_dvd_int @ C2 @ B2 )
     => ( ( times_times_int @ A2 @ ( divide_divide_int @ B2 @ C2 ) )
        = ( divide_divide_int @ ( times_times_int @ A2 @ B2 ) @ C2 ) ) ) ).

% div_mult_swap
thf(fact_6222_dvd__div__mult,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ C2 @ B2 )
     => ( ( times_times_nat @ ( divide_divide_nat @ B2 @ C2 ) @ A2 )
        = ( divide_divide_nat @ ( times_times_nat @ B2 @ A2 ) @ C2 ) ) ) ).

% dvd_div_mult
thf(fact_6223_dvd__div__mult,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( dvd_dvd_int @ C2 @ B2 )
     => ( ( times_times_int @ ( divide_divide_int @ B2 @ C2 ) @ A2 )
        = ( divide_divide_int @ ( times_times_int @ B2 @ A2 ) @ C2 ) ) ) ).

% dvd_div_mult
thf(fact_6224_div__power,axiom,
    ! [B2: code_integer,A2: code_integer,N3: nat] :
      ( ( dvd_dvd_Code_integer @ B2 @ A2 )
     => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ N3 )
        = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ ( power_8256067586552552935nteger @ B2 @ N3 ) ) ) ) ).

% div_power
thf(fact_6225_div__power,axiom,
    ! [B2: nat,A2: nat,N3: nat] :
      ( ( dvd_dvd_nat @ B2 @ A2 )
     => ( ( power_power_nat @ ( divide_divide_nat @ A2 @ B2 ) @ N3 )
        = ( divide_divide_nat @ ( power_power_nat @ A2 @ N3 ) @ ( power_power_nat @ B2 @ N3 ) ) ) ) ).

% div_power
thf(fact_6226_div__power,axiom,
    ! [B2: int,A2: int,N3: nat] :
      ( ( dvd_dvd_int @ B2 @ A2 )
     => ( ( power_power_int @ ( divide_divide_int @ A2 @ B2 ) @ N3 )
        = ( divide_divide_int @ ( power_power_int @ A2 @ N3 ) @ ( power_power_int @ B2 @ N3 ) ) ) ) ).

% div_power
thf(fact_6227_dvd__power__le,axiom,
    ! [X: nat,Y: nat,N3: nat,M: nat] :
      ( ( dvd_dvd_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( dvd_dvd_nat @ ( power_power_nat @ X @ N3 ) @ ( power_power_nat @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_6228_dvd__power__le,axiom,
    ! [X: real,Y: real,N3: nat,M: nat] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( dvd_dvd_real @ ( power_power_real @ X @ N3 ) @ ( power_power_real @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_6229_dvd__power__le,axiom,
    ! [X: int,Y: int,N3: nat,M: nat] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( dvd_dvd_int @ ( power_power_int @ X @ N3 ) @ ( power_power_int @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_6230_dvd__power__le,axiom,
    ! [X: complex,Y: complex,N3: nat,M: nat] :
      ( ( dvd_dvd_complex @ X @ Y )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( dvd_dvd_complex @ ( power_power_complex @ X @ N3 ) @ ( power_power_complex @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_6231_dvd__power__le,axiom,
    ! [X: code_integer,Y: code_integer,N3: nat,M: nat] :
      ( ( dvd_dvd_Code_integer @ X @ Y )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N3 ) @ ( power_8256067586552552935nteger @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_6232_power__le__dvd,axiom,
    ! [A2: nat,N3: nat,B2: nat,M: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A2 @ N3 ) @ B2 )
     => ( ( ord_less_eq_nat @ M @ N3 )
       => ( dvd_dvd_nat @ ( power_power_nat @ A2 @ M ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_6233_power__le__dvd,axiom,
    ! [A2: real,N3: nat,B2: real,M: nat] :
      ( ( dvd_dvd_real @ ( power_power_real @ A2 @ N3 ) @ B2 )
     => ( ( ord_less_eq_nat @ M @ N3 )
       => ( dvd_dvd_real @ ( power_power_real @ A2 @ M ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_6234_power__le__dvd,axiom,
    ! [A2: int,N3: nat,B2: int,M: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A2 @ N3 ) @ B2 )
     => ( ( ord_less_eq_nat @ M @ N3 )
       => ( dvd_dvd_int @ ( power_power_int @ A2 @ M ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_6235_power__le__dvd,axiom,
    ! [A2: complex,N3: nat,B2: complex,M: nat] :
      ( ( dvd_dvd_complex @ ( power_power_complex @ A2 @ N3 ) @ B2 )
     => ( ( ord_less_eq_nat @ M @ N3 )
       => ( dvd_dvd_complex @ ( power_power_complex @ A2 @ M ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_6236_power__le__dvd,axiom,
    ! [A2: code_integer,N3: nat,B2: code_integer,M: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ B2 )
     => ( ( ord_less_eq_nat @ M @ N3 )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A2 @ M ) @ B2 ) ) ) ).

% power_le_dvd
thf(fact_6237_le__imp__power__dvd,axiom,
    ! [M: nat,N3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( dvd_dvd_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N3 ) ) ) ).

% le_imp_power_dvd
thf(fact_6238_le__imp__power__dvd,axiom,
    ! [M: nat,N3: nat,A2: real] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( dvd_dvd_real @ ( power_power_real @ A2 @ M ) @ ( power_power_real @ A2 @ N3 ) ) ) ).

% le_imp_power_dvd
thf(fact_6239_le__imp__power__dvd,axiom,
    ! [M: nat,N3: nat,A2: int] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( dvd_dvd_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N3 ) ) ) ).

% le_imp_power_dvd
thf(fact_6240_le__imp__power__dvd,axiom,
    ! [M: nat,N3: nat,A2: complex] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( dvd_dvd_complex @ ( power_power_complex @ A2 @ M ) @ ( power_power_complex @ A2 @ N3 ) ) ) ).

% le_imp_power_dvd
thf(fact_6241_le__imp__power__dvd,axiom,
    ! [M: nat,N3: nat,A2: code_integer] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A2 @ M ) @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ).

% le_imp_power_dvd
thf(fact_6242_nat__dvd__not__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N3 )
       => ~ ( dvd_dvd_nat @ N3 @ M ) ) ) ).

% nat_dvd_not_less
thf(fact_6243_dvd__minus__self,axiom,
    ! [M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N3 @ M ) )
      = ( ( ord_less_nat @ N3 @ M )
        | ( dvd_dvd_nat @ M @ N3 ) ) ) ).

% dvd_minus_self
thf(fact_6244_less__eq__dvd__minus,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( dvd_dvd_nat @ M @ N3 )
        = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N3 @ M ) ) ) ) ).

% less_eq_dvd_minus
thf(fact_6245_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N3 ) )
     => ( ( dvd_dvd_nat @ K @ M )
       => ( ( ord_less_eq_nat @ N3 @ M )
         => ( dvd_dvd_nat @ K @ N3 ) ) ) ) ).

% dvd_diffD1
thf(fact_6246_dvd__diffD,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N3 ) )
     => ( ( dvd_dvd_nat @ K @ N3 )
       => ( ( ord_less_eq_nat @ N3 @ M )
         => ( dvd_dvd_nat @ K @ M ) ) ) ) ).

% dvd_diffD
thf(fact_6247_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_6248_even__numeral,axiom,
    ! [N3: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N3 ) ) ) ).

% even_numeral
thf(fact_6249_even__numeral,axiom,
    ! [N3: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) ).

% even_numeral
thf(fact_6250_unity__coeff__ex,axiom,
    ! [P: complex > $o,L2: complex] :
      ( ( ? [X3: complex] : ( P @ ( times_times_complex @ L2 @ X3 ) ) )
      = ( ? [X3: complex] :
            ( ( dvd_dvd_complex @ L2 @ ( plus_plus_complex @ X3 @ zero_zero_complex ) )
            & ( P @ X3 ) ) ) ) ).

% unity_coeff_ex
thf(fact_6251_unity__coeff__ex,axiom,
    ! [P: real > $o,L2: real] :
      ( ( ? [X3: real] : ( P @ ( times_times_real @ L2 @ X3 ) ) )
      = ( ? [X3: real] :
            ( ( dvd_dvd_real @ L2 @ ( plus_plus_real @ X3 @ zero_zero_real ) )
            & ( P @ X3 ) ) ) ) ).

% unity_coeff_ex
thf(fact_6252_unity__coeff__ex,axiom,
    ! [P: rat > $o,L2: rat] :
      ( ( ? [X3: rat] : ( P @ ( times_times_rat @ L2 @ X3 ) ) )
      = ( ? [X3: rat] :
            ( ( dvd_dvd_rat @ L2 @ ( plus_plus_rat @ X3 @ zero_zero_rat ) )
            & ( P @ X3 ) ) ) ) ).

% unity_coeff_ex
thf(fact_6253_unity__coeff__ex,axiom,
    ! [P: nat > $o,L2: nat] :
      ( ( ? [X3: nat] : ( P @ ( times_times_nat @ L2 @ X3 ) ) )
      = ( ? [X3: nat] :
            ( ( dvd_dvd_nat @ L2 @ ( plus_plus_nat @ X3 @ zero_zero_nat ) )
            & ( P @ X3 ) ) ) ) ).

% unity_coeff_ex
thf(fact_6254_unity__coeff__ex,axiom,
    ! [P: int > $o,L2: int] :
      ( ( ? [X3: int] : ( P @ ( times_times_int @ L2 @ X3 ) ) )
      = ( ? [X3: int] :
            ( ( dvd_dvd_int @ L2 @ ( plus_plus_int @ X3 @ zero_zero_int ) )
            & ( P @ X3 ) ) ) ) ).

% unity_coeff_ex
thf(fact_6255_unit__dvdE,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ~ ( ( A2 != zero_zero_nat )
         => ! [C4: nat] :
              ( B2
             != ( times_times_nat @ A2 @ C4 ) ) ) ) ).

% unit_dvdE
thf(fact_6256_unit__dvdE,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ~ ( ( A2 != zero_zero_int )
         => ! [C4: int] :
              ( B2
             != ( times_times_int @ A2 @ C4 ) ) ) ) ).

% unit_dvdE
thf(fact_6257_unit__div__eq__0__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( ( divide_divide_nat @ A2 @ B2 )
          = zero_zero_nat )
        = ( A2 = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
thf(fact_6258_unit__div__eq__0__iff,axiom,
    ! [B2: int,A2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( ( divide_divide_int @ A2 @ B2 )
          = zero_zero_int )
        = ( A2 = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
thf(fact_6259_dvd__div__div__eq__mult,axiom,
    ! [A2: nat,C2: nat,B2: nat,D2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( C2 != zero_zero_nat )
       => ( ( dvd_dvd_nat @ A2 @ B2 )
         => ( ( dvd_dvd_nat @ C2 @ D2 )
           => ( ( ( divide_divide_nat @ B2 @ A2 )
                = ( divide_divide_nat @ D2 @ C2 ) )
              = ( ( times_times_nat @ B2 @ C2 )
                = ( times_times_nat @ A2 @ D2 ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_6260_dvd__div__div__eq__mult,axiom,
    ! [A2: int,C2: int,B2: int,D2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( C2 != zero_zero_int )
       => ( ( dvd_dvd_int @ A2 @ B2 )
         => ( ( dvd_dvd_int @ C2 @ D2 )
           => ( ( ( divide_divide_int @ B2 @ A2 )
                = ( divide_divide_int @ D2 @ C2 ) )
              = ( ( times_times_int @ B2 @ C2 )
                = ( times_times_int @ A2 @ D2 ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_6261_dvd__div__iff__mult,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ C2 @ B2 )
       => ( ( dvd_dvd_nat @ A2 @ ( divide_divide_nat @ B2 @ C2 ) )
          = ( dvd_dvd_nat @ ( times_times_nat @ A2 @ C2 ) @ B2 ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_6262_dvd__div__iff__mult,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( C2 != zero_zero_int )
     => ( ( dvd_dvd_int @ C2 @ B2 )
       => ( ( dvd_dvd_int @ A2 @ ( divide_divide_int @ B2 @ C2 ) )
          = ( dvd_dvd_int @ ( times_times_int @ A2 @ C2 ) @ B2 ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_6263_div__dvd__iff__mult,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( B2 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B2 @ A2 )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C2 )
          = ( dvd_dvd_nat @ A2 @ ( times_times_nat @ C2 @ B2 ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_6264_div__dvd__iff__mult,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( dvd_dvd_int @ B2 @ A2 )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ A2 @ B2 ) @ C2 )
          = ( dvd_dvd_int @ A2 @ ( times_times_int @ C2 @ B2 ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_6265_dvd__div__eq__mult,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ A2 @ B2 )
       => ( ( ( divide_divide_nat @ B2 @ A2 )
            = C2 )
          = ( B2
            = ( times_times_nat @ C2 @ A2 ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_6266_dvd__div__eq__mult,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( dvd_dvd_int @ A2 @ B2 )
       => ( ( ( divide_divide_int @ B2 @ A2 )
            = C2 )
          = ( B2
            = ( times_times_int @ C2 @ A2 ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_6267_inf__period_I3_J,axiom,
    ! [D2: real,D: real,T: real] :
      ( ( dvd_dvd_real @ D2 @ D )
     => ! [X5: real,K5: real] :
          ( ( dvd_dvd_real @ D2 @ ( plus_plus_real @ X5 @ T ) )
          = ( dvd_dvd_real @ D2 @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K5 @ D ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_6268_inf__period_I3_J,axiom,
    ! [D2: rat,D: rat,T: rat] :
      ( ( dvd_dvd_rat @ D2 @ D )
     => ! [X5: rat,K5: rat] :
          ( ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X5 @ T ) )
          = ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K5 @ D ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_6269_inf__period_I3_J,axiom,
    ! [D2: int,D: int,T: int] :
      ( ( dvd_dvd_int @ D2 @ D )
     => ! [X5: int,K5: int] :
          ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X5 @ T ) )
          = ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K5 @ D ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_6270_inf__period_I4_J,axiom,
    ! [D2: real,D: real,T: real] :
      ( ( dvd_dvd_real @ D2 @ D )
     => ! [X5: real,K5: real] :
          ( ( ~ ( dvd_dvd_real @ D2 @ ( plus_plus_real @ X5 @ T ) ) )
          = ( ~ ( dvd_dvd_real @ D2 @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K5 @ D ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_6271_inf__period_I4_J,axiom,
    ! [D2: rat,D: rat,T: rat] :
      ( ( dvd_dvd_rat @ D2 @ D )
     => ! [X5: rat,K5: rat] :
          ( ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ X5 @ T ) ) )
          = ( ~ ( dvd_dvd_rat @ D2 @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K5 @ D ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_6272_inf__period_I4_J,axiom,
    ! [D2: int,D: int,T: int] :
      ( ( dvd_dvd_int @ D2 @ D )
     => ! [X5: int,K5: int] :
          ( ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X5 @ T ) ) )
          = ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K5 @ D ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_6273_is__unit__div__mult2__eq,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ C2 @ one_one_nat )
       => ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C2 ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_6274_is__unit__div__mult2__eq,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( dvd_dvd_int @ C2 @ one_one_int )
       => ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ C2 ) )
          = ( divide_divide_int @ ( divide_divide_int @ A2 @ B2 ) @ C2 ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_6275_unit__div__mult__swap,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C2 @ one_one_nat )
     => ( ( times_times_nat @ A2 @ ( divide_divide_nat @ B2 @ C2 ) )
        = ( divide_divide_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 ) ) ) ).

% unit_div_mult_swap
thf(fact_6276_unit__div__mult__swap,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( dvd_dvd_int @ C2 @ one_one_int )
     => ( ( times_times_int @ A2 @ ( divide_divide_int @ B2 @ C2 ) )
        = ( divide_divide_int @ ( times_times_int @ A2 @ B2 ) @ C2 ) ) ) ).

% unit_div_mult_swap
thf(fact_6277_unit__div__commute,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C2 )
        = ( divide_divide_nat @ ( times_times_nat @ A2 @ C2 ) @ B2 ) ) ) ).

% unit_div_commute
thf(fact_6278_unit__div__commute,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ C2 )
        = ( divide_divide_int @ ( times_times_int @ A2 @ C2 ) @ B2 ) ) ) ).

% unit_div_commute
thf(fact_6279_div__mult__unit2,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ C2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ B2 @ A2 )
       => ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C2 ) ) ) ) ).

% div_mult_unit2
thf(fact_6280_div__mult__unit2,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( dvd_dvd_int @ C2 @ one_one_int )
     => ( ( dvd_dvd_int @ B2 @ A2 )
       => ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ C2 ) )
          = ( divide_divide_int @ ( divide_divide_int @ A2 @ B2 ) @ C2 ) ) ) ) ).

% div_mult_unit2
thf(fact_6281_unit__eq__div2,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( A2
          = ( divide_divide_nat @ C2 @ B2 ) )
        = ( ( times_times_nat @ A2 @ B2 )
          = C2 ) ) ) ).

% unit_eq_div2
thf(fact_6282_unit__eq__div2,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( A2
          = ( divide_divide_int @ C2 @ B2 ) )
        = ( ( times_times_int @ A2 @ B2 )
          = C2 ) ) ) ).

% unit_eq_div2
thf(fact_6283_unit__eq__div1,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( ( divide_divide_nat @ A2 @ B2 )
          = C2 )
        = ( A2
          = ( times_times_nat @ C2 @ B2 ) ) ) ) ).

% unit_eq_div1
thf(fact_6284_unit__eq__div1,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( ( divide_divide_int @ A2 @ B2 )
          = C2 )
        = ( A2
          = ( times_times_int @ C2 @ B2 ) ) ) ) ).

% unit_eq_div1
thf(fact_6285_is__unit__power__iff,axiom,
    ! [A2: nat,N3: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A2 @ N3 ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A2 @ one_one_nat )
        | ( N3 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_6286_is__unit__power__iff,axiom,
    ! [A2: int,N3: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A2 @ N3 ) @ one_one_int )
      = ( ( dvd_dvd_int @ A2 @ one_one_int )
        | ( N3 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_6287_is__unit__power__iff,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A2 @ one_one_Code_integer )
        | ( N3 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_6288_dvd__imp__le,axiom,
    ! [K: nat,N3: nat] :
      ( ( dvd_dvd_nat @ K @ N3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_eq_nat @ K @ N3 ) ) ) ).

% dvd_imp_le
thf(fact_6289_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N3 ) )
        = ( dvd_dvd_nat @ M @ N3 ) ) ) ).

% nat_mult_dvd_cancel1
thf(fact_6290_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N3 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( dvd_dvd_nat @ M @ N3 ) ) ) ).

% dvd_mult_cancel
thf(fact_6291_real__of__nat__div,axiom,
    ! [D2: nat,N3: nat] :
      ( ( dvd_dvd_nat @ D2 @ N3 )
     => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N3 @ D2 ) )
        = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( semiri5074537144036343181t_real @ D2 ) ) ) ) ).

% real_of_nat_div
thf(fact_6292_even__zero,axiom,
    dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).

% even_zero
thf(fact_6293_even__zero,axiom,
    dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).

% even_zero
thf(fact_6294_odd__even__add,axiom,
    ! [A2: nat,B2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
       => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% odd_even_add
thf(fact_6295_odd__even__add,axiom,
    ! [A2: int,B2: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 )
       => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).

% odd_even_add
thf(fact_6296_odd__one,axiom,
    ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).

% odd_one
thf(fact_6297_odd__one,axiom,
    ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).

% odd_one
thf(fact_6298_evenE,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ~ ! [B3: nat] :
            ( A2
           != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) ).

% evenE
thf(fact_6299_evenE,axiom,
    ! [A2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ~ ! [B3: int] :
            ( A2
           != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) ) ).

% evenE
thf(fact_6300_bit__eq__rec,axiom,
    ( ( ^ [Y5: nat,Z3: nat] : Y5 = Z3 )
    = ( ^ [A7: nat,B7: nat] :
          ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A7 )
            = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B7 ) )
          & ( ( divide_divide_nat @ A7 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( divide_divide_nat @ B7 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_6301_bit__eq__rec,axiom,
    ( ( ^ [Y5: int,Z3: int] : Y5 = Z3 )
    = ( ^ [A7: int,B7: int] :
          ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A7 )
            = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B7 ) )
          & ( ( divide_divide_int @ A7 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = ( divide_divide_int @ B7 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_6302_is__unit__div__mult__cancel__right,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B2 @ A2 ) )
          = ( divide_divide_nat @ one_one_nat @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_6303_is__unit__div__mult__cancel__right,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( ( divide_divide_int @ A2 @ ( times_times_int @ B2 @ A2 ) )
          = ( divide_divide_int @ one_one_int @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_6304_is__unit__div__mult__cancel__left,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
       => ( ( divide_divide_nat @ A2 @ ( times_times_nat @ A2 @ B2 ) )
          = ( divide_divide_nat @ one_one_nat @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_6305_is__unit__div__mult__cancel__left,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero_int )
     => ( ( dvd_dvd_int @ B2 @ one_one_int )
       => ( ( divide_divide_int @ A2 @ ( times_times_int @ A2 @ B2 ) )
          = ( divide_divide_int @ one_one_int @ B2 ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_6306_is__unitE,axiom,
    ! [A2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A2 @ one_one_nat )
     => ~ ( ( A2 != zero_zero_nat )
         => ! [B3: nat] :
              ( ( B3 != zero_zero_nat )
             => ( ( dvd_dvd_nat @ B3 @ one_one_nat )
               => ( ( ( divide_divide_nat @ one_one_nat @ A2 )
                    = B3 )
                 => ( ( ( divide_divide_nat @ one_one_nat @ B3 )
                      = A2 )
                   => ( ( ( times_times_nat @ A2 @ B3 )
                        = one_one_nat )
                     => ( ( divide_divide_nat @ C2 @ A2 )
                       != ( times_times_nat @ C2 @ B3 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_6307_is__unitE,axiom,
    ! [A2: int,C2: int] :
      ( ( dvd_dvd_int @ A2 @ one_one_int )
     => ~ ( ( A2 != zero_zero_int )
         => ! [B3: int] :
              ( ( B3 != zero_zero_int )
             => ( ( dvd_dvd_int @ B3 @ one_one_int )
               => ( ( ( divide_divide_int @ one_one_int @ A2 )
                    = B3 )
                 => ( ( ( divide_divide_int @ one_one_int @ B3 )
                      = A2 )
                   => ( ( ( times_times_int @ A2 @ B3 )
                        = one_one_int )
                     => ( ( divide_divide_int @ C2 @ A2 )
                       != ( times_times_int @ C2 @ B3 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_6308_odd__numeral,axiom,
    ! [N3: num] :
      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N3 ) ) ) ).

% odd_numeral
thf(fact_6309_odd__numeral,axiom,
    ! [N3: num] :
      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) ).

% odd_numeral
thf(fact_6310_dvd__power__iff,axiom,
    ! [X: nat,M: nat,N3: nat] :
      ( ( X != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N3 ) )
        = ( ( dvd_dvd_nat @ X @ one_one_nat )
          | ( ord_less_eq_nat @ M @ N3 ) ) ) ) ).

% dvd_power_iff
thf(fact_6311_dvd__power__iff,axiom,
    ! [X: int,M: nat,N3: nat] :
      ( ( X != zero_zero_int )
     => ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N3 ) )
        = ( ( dvd_dvd_int @ X @ one_one_int )
          | ( ord_less_eq_nat @ M @ N3 ) ) ) ) ).

% dvd_power_iff
thf(fact_6312_dvd__power__iff,axiom,
    ! [X: code_integer,M: nat,N3: nat] :
      ( ( X != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N3 ) )
        = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
          | ( ord_less_eq_nat @ M @ N3 ) ) ) ) ).

% dvd_power_iff
thf(fact_6313_dvd__power,axiom,
    ! [N3: nat,X: rat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N3 )
        | ( X = one_one_rat ) )
     => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N3 ) ) ) ).

% dvd_power
thf(fact_6314_dvd__power,axiom,
    ! [N3: nat,X: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N3 )
        | ( X = one_one_nat ) )
     => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N3 ) ) ) ).

% dvd_power
thf(fact_6315_dvd__power,axiom,
    ! [N3: nat,X: real] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N3 )
        | ( X = one_one_real ) )
     => ( dvd_dvd_real @ X @ ( power_power_real @ X @ N3 ) ) ) ).

% dvd_power
thf(fact_6316_dvd__power,axiom,
    ! [N3: nat,X: int] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N3 )
        | ( X = one_one_int ) )
     => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N3 ) ) ) ).

% dvd_power
thf(fact_6317_dvd__power,axiom,
    ! [N3: nat,X: complex] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N3 )
        | ( X = one_one_complex ) )
     => ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N3 ) ) ) ).

% dvd_power
thf(fact_6318_dvd__power,axiom,
    ! [N3: nat,X: code_integer] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N3 )
        | ( X = one_one_Code_integer ) )
     => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N3 ) ) ) ).

% dvd_power
thf(fact_6319_dvd__mult__cancel2,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ N3 @ M ) @ M )
        = ( N3 = one_one_nat ) ) ) ).

% dvd_mult_cancel2
thf(fact_6320_dvd__mult__cancel1,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N3 ) @ M )
        = ( N3 = one_one_nat ) ) ) ).

% dvd_mult_cancel1
thf(fact_6321_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N3 ) )
     => ( ( ord_less_nat @ one_one_nat @ I )
       => ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% power_dvd_imp_le
thf(fact_6322_dvd__minus__add,axiom,
    ! [Q3: nat,N3: nat,R3: nat,M: nat] :
      ( ( ord_less_eq_nat @ Q3 @ N3 )
     => ( ( ord_less_eq_nat @ Q3 @ ( times_times_nat @ R3 @ M ) )
       => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N3 @ Q3 ) )
          = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N3 @ ( minus_minus_nat @ ( times_times_nat @ R3 @ M ) @ Q3 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_6323_even__two__times__div__two,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = A2 ) ) ).

% even_two_times_div_two
thf(fact_6324_even__two__times__div__two,axiom,
    ! [A2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
        = A2 ) ) ).

% even_two_times_div_two
thf(fact_6325_power__mono__odd,axiom,
    ! [N3: nat,A2: real,B2: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( ord_less_eq_real @ A2 @ B2 )
       => ( ord_less_eq_real @ ( power_power_real @ A2 @ N3 ) @ ( power_power_real @ B2 @ N3 ) ) ) ) ).

% power_mono_odd
thf(fact_6326_power__mono__odd,axiom,
    ! [N3: nat,A2: code_integer,B2: code_integer] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ ( power_8256067586552552935nteger @ B2 @ N3 ) ) ) ) ).

% power_mono_odd
thf(fact_6327_power__mono__odd,axiom,
    ! [N3: nat,A2: rat,B2: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( ord_less_eq_rat @ A2 @ B2 )
       => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N3 ) @ ( power_power_rat @ B2 @ N3 ) ) ) ) ).

% power_mono_odd
thf(fact_6328_power__mono__odd,axiom,
    ! [N3: nat,A2: int,B2: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( ord_less_eq_int @ A2 @ B2 )
       => ( ord_less_eq_int @ ( power_power_int @ A2 @ N3 ) @ ( power_power_int @ B2 @ N3 ) ) ) ) ).

% power_mono_odd
thf(fact_6329_odd__pos,axiom,
    ! [N3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).

% odd_pos
thf(fact_6330_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N3 ) )
        = ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% dvd_power_iff_le
thf(fact_6331_even__set__bit__iff,axiom,
    ! [M: nat,A2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A2 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_6332_even__set__bit__iff,axiom,
    ! [M: nat,A2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_6333_oddE,axiom,
    ! [A2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ~ ! [B3: nat] :
            ( A2
           != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) @ one_one_nat ) ) ) ).

% oddE
thf(fact_6334_oddE,axiom,
    ! [A2: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ~ ! [B3: int] :
            ( A2
           != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ one_one_int ) ) ) ).

% oddE
thf(fact_6335_zero__le__even__power,axiom,
    ! [N3: nat,A2: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ N3 ) ) ) ).

% zero_le_even_power
thf(fact_6336_zero__le__even__power,axiom,
    ! [N3: nat,A2: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ).

% zero_le_even_power
thf(fact_6337_zero__le__even__power,axiom,
    ! [N3: nat,A2: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ N3 ) ) ) ).

% zero_le_even_power
thf(fact_6338_zero__le__even__power,axiom,
    ! [N3: nat,A2: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ N3 ) ) ) ).

% zero_le_even_power
thf(fact_6339_zero__le__odd__power,axiom,
    ! [N3: nat,A2: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ N3 ) )
        = ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ) ).

% zero_le_odd_power
thf(fact_6340_zero__le__odd__power,axiom,
    ! [N3: nat,A2: code_integer] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) )
        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ) ).

% zero_le_odd_power
thf(fact_6341_zero__le__odd__power,axiom,
    ! [N3: nat,A2: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ N3 ) )
        = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ) ).

% zero_le_odd_power
thf(fact_6342_zero__le__odd__power,axiom,
    ! [N3: nat,A2: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ N3 ) )
        = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ) ).

% zero_le_odd_power
thf(fact_6343_zero__le__power__eq,axiom,
    ! [A2: real,N3: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ N3 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          & ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ) ) ).

% zero_le_power_eq
thf(fact_6344_zero__le__power__eq,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ) ) ).

% zero_le_power_eq
thf(fact_6345_zero__le__power__eq,axiom,
    ! [A2: rat,N3: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ N3 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          & ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ) ) ).

% zero_le_power_eq
thf(fact_6346_zero__le__power__eq,axiom,
    ! [A2: int,N3: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ N3 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          & ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ) ) ).

% zero_le_power_eq
thf(fact_6347_zero__less__power__eq,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) )
      = ( ( N3 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          & ( A2 != zero_z3403309356797280102nteger ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ) ) ).

% zero_less_power_eq
thf(fact_6348_zero__less__power__eq,axiom,
    ! [A2: real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A2 @ N3 ) )
      = ( ( N3 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          & ( A2 != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          & ( ord_less_real @ zero_zero_real @ A2 ) ) ) ) ).

% zero_less_power_eq
thf(fact_6349_zero__less__power__eq,axiom,
    ! [A2: rat,N3: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ N3 ) )
      = ( ( N3 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          & ( A2 != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          & ( ord_less_rat @ zero_zero_rat @ A2 ) ) ) ) ).

% zero_less_power_eq
thf(fact_6350_zero__less__power__eq,axiom,
    ! [A2: int,N3: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A2 @ N3 ) )
      = ( ( N3 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          & ( A2 != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          & ( ord_less_int @ zero_zero_int @ A2 ) ) ) ) ).

% zero_less_power_eq
thf(fact_6351_even__mask__div__iff_H,axiom,
    ! [M: nat,N3: 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 ) ) @ N3 ) ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% even_mask_div_iff'
thf(fact_6352_even__mask__div__iff_H,axiom,
    ! [M: nat,N3: 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 ) ) @ N3 ) ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% even_mask_div_iff'
thf(fact_6353_even__mask__div__iff_H,axiom,
    ! [M: nat,N3: 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 ) ) @ N3 ) ) )
      = ( ord_less_eq_nat @ M @ N3 ) ) ).

% even_mask_div_iff'
thf(fact_6354_power__le__zero__eq,axiom,
    ! [A2: real,N3: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ A2 @ N3 ) @ zero_zero_real )
      = ( ( ord_less_nat @ zero_zero_nat @ N3 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( ord_less_eq_real @ A2 @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( A2 = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_6355_power__le__zero__eq,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ zero_z3403309356797280102nteger )
      = ( ( ord_less_nat @ zero_zero_nat @ N3 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( A2 = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_6356_power__le__zero__eq,axiom,
    ! [A2: rat,N3: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N3 ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ N3 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( ord_less_eq_rat @ A2 @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( A2 = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_6357_power__le__zero__eq,axiom,
    ! [A2: int,N3: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ A2 @ N3 ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ N3 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( ord_less_eq_int @ A2 @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( A2 = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_6358_even__mask__div__iff,axiom,
    ! [M: nat,N3: 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 ) ) @ N3 ) ) )
      = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 )
          = zero_z3403309356797280102nteger )
        | ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% even_mask_div_iff
thf(fact_6359_even__mask__div__iff,axiom,
    ! [M: nat,N3: 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 ) ) @ N3 ) ) )
      = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          = zero_zero_nat )
        | ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% even_mask_div_iff
thf(fact_6360_even__mask__div__iff,axiom,
    ! [M: nat,N3: 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 ) ) @ N3 ) ) )
      = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 )
          = zero_zero_int )
        | ( ord_less_eq_nat @ M @ N3 ) ) ) ).

% even_mask_div_iff
thf(fact_6361_divmod__divmod__step,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M5: num,N2: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M5 @ N2 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M5 ) ) @ ( unique5026877609467782581ep_nat @ N2 @ ( unique5055182867167087721od_nat @ M5 @ ( bit0 @ N2 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_6362_divmod__divmod__step,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M5: num,N2: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M5 @ N2 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M5 ) ) @ ( unique5024387138958732305ep_int @ N2 @ ( unique5052692396658037445od_int @ M5 @ ( bit0 @ N2 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_6363_divmod__divmod__step,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M5: num,N2: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M5 @ N2 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M5 ) ) @ ( unique4921790084139445826nteger @ N2 @ ( unique3479559517661332726nteger @ M5 @ ( bit0 @ N2 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_6364_Bernoulli__inequality__even,axiom,
    ! [N3: nat,X: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N3 ) ) ) ).

% Bernoulli_inequality_even
thf(fact_6365_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
    ! [N3: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ N3 ) ) )
          = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ N3 ) ) )
          = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.simps(3)
thf(fact_6366_even__mult__exp__div__exp__iff,axiom,
    ! [A2: code_integer,M: nat,N3: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( ( ord_less_nat @ N3 @ M )
        | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 )
          = zero_z3403309356797280102nteger )
        | ( ( ord_less_eq_nat @ M @ N3 )
          & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_6367_even__mult__exp__div__exp__iff,axiom,
    ! [A2: nat,M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( ( ord_less_nat @ N3 @ M )
        | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
          = zero_zero_nat )
        | ( ( ord_less_eq_nat @ M @ N3 )
          & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_6368_even__mult__exp__div__exp__iff,axiom,
    ! [A2: int,M: nat,N3: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( ( ord_less_nat @ N3 @ M )
        | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 )
          = zero_zero_int )
        | ( ( ord_less_eq_nat @ M @ N3 )
          & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_6369_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_V441764108873111860ildupi @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( suc @ zero_zero_nat ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( suc @ zero_zero_nat ) ) )
         => ~ ! [N: nat] :
                ( ( X
                  = ( suc @ ( suc @ N ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                     => ( Y
                        = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                     => ( Y
                        = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.elims
thf(fact_6370_VEBT__internal_OTb_H_Osimps_I3_J,axiom,
    ! [N3: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ N3 ) ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ N3 ) ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.simps(3)
thf(fact_6371_VEBT__internal_OTb_H_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_VEBT_Tb2 @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
         => ~ ! [N: nat] :
                ( ( X
                  = ( suc @ ( suc @ N ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                     => ( Y
                        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                     => ( Y
                        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.elims
thf(fact_6372_VEBT__internal_OTb_Osimps_I3_J,axiom,
    ! [N3: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( vEBT_VEBT_Tb @ ( suc @ ( suc @ N3 ) ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( vEBT_VEBT_Tb @ ( suc @ ( suc @ N3 ) ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb.simps(3)
thf(fact_6373_VEBT__internal_OT__vebt__buildupi_H_Osimps_I3_J,axiom,
    ! [N3: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ N3 ) ) )
          = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ N3 ) ) )
          = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.simps(3)
thf(fact_6374_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I3_J,axiom,
    ! [Va: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va ) ) )
          = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va ) ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
thf(fact_6375_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [Va: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va ) ) )
          = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va ) ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
thf(fact_6376_VEBT__internal_OTb_Oelims,axiom,
    ! [X: nat,Y: int] :
      ( ( ( vEBT_VEBT_Tb @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( numeral_numeral_int @ ( bit1 @ one ) ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( numeral_numeral_int @ ( bit1 @ one ) ) ) )
         => ~ ! [N: nat] :
                ( ( X
                  = ( suc @ ( suc @ N ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                     => ( Y
                        = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                     => ( Y
                        = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb.elims
thf(fact_6377_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_V8346862874174094_d_u_p @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
         => ~ ! [Va3: nat] :
                ( ( X
                  = ( suc @ ( suc @ Va3 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y
                        = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y
                        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
thf(fact_6378_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_V8646137997579335489_i_l_d @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
         => ~ ! [Va3: nat] :
                ( ( X
                  = ( suc @ ( suc @ Va3 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y
                        = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y
                        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
thf(fact_6379_pow__divides__pow__iff,axiom,
    ! [N3: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ A2 @ N3 ) @ ( power_power_nat @ B2 @ N3 ) )
        = ( dvd_dvd_nat @ A2 @ B2 ) ) ) ).

% pow_divides_pow_iff
thf(fact_6380_pow__divides__pow__iff,axiom,
    ! [N3: nat,A2: int,B2: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( dvd_dvd_int @ ( power_power_int @ A2 @ N3 ) @ ( power_power_int @ B2 @ N3 ) )
        = ( dvd_dvd_int @ A2 @ B2 ) ) ) ).

% pow_divides_pow_iff
thf(fact_6381_artanh__def,axiom,
    ( artanh_real
    = ( ^ [X3: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% artanh_def
thf(fact_6382_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_6383_bezout__add__strong__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero_nat )
     => ? [D5: nat,X4: nat,Y3: nat] :
          ( ( dvd_dvd_nat @ D5 @ A2 )
          & ( dvd_dvd_nat @ D5 @ B2 )
          & ( ( times_times_nat @ A2 @ X4 )
            = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y3 ) @ D5 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_6384_highi__def,axiom,
    ( vEBT_VEBT_highi
    = ( ^ [X3: nat,N2: nat] : ( heap_Time_return_nat @ ( divide_divide_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% highi_def
thf(fact_6385_nth__rule,axiom,
    ! [I: nat,Xs2: list_o,A2: array_o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( hoare_hoare_triple_o @ ( snga_assn_o @ A2 @ Xs2 ) @ ( array_nth_o @ A2 @ I )
        @ ^ [R5: $o] :
            ( times_times_assn @ ( snga_assn_o @ A2 @ Xs2 )
            @ ( pure_assn
              @ ( R5
                = ( nth_o @ Xs2 @ I ) ) ) ) ) ) ).

% nth_rule
thf(fact_6386_nth__rule,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,A2: array_VEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( hoare_1429296392585015714_VEBTi @ ( snga_assn_VEBT_VEBTi @ A2 @ Xs2 ) @ ( array_nth_VEBT_VEBTi @ A2 @ I )
        @ ^ [R5: vEBT_VEBTi] :
            ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ A2 @ Xs2 )
            @ ( pure_assn
              @ ( R5
                = ( nth_VEBT_VEBTi @ Xs2 @ I ) ) ) ) ) ) ).

% nth_rule
thf(fact_6387_nth__rule,axiom,
    ! [I: nat,Xs2: list_option_nat,A2: array_option_nat] :
      ( ( ord_less_nat @ I @ ( size_s6086282163384603972on_nat @ Xs2 ) )
     => ( hoare_7629718768684598413on_nat @ ( snga_assn_option_nat @ A2 @ Xs2 ) @ ( array_nth_option_nat @ A2 @ I )
        @ ^ [R5: option_nat] :
            ( times_times_assn @ ( snga_assn_option_nat @ A2 @ Xs2 )
            @ ( pure_assn
              @ ( R5
                = ( nth_option_nat @ Xs2 @ I ) ) ) ) ) ) ).

% nth_rule
thf(fact_6388_nth__rule,axiom,
    ! [I: nat,Xs2: list_nat,A2: array_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( hoare_3067605981109127869le_nat @ ( snga_assn_nat @ A2 @ Xs2 ) @ ( array_nth_nat @ A2 @ I )
        @ ^ [R5: nat] :
            ( times_times_assn @ ( snga_assn_nat @ A2 @ Xs2 )
            @ ( pure_assn
              @ ( R5
                = ( nth_nat @ Xs2 @ I ) ) ) ) ) ) ).

% nth_rule
thf(fact_6389_highsimp,axiom,
    ! [X: nat,N3: nat] :
      ( ( heap_Time_return_nat @ ( vEBT_VEBT_high @ X @ N3 ) )
      = ( vEBT_VEBT_highi @ X @ N3 ) ) ).

% highsimp
thf(fact_6390_lowsimp,axiom,
    ! [X: nat,N3: nat] :
      ( ( heap_Time_return_nat @ ( vEBT_VEBT_low @ X @ N3 ) )
      = ( vEBT_VEBT_lowi @ X @ N3 ) ) ).

% lowsimp
thf(fact_6391_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_6392_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_6393_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_6394_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_6395_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_6396_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_6397_ln__one,axiom,
    ( ( ln_ln_real @ one_one_real )
    = zero_zero_real ) ).

% ln_one
thf(fact_6398_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_6399_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_6400_zdvd__zdiffD,axiom,
    ! [K: int,M: int,N3: int] :
      ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N3 ) )
     => ( ( dvd_dvd_int @ K @ N3 )
       => ( dvd_dvd_int @ K @ M ) ) ) ).

% zdvd_zdiffD
thf(fact_6401_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_6402_log__def,axiom,
    ( log
    = ( ^ [A7: real,X3: real] : ( divide_divide_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ A7 ) ) ) ) ).

% log_def
thf(fact_6403_zdvd__not__zless,axiom,
    ! [M: int,N3: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_int @ M @ N3 )
       => ~ ( dvd_dvd_int @ N3 @ M ) ) ) ).

% zdvd_not_zless
thf(fact_6404_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_6405_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_6406_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_6407_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_6408_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_6409_zdvd__imp__le,axiom,
    ! [Z: int,N3: int] :
      ( ( dvd_dvd_int @ Z @ N3 )
     => ( ( ord_less_int @ zero_zero_int @ N3 )
       => ( ord_less_eq_int @ Z @ N3 ) ) ) ).

% zdvd_imp_le
thf(fact_6410_real__of__int__div,axiom,
    ! [D2: int,N3: int] :
      ( ( dvd_dvd_int @ D2 @ N3 )
     => ( ( ring_1_of_int_real @ ( divide_divide_int @ N3 @ D2 ) )
        = ( divide_divide_real @ ( ring_1_of_int_real @ N3 ) @ ( ring_1_of_int_real @ D2 ) ) ) ) ).

% real_of_int_div
thf(fact_6411_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K2: nat,M4: nat] :
          ( X
         != ( product_Pair_nat_nat @ K2 @ M4 ) ) ).

% prod_decode_aux.cases
thf(fact_6412_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_6413_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_6414_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_6415_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_6416_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_6417_int__div__sub__1,axiom,
    ! [M: int,N3: int] :
      ( ( ord_less_eq_int @ one_one_int @ M )
     => ( ( ( dvd_dvd_int @ M @ N3 )
         => ( ( divide_divide_int @ ( minus_minus_int @ N3 @ one_one_int ) @ M )
            = ( minus_minus_int @ ( divide_divide_int @ N3 @ M ) @ one_one_int ) ) )
        & ( ~ ( dvd_dvd_int @ M @ N3 )
         => ( ( divide_divide_int @ ( minus_minus_int @ N3 @ one_one_int ) @ M )
            = ( divide_divide_int @ N3 @ M ) ) ) ) ) ).

% int_div_sub_1
thf(fact_6418_bset_I9_J,axiom,
    ! [D2: int,D: int,B: set_int,T: int] :
      ( ( dvd_dvd_int @ D2 @ D )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B )
                 => ( X5
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X5 @ T ) )
           => ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D ) @ T ) ) ) ) ) ).

% bset(9)
thf(fact_6419_bset_I10_J,axiom,
    ! [D2: int,D: int,B: set_int,T: int] :
      ( ( dvd_dvd_int @ D2 @ D )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B )
                 => ( X5
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X5 @ T ) )
           => ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D ) @ T ) ) ) ) ) ).

% bset(10)
thf(fact_6420_aset_I9_J,axiom,
    ! [D2: int,D: int,A: set_int,T: int] :
      ( ( dvd_dvd_int @ D2 @ D )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A )
                 => ( X5
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X5 @ T ) )
           => ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D ) @ T ) ) ) ) ) ).

% aset(9)
thf(fact_6421_aset_I10_J,axiom,
    ! [D2: int,D: int,A: set_int,T: int] :
      ( ( dvd_dvd_int @ D2 @ D )
     => ! [X5: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A )
                 => ( X5
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ X5 @ T ) )
           => ~ ( dvd_dvd_int @ D2 @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D ) @ T ) ) ) ) ) ).

% aset(10)
thf(fact_6422_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_6423_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_6424_ln__realpow,axiom,
    ! [X: real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ln_ln_real @ ( power_power_real @ X @ N3 ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( ln_ln_real @ X ) ) ) ) ).

% ln_realpow
thf(fact_6425_even__diff__iff,axiom,
    ! [K: int,L2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L2 ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).

% even_diff_iff
thf(fact_6426_return__sp__rule,axiom,
    ! [P: assn,X: $o] :
      ( hoare_hoare_triple_o @ P @ ( heap_Time_return_o @ X )
      @ ^ [R5: $o] : ( times_times_assn @ P @ ( pure_assn @ ( R5 = X ) ) ) ) ).

% return_sp_rule
thf(fact_6427_return__sp__rule,axiom,
    ! [P: assn,X: vEBT_VEBTi] :
      ( hoare_1429296392585015714_VEBTi @ P @ ( heap_T3630416162098727440_VEBTi @ X )
      @ ^ [R5: vEBT_VEBTi] : ( times_times_assn @ P @ ( pure_assn @ ( R5 = X ) ) ) ) ).

% return_sp_rule
thf(fact_6428_return__sp__rule,axiom,
    ! [P: assn,X: option_nat] :
      ( hoare_7629718768684598413on_nat @ P @ ( heap_T3487192422709364219on_nat @ X )
      @ ^ [R5: option_nat] : ( times_times_assn @ P @ ( pure_assn @ ( R5 = X ) ) ) ) ).

% return_sp_rule
thf(fact_6429_return__sp__rule,axiom,
    ! [P: assn,X: nat] :
      ( hoare_3067605981109127869le_nat @ P @ ( heap_Time_return_nat @ X )
      @ ^ [R5: nat] : ( times_times_assn @ P @ ( pure_assn @ ( R5 = X ) ) ) ) ).

% return_sp_rule
thf(fact_6430_log__eq__div__ln__mult__log,axiom,
    ! [A2: real,B2: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( A2 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ B2 )
         => ( ( B2 != one_one_real )
           => ( ( ord_less_real @ zero_zero_real @ X )
             => ( ( log @ A2 @ X )
                = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B2 ) @ ( ln_ln_real @ A2 ) ) @ ( log @ B2 @ X ) ) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
thf(fact_6431_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( ( dvd_dvd_nat @ ( nat2 @ Z ) @ M )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
         => ( dvd_dvd_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
thf(fact_6432_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N: nat] :
            ( X
           != ( suc @ N ) ) ) ).

% list_decode.cases
thf(fact_6433_dvd__productE,axiom,
    ! [P6: nat,A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ P6 @ ( times_times_nat @ A2 @ B2 ) )
     => ~ ! [X4: nat,Y3: nat] :
            ( ( P6
              = ( times_times_nat @ X4 @ Y3 ) )
           => ( ( dvd_dvd_nat @ X4 @ A2 )
             => ~ ( dvd_dvd_nat @ Y3 @ B2 ) ) ) ) ).

% dvd_productE
thf(fact_6434_dvd__productE,axiom,
    ! [P6: int,A2: int,B2: int] :
      ( ( dvd_dvd_int @ P6 @ ( times_times_int @ A2 @ B2 ) )
     => ~ ! [X4: int,Y3: int] :
            ( ( P6
              = ( times_times_int @ X4 @ Y3 ) )
           => ( ( dvd_dvd_int @ X4 @ A2 )
             => ~ ( dvd_dvd_int @ Y3 @ B2 ) ) ) ) ).

% dvd_productE
thf(fact_6435_division__decomp,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) )
     => ? [B8: nat,C6: nat] :
          ( ( A2
            = ( times_times_nat @ B8 @ C6 ) )
          & ( dvd_dvd_nat @ B8 @ B2 )
          & ( dvd_dvd_nat @ C6 @ C2 ) ) ) ).

% division_decomp
thf(fact_6436_division__decomp,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( dvd_dvd_int @ A2 @ ( times_times_int @ B2 @ C2 ) )
     => ? [B8: int,C6: int] :
          ( ( A2
            = ( times_times_int @ B8 @ C6 ) )
          & ( dvd_dvd_int @ B8 @ B2 )
          & ( dvd_dvd_int @ C6 @ C2 ) ) ) ).

% division_decomp
thf(fact_6437_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A2: nat,B2: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( P @ A3 @ B3 )
          = ( P @ B3 @ A3 ) )
     => ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
       => ( ! [A3: nat,B3: nat] :
              ( ( P @ A3 @ B3 )
             => ( P @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% Euclid_induct
thf(fact_6438_gcd__nat_Oextremum,axiom,
    ! [A2: nat] : ( dvd_dvd_nat @ A2 @ zero_zero_nat ) ).

% gcd_nat.extremum
thf(fact_6439_gcd__nat_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
        & ( zero_zero_nat != A2 ) ) ).

% gcd_nat.extremum_strict
thf(fact_6440_gcd__nat_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
      = ( A2 = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
thf(fact_6441_gcd__nat_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
      = ( ( dvd_dvd_nat @ A2 @ zero_zero_nat )
        & ( A2 != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
thf(fact_6442_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
     => ( A2 = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
thf(fact_6443_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_6444_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_6445_dvd__pos__nat,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( dvd_dvd_nat @ M @ N3 )
       => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).

% dvd_pos_nat
thf(fact_6446_bezout__add__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D5: nat,X4: nat,Y3: nat] :
      ( ( dvd_dvd_nat @ D5 @ A2 )
      & ( dvd_dvd_nat @ D5 @ B2 )
      & ( ( ( times_times_nat @ A2 @ X4 )
          = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y3 ) @ D5 ) )
        | ( ( times_times_nat @ B2 @ X4 )
          = ( plus_plus_nat @ ( times_times_nat @ A2 @ Y3 ) @ D5 ) ) ) ) ).

% bezout_add_nat
thf(fact_6447_bezout__lemma__nat,axiom,
    ! [D2: nat,A2: nat,B2: nat,X: nat,Y: nat] :
      ( ( dvd_dvd_nat @ D2 @ A2 )
     => ( ( dvd_dvd_nat @ D2 @ B2 )
       => ( ( ( ( times_times_nat @ A2 @ X )
              = ( plus_plus_nat @ ( times_times_nat @ B2 @ Y ) @ D2 ) )
            | ( ( times_times_nat @ B2 @ X )
              = ( plus_plus_nat @ ( times_times_nat @ A2 @ Y ) @ D2 ) ) )
         => ? [X4: nat,Y3: nat] :
              ( ( dvd_dvd_nat @ D2 @ A2 )
              & ( dvd_dvd_nat @ D2 @ ( plus_plus_nat @ A2 @ B2 ) )
              & ( ( ( times_times_nat @ A2 @ X4 )
                  = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ Y3 ) @ D2 ) )
                | ( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ X4 )
                  = ( plus_plus_nat @ ( times_times_nat @ A2 @ Y3 ) @ D2 ) ) ) ) ) ) ) ).

% bezout_lemma_nat
thf(fact_6448_bezout1__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D5: nat,X4: nat,Y3: nat] :
      ( ( dvd_dvd_nat @ D5 @ A2 )
      & ( dvd_dvd_nat @ D5 @ B2 )
      & ( ( ( minus_minus_nat @ ( times_times_nat @ A2 @ X4 ) @ ( times_times_nat @ B2 @ Y3 ) )
          = D5 )
        | ( ( minus_minus_nat @ ( times_times_nat @ B2 @ X4 ) @ ( times_times_nat @ A2 @ Y3 ) )
          = D5 ) ) ) ).

% bezout1_nat
thf(fact_6449_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_6450_lowi__def,axiom,
    ( vEBT_VEBT_lowi
    = ( ^ [X3: nat,N2: nat] : ( heap_Time_return_nat @ ( modulo_modulo_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% lowi_def
thf(fact_6451_unset__bit__0,axiom,
    ! [A2: int] :
      ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A2 )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_6452_unset__bit__0,axiom,
    ! [A2: nat] :
      ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A2 )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_6453_fi__rule,axiom,
    ! [P: assn,C2: heap_Time_Heap_o,Q: $o > assn,Ps: assn,F: assn] :
      ( ( hoare_hoare_triple_o @ P @ C2 @ Q )
     => ( ( entails @ Ps @ ( times_times_assn @ P @ F ) )
       => ( hoare_hoare_triple_o @ Ps @ C2
          @ ^ [X3: $o] : ( times_times_assn @ ( Q @ X3 ) @ F ) ) ) ) ).

% fi_rule
thf(fact_6454_fi__rule,axiom,
    ! [P: assn,C2: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,Ps: assn,F: assn] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C2 @ Q )
     => ( ( entails @ Ps @ ( times_times_assn @ P @ F ) )
       => ( hoare_1429296392585015714_VEBTi @ Ps @ C2
          @ ^ [X3: vEBT_VEBTi] : ( times_times_assn @ ( Q @ X3 ) @ F ) ) ) ) ).

% fi_rule
thf(fact_6455_fi__rule,axiom,
    ! [P: assn,C2: heap_T2636463487746394924on_nat,Q: option_nat > assn,Ps: assn,F: assn] :
      ( ( hoare_7629718768684598413on_nat @ P @ C2 @ Q )
     => ( ( entails @ Ps @ ( times_times_assn @ P @ F ) )
       => ( hoare_7629718768684598413on_nat @ Ps @ C2
          @ ^ [X3: option_nat] : ( times_times_assn @ ( Q @ X3 ) @ F ) ) ) ) ).

% fi_rule
thf(fact_6456_fi__rule,axiom,
    ! [P: assn,C2: heap_Time_Heap_nat,Q: nat > assn,Ps: assn,F: assn] :
      ( ( hoare_3067605981109127869le_nat @ P @ C2 @ Q )
     => ( ( entails @ Ps @ ( times_times_assn @ P @ F ) )
       => ( hoare_3067605981109127869le_nat @ Ps @ C2
          @ ^ [X3: nat] : ( times_times_assn @ ( Q @ X3 ) @ F ) ) ) ) ).

% fi_rule
thf(fact_6457_low__def,axiom,
    ( vEBT_VEBT_low
    = ( ^ [X3: nat,N2: nat] : ( modulo_modulo_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% low_def
thf(fact_6458_return__cons__rule,axiom,
    ! [P: assn,Q: $o > assn,X: $o] :
      ( ( entails @ P @ ( Q @ X ) )
     => ( hoare_hoare_triple_o @ P @ ( heap_Time_return_o @ X ) @ Q ) ) ).

% return_cons_rule
thf(fact_6459_return__cons__rule,axiom,
    ! [P: assn,Q: vEBT_VEBTi > assn,X: vEBT_VEBTi] :
      ( ( entails @ P @ ( Q @ X ) )
     => ( hoare_1429296392585015714_VEBTi @ P @ ( heap_T3630416162098727440_VEBTi @ X ) @ Q ) ) ).

% return_cons_rule
thf(fact_6460_return__cons__rule,axiom,
    ! [P: assn,Q: option_nat > assn,X: option_nat] :
      ( ( entails @ P @ ( Q @ X ) )
     => ( hoare_7629718768684598413on_nat @ P @ ( heap_T3487192422709364219on_nat @ X ) @ Q ) ) ).

% return_cons_rule
thf(fact_6461_return__cons__rule,axiom,
    ! [P: assn,Q: nat > assn,X: nat] :
      ( ( entails @ P @ ( Q @ X ) )
     => ( hoare_3067605981109127869le_nat @ P @ ( heap_Time_return_nat @ X ) @ Q ) ) ).

% return_cons_rule
thf(fact_6462_mod__mod__trivial,axiom,
    ! [A2: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A2 @ B2 ) @ B2 )
      = ( modulo_modulo_nat @ A2 @ B2 ) ) ).

% mod_mod_trivial
thf(fact_6463_mod__mod__trivial,axiom,
    ! [A2: int,B2: int] :
      ( ( modulo_modulo_int @ ( modulo_modulo_int @ A2 @ B2 ) @ B2 )
      = ( modulo_modulo_int @ A2 @ B2 ) ) ).

% mod_mod_trivial
thf(fact_6464_mod__mod__trivial,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A2 @ B2 ) @ B2 )
      = ( modulo364778990260209775nteger @ A2 @ B2 ) ) ).

% mod_mod_trivial
thf(fact_6465_bits__mod__0,axiom,
    ! [A2: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% bits_mod_0
thf(fact_6466_bits__mod__0,axiom,
    ! [A2: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A2 )
      = zero_zero_int ) ).

% bits_mod_0
thf(fact_6467_bits__mod__0,axiom,
    ! [A2: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A2 )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_0
thf(fact_6468_mod__self,axiom,
    ! [A2: nat] :
      ( ( modulo_modulo_nat @ A2 @ A2 )
      = zero_zero_nat ) ).

% mod_self
thf(fact_6469_mod__self,axiom,
    ! [A2: int] :
      ( ( modulo_modulo_int @ A2 @ A2 )
      = zero_zero_int ) ).

% mod_self
thf(fact_6470_mod__self,axiom,
    ! [A2: code_integer] :
      ( ( modulo364778990260209775nteger @ A2 @ A2 )
      = zero_z3403309356797280102nteger ) ).

% mod_self
thf(fact_6471_mod__by__0,axiom,
    ! [A2: nat] :
      ( ( modulo_modulo_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% mod_by_0
thf(fact_6472_mod__by__0,axiom,
    ! [A2: int] :
      ( ( modulo_modulo_int @ A2 @ zero_zero_int )
      = A2 ) ).

% mod_by_0
thf(fact_6473_mod__by__0,axiom,
    ! [A2: code_integer] :
      ( ( modulo364778990260209775nteger @ A2 @ zero_z3403309356797280102nteger )
      = A2 ) ).

% mod_by_0
thf(fact_6474_mod__0,axiom,
    ! [A2: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% mod_0
thf(fact_6475_mod__0,axiom,
    ! [A2: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A2 )
      = zero_zero_int ) ).

% mod_0
thf(fact_6476_mod__0,axiom,
    ! [A2: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A2 )
      = zero_z3403309356797280102nteger ) ).

% mod_0
thf(fact_6477_mod__add__self2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
      = ( modulo_modulo_nat @ A2 @ B2 ) ) ).

% mod_add_self2
thf(fact_6478_mod__add__self2,axiom,
    ! [A2: int,B2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
      = ( modulo_modulo_int @ A2 @ B2 ) ) ).

% mod_add_self2
thf(fact_6479_mod__add__self2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ B2 )
      = ( modulo364778990260209775nteger @ A2 @ B2 ) ) ).

% mod_add_self2
thf(fact_6480_mod__add__self1,axiom,
    ! [B2: nat,A2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
      = ( modulo_modulo_nat @ A2 @ B2 ) ) ).

% mod_add_self1
thf(fact_6481_mod__add__self1,axiom,
    ! [B2: int,A2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
      = ( modulo_modulo_int @ A2 @ B2 ) ) ).

% mod_add_self1
thf(fact_6482_mod__add__self1,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B2 @ A2 ) @ B2 )
      = ( modulo364778990260209775nteger @ A2 @ B2 ) ) ).

% mod_add_self1
thf(fact_6483_minus__mod__self2,axiom,
    ! [A2: int,B2: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ A2 @ B2 ) @ B2 )
      = ( modulo_modulo_int @ A2 @ B2 ) ) ).

% minus_mod_self2
thf(fact_6484_minus__mod__self2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ B2 )
      = ( modulo364778990260209775nteger @ A2 @ B2 ) ) ).

% minus_mod_self2
thf(fact_6485_unset__bit__nonnegative__int__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N3 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% unset_bit_nonnegative_int_iff
thf(fact_6486_unset__bit__negative__int__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N3 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% unset_bit_negative_int_iff
thf(fact_6487_nat__mod__eq_H,axiom,
    ! [A2: nat,N3: nat] :
      ( ( ord_less_nat @ A2 @ N3 )
     => ( ( modulo_modulo_nat @ A2 @ N3 )
        = A2 ) ) ).

% nat_mod_eq'
thf(fact_6488_mod__less,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ N3 )
     => ( ( modulo_modulo_nat @ M @ N3 )
        = M ) ) ).

% mod_less
thf(fact_6489_bits__mod__by__1,axiom,
    ! [A2: nat] :
      ( ( modulo_modulo_nat @ A2 @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_6490_bits__mod__by__1,axiom,
    ! [A2: int] :
      ( ( modulo_modulo_int @ A2 @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_6491_bits__mod__by__1,axiom,
    ! [A2: code_integer] :
      ( ( modulo364778990260209775nteger @ A2 @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_by_1
thf(fact_6492_mod__by__1,axiom,
    ! [A2: nat] :
      ( ( modulo_modulo_nat @ A2 @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_6493_mod__by__1,axiom,
    ! [A2: int] :
      ( ( modulo_modulo_int @ A2 @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_6494_mod__by__1,axiom,
    ! [A2: code_integer] :
      ( ( modulo364778990260209775nteger @ A2 @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% mod_by_1
thf(fact_6495_mod__mult__self2__is__0,axiom,
    ! [A2: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B2 ) @ B2 )
      = zero_zero_nat ) ).

% mod_mult_self2_is_0
thf(fact_6496_mod__mult__self2__is__0,axiom,
    ! [A2: int,B2: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A2 @ B2 ) @ B2 )
      = zero_zero_int ) ).

% mod_mult_self2_is_0
thf(fact_6497_mod__mult__self2__is__0,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ B2 )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self2_is_0
thf(fact_6498_mod__mult__self1__is__0,axiom,
    ! [B2: nat,A2: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ B2 @ A2 ) @ B2 )
      = zero_zero_nat ) ).

% mod_mult_self1_is_0
thf(fact_6499_mod__mult__self1__is__0,axiom,
    ! [B2: int,A2: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ B2 @ A2 ) @ B2 )
      = zero_zero_int ) ).

% mod_mult_self1_is_0
thf(fact_6500_mod__mult__self1__is__0,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B2 @ A2 ) @ B2 )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self1_is_0
thf(fact_6501_bits__mod__div__trivial,axiom,
    ! [A2: nat,B2: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A2 @ B2 ) @ B2 )
      = zero_zero_nat ) ).

% bits_mod_div_trivial
thf(fact_6502_bits__mod__div__trivial,axiom,
    ! [A2: int,B2: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A2 @ B2 ) @ B2 )
      = zero_zero_int ) ).

% bits_mod_div_trivial
thf(fact_6503_bits__mod__div__trivial,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A2 @ B2 ) @ B2 )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_div_trivial
thf(fact_6504_mod__div__trivial,axiom,
    ! [A2: nat,B2: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A2 @ B2 ) @ B2 )
      = zero_zero_nat ) ).

% mod_div_trivial
thf(fact_6505_mod__div__trivial,axiom,
    ! [A2: int,B2: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A2 @ B2 ) @ B2 )
      = zero_zero_int ) ).

% mod_div_trivial
thf(fact_6506_mod__div__trivial,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A2 @ B2 ) @ B2 )
      = zero_z3403309356797280102nteger ) ).

% mod_div_trivial
thf(fact_6507_mod__mult__self4,axiom,
    ! [B2: nat,C2: nat,A2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ C2 ) @ A2 ) @ B2 )
      = ( modulo_modulo_nat @ A2 @ B2 ) ) ).

% mod_mult_self4
thf(fact_6508_mod__mult__self4,axiom,
    ! [B2: int,C2: int,A2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B2 @ C2 ) @ A2 ) @ B2 )
      = ( modulo_modulo_int @ A2 @ B2 ) ) ).

% mod_mult_self4
thf(fact_6509_mod__mult__self4,axiom,
    ! [B2: code_integer,C2: code_integer,A2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ C2 ) @ A2 ) @ B2 )
      = ( modulo364778990260209775nteger @ A2 @ B2 ) ) ).

% mod_mult_self4
thf(fact_6510_mod__mult__self3,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C2 @ B2 ) @ A2 ) @ B2 )
      = ( modulo_modulo_nat @ A2 @ B2 ) ) ).

% mod_mult_self3
thf(fact_6511_mod__mult__self3,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C2 @ B2 ) @ A2 ) @ B2 )
      = ( modulo_modulo_int @ A2 @ B2 ) ) ).

% mod_mult_self3
thf(fact_6512_mod__mult__self3,axiom,
    ! [C2: code_integer,B2: code_integer,A2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C2 @ B2 ) @ A2 ) @ B2 )
      = ( modulo364778990260209775nteger @ A2 @ B2 ) ) ).

% mod_mult_self3
thf(fact_6513_mod__mult__self2,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) ) @ B2 )
      = ( modulo_modulo_nat @ A2 @ B2 ) ) ).

% mod_mult_self2
thf(fact_6514_mod__mult__self2,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A2 @ ( times_times_int @ B2 @ C2 ) ) @ B2 )
      = ( modulo_modulo_int @ A2 @ B2 ) ) ).

% mod_mult_self2
thf(fact_6515_mod__mult__self2,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A2 @ ( times_3573771949741848930nteger @ B2 @ C2 ) ) @ B2 )
      = ( modulo364778990260209775nteger @ A2 @ B2 ) ) ).

% mod_mult_self2
thf(fact_6516_mod__mult__self1,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ C2 @ B2 ) ) @ B2 )
      = ( modulo_modulo_nat @ A2 @ B2 ) ) ).

% mod_mult_self1
thf(fact_6517_mod__mult__self1,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A2 @ ( times_times_int @ C2 @ B2 ) ) @ B2 )
      = ( modulo_modulo_int @ A2 @ B2 ) ) ).

% mod_mult_self1
thf(fact_6518_mod__mult__self1,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A2 @ ( times_3573771949741848930nteger @ C2 @ B2 ) ) @ B2 )
      = ( modulo364778990260209775nteger @ A2 @ B2 ) ) ).

% mod_mult_self1
thf(fact_6519_dvd__imp__mod__0,axiom,
    ! [A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ A2 @ B2 )
     => ( ( modulo_modulo_nat @ B2 @ A2 )
        = zero_zero_nat ) ) ).

% dvd_imp_mod_0
thf(fact_6520_dvd__imp__mod__0,axiom,
    ! [A2: int,B2: int] :
      ( ( dvd_dvd_int @ A2 @ B2 )
     => ( ( modulo_modulo_int @ B2 @ A2 )
        = zero_zero_int ) ) ).

% dvd_imp_mod_0
thf(fact_6521_dvd__imp__mod__0,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A2 @ B2 )
     => ( ( modulo364778990260209775nteger @ B2 @ A2 )
        = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
thf(fact_6522_mod__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_6523_Suc__mod__mult__self4,axiom,
    ! [N3: nat,K: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N3 @ K ) @ M ) ) @ N3 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N3 ) ) ).

% Suc_mod_mult_self4
thf(fact_6524_Suc__mod__mult__self3,axiom,
    ! [K: nat,N3: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N3 ) @ M ) ) @ N3 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N3 ) ) ).

% Suc_mod_mult_self3
thf(fact_6525_Suc__mod__mult__self2,axiom,
    ! [M: nat,N3: nat,K: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N3 @ K ) ) ) @ N3 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N3 ) ) ).

% Suc_mod_mult_self2
thf(fact_6526_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N3: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N3 ) ) ) @ N3 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N3 ) ) ).

% Suc_mod_mult_self1
thf(fact_6527_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_6528_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_6529_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_6530_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_6531_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_6532_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_6533_even__mod__2__iff,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) ) ).

% even_mod_2_iff
thf(fact_6534_even__mod__2__iff,axiom,
    ! [A2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) ).

% even_mod_2_iff
thf(fact_6535_even__mod__2__iff,axiom,
    ! [A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) ) ).

% even_mod_2_iff
thf(fact_6536_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_6537_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N3: nat] :
      ( ( ( numeral_numeral_nat @ K )
       != one_one_nat )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N3 ) ) @ ( numeral_numeral_nat @ K ) )
        = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
thf(fact_6538_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: nat] :
      ( ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != one_one_nat )
      = ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_6539_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: int] :
      ( ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != one_one_int )
      = ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_6540_not__mod__2__eq__1__eq__0,axiom,
    ! [A2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != one_one_Code_integer )
      = ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_6541_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: nat] :
      ( ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != zero_zero_nat )
      = ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_6542_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: int] :
      ( ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != zero_zero_int )
      = ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_6543_not__mod__2__eq__0__eq__1,axiom,
    ! [A2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_6544_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N3: nat] :
      ( ( ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != ( suc @ zero_zero_nat ) )
      = ( ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
thf(fact_6545_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_6546_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_6547_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N3: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N3 ) ) ) )
      = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N3 ) ) ) ).

% mod_Suc_eq_mod_add3
thf(fact_6548_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_6549_even__succ__mod__exp,axiom,
    ! [A2: nat,N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
          = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_6550_even__succ__mod__exp,axiom,
    ! [A2: int,N3: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
          = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_6551_even__succ__mod__exp,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) )
          = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_6552_of__nat__mod,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N3 ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% of_nat_mod
thf(fact_6553_of__nat__mod,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N3 ) )
      = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N3 ) ) ) ).

% of_nat_mod
thf(fact_6554_of__nat__mod,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N3 ) )
      = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N3 ) ) ) ).

% of_nat_mod
thf(fact_6555_mod__add__right__eq,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ ( modulo_modulo_nat @ B2 @ C2 ) ) @ C2 )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 ) ) ).

% mod_add_right_eq
thf(fact_6556_mod__add__right__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A2 @ ( modulo_modulo_int @ B2 @ C2 ) ) @ C2 )
      = ( modulo_modulo_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 ) ) ).

% mod_add_right_eq
thf(fact_6557_mod__add__right__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A2 @ ( modulo364778990260209775nteger @ B2 @ C2 ) ) @ C2 )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ C2 ) ) ).

% mod_add_right_eq
thf(fact_6558_mod__add__left__eq,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A2 @ C2 ) @ B2 ) @ C2 )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 ) ) ).

% mod_add_left_eq
thf(fact_6559_mod__add__left__eq,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A2 @ C2 ) @ B2 ) @ C2 )
      = ( modulo_modulo_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 ) ) ).

% mod_add_left_eq
thf(fact_6560_mod__add__left__eq,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ C2 ) @ B2 ) @ C2 )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ C2 ) ) ).

% mod_add_left_eq
thf(fact_6561_mod__add__cong,axiom,
    ! [A2: nat,C2: nat,A4: nat,B2: nat,B4: nat] :
      ( ( ( modulo_modulo_nat @ A2 @ C2 )
        = ( modulo_modulo_nat @ A4 @ C2 ) )
     => ( ( ( modulo_modulo_nat @ B2 @ C2 )
          = ( modulo_modulo_nat @ B4 @ C2 ) )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 )
          = ( modulo_modulo_nat @ ( plus_plus_nat @ A4 @ B4 ) @ C2 ) ) ) ) ).

% mod_add_cong
thf(fact_6562_mod__add__cong,axiom,
    ! [A2: int,C2: int,A4: int,B2: int,B4: int] :
      ( ( ( modulo_modulo_int @ A2 @ C2 )
        = ( modulo_modulo_int @ A4 @ C2 ) )
     => ( ( ( modulo_modulo_int @ B2 @ C2 )
          = ( modulo_modulo_int @ B4 @ C2 ) )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 )
          = ( modulo_modulo_int @ ( plus_plus_int @ A4 @ B4 ) @ C2 ) ) ) ) ).

% mod_add_cong
thf(fact_6563_mod__add__cong,axiom,
    ! [A2: code_integer,C2: code_integer,A4: code_integer,B2: code_integer,B4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A2 @ C2 )
        = ( modulo364778990260209775nteger @ A4 @ C2 ) )
     => ( ( ( modulo364778990260209775nteger @ B2 @ C2 )
          = ( modulo364778990260209775nteger @ B4 @ C2 ) )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ C2 )
          = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A4 @ B4 ) @ C2 ) ) ) ) ).

% mod_add_cong
thf(fact_6564_mod__add__eq,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A2 @ C2 ) @ ( modulo_modulo_nat @ B2 @ C2 ) ) @ C2 )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 ) ) ).

% mod_add_eq
thf(fact_6565_mod__add__eq,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A2 @ C2 ) @ ( modulo_modulo_int @ B2 @ C2 ) ) @ C2 )
      = ( modulo_modulo_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 ) ) ).

% mod_add_eq
thf(fact_6566_mod__add__eq,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ C2 ) @ ( modulo364778990260209775nteger @ B2 @ C2 ) ) @ C2 )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ C2 ) ) ).

% mod_add_eq
thf(fact_6567_mod__mult__right__eq,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A2 @ ( modulo_modulo_nat @ B2 @ C2 ) ) @ C2 )
      = ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_right_eq
thf(fact_6568_mod__mult__right__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A2 @ ( modulo_modulo_int @ B2 @ C2 ) ) @ C2 )
      = ( modulo_modulo_int @ ( times_times_int @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_right_eq
thf(fact_6569_mod__mult__right__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ ( modulo364778990260209775nteger @ B2 @ C2 ) ) @ C2 )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_right_eq
thf(fact_6570_mod__mult__left__eq,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A2 @ C2 ) @ B2 ) @ C2 )
      = ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_left_eq
thf(fact_6571_mod__mult__left__eq,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A2 @ C2 ) @ B2 ) @ C2 )
      = ( modulo_modulo_int @ ( times_times_int @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_left_eq
thf(fact_6572_mod__mult__left__eq,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A2 @ C2 ) @ B2 ) @ C2 )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_left_eq
thf(fact_6573_mult__mod__right,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( times_times_nat @ C2 @ ( modulo_modulo_nat @ A2 @ B2 ) )
      = ( modulo_modulo_nat @ ( times_times_nat @ C2 @ A2 ) @ ( times_times_nat @ C2 @ B2 ) ) ) ).

% mult_mod_right
thf(fact_6574_mult__mod__right,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( times_times_int @ C2 @ ( modulo_modulo_int @ A2 @ B2 ) )
      = ( modulo_modulo_int @ ( times_times_int @ C2 @ A2 ) @ ( times_times_int @ C2 @ B2 ) ) ) ).

% mult_mod_right
thf(fact_6575_mult__mod__right,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( times_3573771949741848930nteger @ C2 @ ( modulo364778990260209775nteger @ A2 @ B2 ) )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C2 @ A2 ) @ ( times_3573771949741848930nteger @ C2 @ B2 ) ) ) ).

% mult_mod_right
thf(fact_6576_mod__mult__mult2,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A2 @ C2 ) @ ( times_times_nat @ B2 @ C2 ) )
      = ( times_times_nat @ ( modulo_modulo_nat @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_mult2
thf(fact_6577_mod__mult__mult2,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A2 @ C2 ) @ ( times_times_int @ B2 @ C2 ) )
      = ( times_times_int @ ( modulo_modulo_int @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_mult2
thf(fact_6578_mod__mult__mult2,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ C2 ) @ ( times_3573771949741848930nteger @ B2 @ C2 ) )
      = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_mult2
thf(fact_6579_mod__mult__cong,axiom,
    ! [A2: nat,C2: nat,A4: nat,B2: nat,B4: nat] :
      ( ( ( modulo_modulo_nat @ A2 @ C2 )
        = ( modulo_modulo_nat @ A4 @ C2 ) )
     => ( ( ( modulo_modulo_nat @ B2 @ C2 )
          = ( modulo_modulo_nat @ B4 @ C2 ) )
       => ( ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 )
          = ( modulo_modulo_nat @ ( times_times_nat @ A4 @ B4 ) @ C2 ) ) ) ) ).

% mod_mult_cong
thf(fact_6580_mod__mult__cong,axiom,
    ! [A2: int,C2: int,A4: int,B2: int,B4: int] :
      ( ( ( modulo_modulo_int @ A2 @ C2 )
        = ( modulo_modulo_int @ A4 @ C2 ) )
     => ( ( ( modulo_modulo_int @ B2 @ C2 )
          = ( modulo_modulo_int @ B4 @ C2 ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ A2 @ B2 ) @ C2 )
          = ( modulo_modulo_int @ ( times_times_int @ A4 @ B4 ) @ C2 ) ) ) ) ).

% mod_mult_cong
thf(fact_6581_mod__mult__cong,axiom,
    ! [A2: code_integer,C2: code_integer,A4: code_integer,B2: code_integer,B4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A2 @ C2 )
        = ( modulo364778990260209775nteger @ A4 @ C2 ) )
     => ( ( ( modulo364778990260209775nteger @ B2 @ C2 )
          = ( modulo364778990260209775nteger @ B4 @ C2 ) )
       => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C2 )
          = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A4 @ B4 ) @ C2 ) ) ) ) ).

% mod_mult_cong
thf(fact_6582_mod__mult__eq,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A2 @ C2 ) @ ( modulo_modulo_nat @ B2 @ C2 ) ) @ C2 )
      = ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_eq
thf(fact_6583_mod__mult__eq,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A2 @ C2 ) @ ( modulo_modulo_int @ B2 @ C2 ) ) @ C2 )
      = ( modulo_modulo_int @ ( times_times_int @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_eq
thf(fact_6584_mod__mult__eq,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A2 @ C2 ) @ ( modulo364778990260209775nteger @ B2 @ C2 ) ) @ C2 )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C2 ) ) ).

% mod_mult_eq
thf(fact_6585_mod__diff__right__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ A2 @ ( modulo_modulo_int @ B2 @ C2 ) ) @ C2 )
      = ( modulo_modulo_int @ ( minus_minus_int @ A2 @ B2 ) @ C2 ) ) ).

% mod_diff_right_eq
thf(fact_6586_mod__diff__right__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A2 @ ( modulo364778990260209775nteger @ B2 @ C2 ) ) @ C2 )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ C2 ) ) ).

% mod_diff_right_eq
thf(fact_6587_mod__diff__left__eq,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A2 @ C2 ) @ B2 ) @ C2 )
      = ( modulo_modulo_int @ ( minus_minus_int @ A2 @ B2 ) @ C2 ) ) ).

% mod_diff_left_eq
thf(fact_6588_mod__diff__left__eq,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A2 @ C2 ) @ B2 ) @ C2 )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ C2 ) ) ).

% mod_diff_left_eq
thf(fact_6589_mod__diff__cong,axiom,
    ! [A2: int,C2: int,A4: int,B2: int,B4: int] :
      ( ( ( modulo_modulo_int @ A2 @ C2 )
        = ( modulo_modulo_int @ A4 @ C2 ) )
     => ( ( ( modulo_modulo_int @ B2 @ C2 )
          = ( modulo_modulo_int @ B4 @ C2 ) )
       => ( ( modulo_modulo_int @ ( minus_minus_int @ A2 @ B2 ) @ C2 )
          = ( modulo_modulo_int @ ( minus_minus_int @ A4 @ B4 ) @ C2 ) ) ) ) ).

% mod_diff_cong
thf(fact_6590_mod__diff__cong,axiom,
    ! [A2: code_integer,C2: code_integer,A4: code_integer,B2: code_integer,B4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A2 @ C2 )
        = ( modulo364778990260209775nteger @ A4 @ C2 ) )
     => ( ( ( modulo364778990260209775nteger @ B2 @ C2 )
          = ( modulo364778990260209775nteger @ B4 @ C2 ) )
       => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ C2 )
          = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A4 @ B4 ) @ C2 ) ) ) ) ).

% mod_diff_cong
thf(fact_6591_mod__diff__eq,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A2 @ C2 ) @ ( modulo_modulo_int @ B2 @ C2 ) ) @ C2 )
      = ( modulo_modulo_int @ ( minus_minus_int @ A2 @ B2 ) @ C2 ) ) ).

% mod_diff_eq
thf(fact_6592_mod__diff__eq,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A2 @ C2 ) @ ( modulo364778990260209775nteger @ B2 @ C2 ) ) @ C2 )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) @ C2 ) ) ).

% mod_diff_eq
thf(fact_6593_power__mod,axiom,
    ! [A2: nat,B2: nat,N3: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A2 @ B2 ) @ N3 ) @ B2 )
      = ( modulo_modulo_nat @ ( power_power_nat @ A2 @ N3 ) @ B2 ) ) ).

% power_mod
thf(fact_6594_power__mod,axiom,
    ! [A2: int,B2: int,N3: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A2 @ B2 ) @ N3 ) @ B2 )
      = ( modulo_modulo_int @ ( power_power_int @ A2 @ N3 ) @ B2 ) ) ).

% power_mod
thf(fact_6595_power__mod,axiom,
    ! [A2: code_integer,B2: code_integer,N3: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A2 @ B2 ) @ N3 ) @ B2 )
      = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ B2 ) ) ).

% power_mod
thf(fact_6596_mod__mod__cancel,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ C2 @ B2 )
     => ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A2 @ B2 ) @ C2 )
        = ( modulo_modulo_nat @ A2 @ C2 ) ) ) ).

% mod_mod_cancel
thf(fact_6597_mod__mod__cancel,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( dvd_dvd_int @ C2 @ B2 )
     => ( ( modulo_modulo_int @ ( modulo_modulo_int @ A2 @ B2 ) @ C2 )
        = ( modulo_modulo_int @ A2 @ C2 ) ) ) ).

% mod_mod_cancel
thf(fact_6598_mod__mod__cancel,axiom,
    ! [C2: code_integer,B2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C2 @ B2 )
     => ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A2 @ B2 ) @ C2 )
        = ( modulo364778990260209775nteger @ A2 @ C2 ) ) ) ).

% mod_mod_cancel
thf(fact_6599_dvd__mod,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N3 )
       => ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N3 ) ) ) ) ).

% dvd_mod
thf(fact_6600_dvd__mod,axiom,
    ! [K: int,M: int,N3: int] :
      ( ( dvd_dvd_int @ K @ M )
     => ( ( dvd_dvd_int @ K @ N3 )
       => ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N3 ) ) ) ) ).

% dvd_mod
thf(fact_6601_dvd__mod,axiom,
    ! [K: code_integer,M: code_integer,N3: code_integer] :
      ( ( dvd_dvd_Code_integer @ K @ M )
     => ( ( dvd_dvd_Code_integer @ K @ N3 )
       => ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N3 ) ) ) ) ).

% dvd_mod
thf(fact_6602_dvd__mod__imp__dvd,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( dvd_dvd_nat @ C2 @ ( modulo_modulo_nat @ A2 @ B2 ) )
     => ( ( dvd_dvd_nat @ C2 @ B2 )
       => ( dvd_dvd_nat @ C2 @ A2 ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_6603_dvd__mod__imp__dvd,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( dvd_dvd_int @ C2 @ ( modulo_modulo_int @ A2 @ B2 ) )
     => ( ( dvd_dvd_int @ C2 @ B2 )
       => ( dvd_dvd_int @ C2 @ A2 ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_6604_dvd__mod__imp__dvd,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C2 @ ( modulo364778990260209775nteger @ A2 @ B2 ) )
     => ( ( dvd_dvd_Code_integer @ C2 @ B2 )
       => ( dvd_dvd_Code_integer @ C2 @ A2 ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_6605_dvd__mod__iff,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ C2 @ B2 )
     => ( ( dvd_dvd_nat @ C2 @ ( modulo_modulo_nat @ A2 @ B2 ) )
        = ( dvd_dvd_nat @ C2 @ A2 ) ) ) ).

% dvd_mod_iff
thf(fact_6606_dvd__mod__iff,axiom,
    ! [C2: int,B2: int,A2: int] :
      ( ( dvd_dvd_int @ C2 @ B2 )
     => ( ( dvd_dvd_int @ C2 @ ( modulo_modulo_int @ A2 @ B2 ) )
        = ( dvd_dvd_int @ C2 @ A2 ) ) ) ).

% dvd_mod_iff
thf(fact_6607_dvd__mod__iff,axiom,
    ! [C2: code_integer,B2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ C2 @ B2 )
     => ( ( dvd_dvd_Code_integer @ C2 @ ( modulo364778990260209775nteger @ A2 @ B2 ) )
        = ( dvd_dvd_Code_integer @ C2 @ A2 ) ) ) ).

% dvd_mod_iff
thf(fact_6608_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N3 ) ) ) @ N3 )
      = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N3 ) ) ).

% mod_Suc_Suc_eq
thf(fact_6609_mod__Suc__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N3 ) ) @ N3 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N3 ) ) ).

% mod_Suc_eq
thf(fact_6610_nat__mod__eq,axiom,
    ! [B2: nat,N3: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ N3 )
     => ( ( ( modulo_modulo_nat @ A2 @ N3 )
          = ( modulo_modulo_nat @ B2 @ N3 ) )
       => ( ( modulo_modulo_nat @ A2 @ N3 )
          = B2 ) ) ) ).

% nat_mod_eq
thf(fact_6611_mod__plus__right,axiom,
    ! [A2: nat,X: nat,M: nat,B2: nat] :
      ( ( ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ X ) @ M )
        = ( modulo_modulo_nat @ ( plus_plus_nat @ B2 @ X ) @ M ) )
      = ( ( modulo_modulo_nat @ A2 @ M )
        = ( modulo_modulo_nat @ B2 @ M ) ) ) ).

% mod_plus_right
thf(fact_6612_mod__less__eq__dividend,axiom,
    ! [M: nat,N3: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N3 ) @ M ) ).

% mod_less_eq_dividend
thf(fact_6613_unset__bit__nat__def,axiom,
    ( bit_se4205575877204974255it_nat
    = ( ^ [M5: nat,N2: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M5 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% unset_bit_nat_def
thf(fact_6614_unset__bit__less__eq,axiom,
    ! [N3: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N3 @ K ) @ K ) ).

% unset_bit_less_eq
thf(fact_6615_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A2 @ B2 ) @ A2 ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_6616_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A2 @ B2 ) @ A2 ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_6617_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ A2 @ B2 ) @ A2 ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_6618_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ord_less_nat @ ( modulo_modulo_nat @ A2 @ B2 ) @ B2 ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_6619_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ord_less_int @ ( modulo_modulo_int @ A2 @ B2 ) @ B2 ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_6620_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
     => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A2 @ B2 ) @ B2 ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_6621_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q3: num,N3: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N3 ) @ ( numeral_numeral_nat @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_6622_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q3: num,N3: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N3 ) @ ( numeral_numeral_int @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_6623_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q3: num,N3: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N3 ) @ ( numera6620942414471956472nteger @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_6624_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N3: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
      = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N3 ) @ ( numeral_numeral_nat @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_6625_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N3: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
      = ( modulo_modulo_int @ ( numeral_numeral_int @ N3 ) @ ( numeral_numeral_int @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_6626_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N3: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
      = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N3 ) @ ( numera6620942414471956472nteger @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_6627_mod__eq__self__iff__div__eq__0,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( modulo_modulo_nat @ A2 @ B2 )
        = A2 )
      = ( ( divide_divide_nat @ A2 @ B2 )
        = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_6628_mod__eq__self__iff__div__eq__0,axiom,
    ! [A2: int,B2: int] :
      ( ( ( modulo_modulo_int @ A2 @ B2 )
        = A2 )
      = ( ( divide_divide_int @ A2 @ B2 )
        = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_6629_mod__eq__self__iff__div__eq__0,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A2 @ B2 )
        = A2 )
      = ( ( divide6298287555418463151nteger @ A2 @ B2 )
        = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_6630_mod__eqE,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ( modulo_modulo_int @ A2 @ C2 )
        = ( modulo_modulo_int @ B2 @ C2 ) )
     => ~ ! [D5: int] :
            ( B2
           != ( plus_plus_int @ A2 @ ( times_times_int @ C2 @ D5 ) ) ) ) ).

% mod_eqE
thf(fact_6631_mod__eqE,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A2 @ C2 )
        = ( modulo364778990260209775nteger @ B2 @ C2 ) )
     => ~ ! [D5: code_integer] :
            ( B2
           != ( plus_p5714425477246183910nteger @ A2 @ ( times_3573771949741848930nteger @ C2 @ D5 ) ) ) ) ).

% mod_eqE
thf(fact_6632_div__add1__eq,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C2 )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ C2 ) @ ( divide_divide_nat @ B2 @ C2 ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A2 @ C2 ) @ ( modulo_modulo_nat @ B2 @ C2 ) ) @ C2 ) ) ) ).

% div_add1_eq
thf(fact_6633_div__add1__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( divide_divide_int @ ( plus_plus_int @ A2 @ B2 ) @ C2 )
      = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A2 @ C2 ) @ ( divide_divide_int @ B2 @ C2 ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A2 @ C2 ) @ ( modulo_modulo_int @ B2 @ C2 ) ) @ C2 ) ) ) ).

% div_add1_eq
thf(fact_6634_div__add1__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ C2 )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A2 @ C2 ) @ ( divide6298287555418463151nteger @ B2 @ C2 ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ C2 ) @ ( modulo364778990260209775nteger @ B2 @ C2 ) ) @ C2 ) ) ) ).

% div_add1_eq
thf(fact_6635_mod__0__imp__dvd,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( modulo_modulo_nat @ A2 @ B2 )
        = zero_zero_nat )
     => ( dvd_dvd_nat @ B2 @ A2 ) ) ).

% mod_0_imp_dvd
thf(fact_6636_mod__0__imp__dvd,axiom,
    ! [A2: int,B2: int] :
      ( ( ( modulo_modulo_int @ A2 @ B2 )
        = zero_zero_int )
     => ( dvd_dvd_int @ B2 @ A2 ) ) ).

% mod_0_imp_dvd
thf(fact_6637_mod__0__imp__dvd,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A2 @ B2 )
        = zero_z3403309356797280102nteger )
     => ( dvd_dvd_Code_integer @ B2 @ A2 ) ) ).

% mod_0_imp_dvd
thf(fact_6638_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_nat
    = ( ^ [A7: nat,B7: nat] :
          ( ( modulo_modulo_nat @ B7 @ A7 )
          = zero_zero_nat ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_6639_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_int
    = ( ^ [A7: int,B7: int] :
          ( ( modulo_modulo_int @ B7 @ A7 )
          = zero_zero_int ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_6640_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [A7: code_integer,B7: code_integer] :
          ( ( modulo364778990260209775nteger @ B7 @ A7 )
          = zero_z3403309356797280102nteger ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_6641_mod__eq__0__iff__dvd,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( modulo_modulo_nat @ A2 @ B2 )
        = zero_zero_nat )
      = ( dvd_dvd_nat @ B2 @ A2 ) ) ).

% mod_eq_0_iff_dvd
thf(fact_6642_mod__eq__0__iff__dvd,axiom,
    ! [A2: int,B2: int] :
      ( ( ( modulo_modulo_int @ A2 @ B2 )
        = zero_zero_int )
      = ( dvd_dvd_int @ B2 @ A2 ) ) ).

% mod_eq_0_iff_dvd
thf(fact_6643_mod__eq__0__iff__dvd,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A2 @ B2 )
        = zero_z3403309356797280102nteger )
      = ( dvd_dvd_Code_integer @ B2 @ A2 ) ) ).

% mod_eq_0_iff_dvd
thf(fact_6644_mod__eq__dvd__iff,axiom,
    ! [A2: int,C2: int,B2: int] :
      ( ( ( modulo_modulo_int @ A2 @ C2 )
        = ( modulo_modulo_int @ B2 @ C2 ) )
      = ( dvd_dvd_int @ C2 @ ( minus_minus_int @ A2 @ B2 ) ) ) ).

% mod_eq_dvd_iff
thf(fact_6645_mod__eq__dvd__iff,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A2 @ C2 )
        = ( modulo364778990260209775nteger @ B2 @ C2 ) )
      = ( dvd_dvd_Code_integer @ C2 @ ( minus_8373710615458151222nteger @ A2 @ B2 ) ) ) ).

% mod_eq_dvd_iff
thf(fact_6646_dvd__minus__mod,axiom,
    ! [B2: nat,A2: nat] : ( dvd_dvd_nat @ B2 @ ( minus_minus_nat @ A2 @ ( modulo_modulo_nat @ A2 @ B2 ) ) ) ).

% dvd_minus_mod
thf(fact_6647_dvd__minus__mod,axiom,
    ! [B2: int,A2: int] : ( dvd_dvd_int @ B2 @ ( minus_minus_int @ A2 @ ( modulo_modulo_int @ A2 @ B2 ) ) ) ).

% dvd_minus_mod
thf(fact_6648_dvd__minus__mod,axiom,
    ! [B2: code_integer,A2: code_integer] : ( dvd_dvd_Code_integer @ B2 @ ( minus_8373710615458151222nteger @ A2 @ ( modulo364778990260209775nteger @ A2 @ B2 ) ) ) ).

% dvd_minus_mod
thf(fact_6649_mod__Suc,axiom,
    ! [M: nat,N3: nat] :
      ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N3 ) )
          = N3 )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N3 )
          = zero_zero_nat ) )
      & ( ( ( suc @ ( modulo_modulo_nat @ M @ N3 ) )
         != N3 )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N3 )
          = ( suc @ ( modulo_modulo_nat @ M @ N3 ) ) ) ) ) ).

% mod_Suc
thf(fact_6650_mod__induct,axiom,
    ! [P: nat > $o,N3: nat,P6: nat,M: nat] :
      ( ( P @ N3 )
     => ( ( ord_less_nat @ N3 @ P6 )
       => ( ( ord_less_nat @ M @ P6 )
         => ( ! [N: nat] :
                ( ( ord_less_nat @ N @ P6 )
               => ( ( P @ N )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N ) @ P6 ) ) ) )
           => ( P @ M ) ) ) ) ) ).

% mod_induct
thf(fact_6651_nat__mod__lem,axiom,
    ! [N3: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_nat @ B2 @ N3 )
        = ( ( modulo_modulo_nat @ B2 @ N3 )
          = B2 ) ) ) ).

% nat_mod_lem
thf(fact_6652_mod__less__divisor,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N3 ) @ N3 ) ) ).

% mod_less_divisor
thf(fact_6653_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N3: nat] :
      ( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
     => ( ! [M4: nat,N: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( ( P @ N @ ( modulo_modulo_nat @ M4 @ N ) )
             => ( P @ M4 @ N ) ) )
       => ( P @ M @ N3 ) ) ) ).

% gcd_nat_induct
thf(fact_6654_mod__Suc__le__divisor,axiom,
    ! [M: nat,N3: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N3 ) ) @ N3 ) ).

% mod_Suc_le_divisor
thf(fact_6655_word__rot__lem,axiom,
    ! [L2: nat,K: nat,D2: nat,N3: nat] :
      ( ( ( plus_plus_nat @ L2 @ K )
        = ( plus_plus_nat @ D2 @ ( modulo_modulo_nat @ K @ L2 ) ) )
     => ( ( ord_less_nat @ N3 @ L2 )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ D2 @ N3 ) @ L2 )
          = N3 ) ) ) ).

% word_rot_lem
thf(fact_6656_nat__minus__mod,axiom,
    ! [N3: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( minus_minus_nat @ N3 @ ( modulo_modulo_nat @ N3 @ M ) ) @ M )
      = zero_zero_nat ) ).

% nat_minus_mod
thf(fact_6657_mod__nat__sub,axiom,
    ! [X: nat,Z: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Z )
     => ( ( modulo_modulo_nat @ ( minus_minus_nat @ X @ Y ) @ Z )
        = ( minus_minus_nat @ X @ Y ) ) ) ).

% mod_nat_sub
thf(fact_6658_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M5: nat,N2: nat] : ( if_nat @ ( ord_less_nat @ M5 @ N2 ) @ M5 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M5 @ N2 ) @ N2 ) ) ) ) ).

% mod_if
thf(fact_6659_mod__geq,axiom,
    ! [M: nat,N3: nat] :
      ( ~ ( ord_less_nat @ M @ N3 )
     => ( ( modulo_modulo_nat @ M @ N3 )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N3 ) @ N3 ) ) ) ).

% mod_geq
thf(fact_6660_mod__eq__0D,axiom,
    ! [M: nat,D2: nat] :
      ( ( ( modulo_modulo_nat @ M @ D2 )
        = zero_zero_nat )
     => ? [Q5: nat] :
          ( M
          = ( times_times_nat @ D2 @ Q5 ) ) ) ).

% mod_eq_0D
thf(fact_6661_nat__minus__mod__plus__right,axiom,
    ! [N3: nat,X: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( minus_minus_nat @ ( plus_plus_nat @ N3 @ X ) @ ( modulo_modulo_nat @ N3 @ M ) ) @ M )
      = ( modulo_modulo_nat @ X @ M ) ) ).

% nat_minus_mod_plus_right
thf(fact_6662_le__mod__geq,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( modulo_modulo_nat @ M @ N3 )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N3 ) @ N3 ) ) ) ).

% le_mod_geq
thf(fact_6663_msrevs_I2_J,axiom,
    ! [K: nat,N3: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ K @ N3 ) @ M ) @ N3 )
      = ( modulo_modulo_nat @ M @ N3 ) ) ).

% msrevs(2)
thf(fact_6664_nat__mod__eq__iff,axiom,
    ! [X: nat,N3: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N3 )
        = ( modulo_modulo_nat @ Y @ N3 ) )
      = ( ? [Q1: nat,Q22: nat] :
            ( ( plus_plus_nat @ X @ ( times_times_nat @ N3 @ Q1 ) )
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N3 @ Q22 ) ) ) ) ) ).

% nat_mod_eq_iff
thf(fact_6665_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A2 @ B2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_6666_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A2 @ B2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_6667_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A2 @ B2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_6668_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
       => ( ( modulo364778990260209775nteger @ A2 @ B2 )
          = A2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_6669_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ A2 @ B2 )
       => ( ( modulo_modulo_nat @ A2 @ B2 )
          = A2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_6670_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ A2 @ B2 )
       => ( ( modulo_modulo_int @ A2 @ B2 )
          = A2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_6671_cong__exp__iff__simps_I2_J,axiom,
    ! [N3: num,Q3: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = zero_zero_nat )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N3 ) @ ( numeral_numeral_nat @ Q3 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
thf(fact_6672_cong__exp__iff__simps_I2_J,axiom,
    ! [N3: num,Q3: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = zero_zero_int )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N3 ) @ ( numeral_numeral_int @ Q3 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
thf(fact_6673_cong__exp__iff__simps_I2_J,axiom,
    ! [N3: num,Q3: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N3 ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
thf(fact_6674_cong__exp__iff__simps_I1_J,axiom,
    ! [N3: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N3 ) @ ( numeral_numeral_nat @ one ) )
      = zero_zero_nat ) ).

% cong_exp_iff_simps(1)
thf(fact_6675_cong__exp__iff__simps_I1_J,axiom,
    ! [N3: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ N3 ) @ ( numeral_numeral_int @ one ) )
      = zero_zero_int ) ).

% cong_exp_iff_simps(1)
thf(fact_6676_cong__exp__iff__simps_I1_J,axiom,
    ! [N3: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N3 ) @ ( numera6620942414471956472nteger @ one ) )
      = zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(1)
thf(fact_6677_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,N3: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_6678_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,N3: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_6679_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,N3: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_6680_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q3: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_6681_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q3: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_6682_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q3: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_6683_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q3: num,N3: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_6684_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q3: num,N3: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_6685_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q3: num,N3: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_6686_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q3: num,N3: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_6687_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q3: num,N3: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_6688_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q3: num,N3: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_6689_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q3: num,N3: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N3 ) @ ( numeral_numeral_nat @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_6690_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q3: num,N3: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N3 ) @ ( numeral_numeral_int @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_6691_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q3: num,N3: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N3 ) @ ( numera6620942414471956472nteger @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_6692_mult__div__mod__eq,axiom,
    ! [B2: nat,A2: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A2 @ B2 ) ) @ ( modulo_modulo_nat @ A2 @ B2 ) )
      = A2 ) ).

% mult_div_mod_eq
thf(fact_6693_mult__div__mod__eq,axiom,
    ! [B2: int,A2: int] :
      ( ( plus_plus_int @ ( times_times_int @ B2 @ ( divide_divide_int @ A2 @ B2 ) ) @ ( modulo_modulo_int @ A2 @ B2 ) )
      = A2 ) ).

% mult_div_mod_eq
thf(fact_6694_mult__div__mod__eq,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A2 @ B2 ) ) @ ( modulo364778990260209775nteger @ A2 @ B2 ) )
      = A2 ) ).

% mult_div_mod_eq
thf(fact_6695_mod__mult__div__eq,axiom,
    ! [A2: nat,B2: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A2 @ B2 ) @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A2 @ B2 ) ) )
      = A2 ) ).

% mod_mult_div_eq
thf(fact_6696_mod__mult__div__eq,axiom,
    ! [A2: int,B2: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A2 @ B2 ) @ ( times_times_int @ B2 @ ( divide_divide_int @ A2 @ B2 ) ) )
      = A2 ) ).

% mod_mult_div_eq
thf(fact_6697_mod__mult__div__eq,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ B2 ) @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A2 @ B2 ) ) )
      = A2 ) ).

% mod_mult_div_eq
thf(fact_6698_mod__div__mult__eq,axiom,
    ! [A2: nat,B2: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A2 @ B2 ) @ ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ B2 ) )
      = A2 ) ).

% mod_div_mult_eq
thf(fact_6699_mod__div__mult__eq,axiom,
    ! [A2: int,B2: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A2 @ B2 ) @ ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ B2 ) )
      = A2 ) ).

% mod_div_mult_eq
thf(fact_6700_mod__div__mult__eq,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ B2 ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ B2 ) )
      = A2 ) ).

% mod_div_mult_eq
thf(fact_6701_div__mult__mod__eq,axiom,
    ! [A2: nat,B2: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A2 @ B2 ) )
      = A2 ) ).

% div_mult_mod_eq
thf(fact_6702_div__mult__mod__eq,axiom,
    ! [A2: int,B2: int] :
      ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A2 @ B2 ) )
      = A2 ) ).

% div_mult_mod_eq
thf(fact_6703_div__mult__mod__eq,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A2 @ B2 ) )
      = A2 ) ).

% div_mult_mod_eq
thf(fact_6704_mod__div__decomp,axiom,
    ! [A2: nat,B2: nat] :
      ( A2
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A2 @ B2 ) ) ) ).

% mod_div_decomp
thf(fact_6705_mod__div__decomp,axiom,
    ! [A2: int,B2: int] :
      ( A2
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A2 @ B2 ) ) ) ).

% mod_div_decomp
thf(fact_6706_mod__div__decomp,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( A2
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A2 @ B2 ) ) ) ).

% mod_div_decomp
thf(fact_6707_cancel__div__mod__rules_I1_J,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo_nat @ A2 @ B2 ) ) @ C2 )
      = ( plus_plus_nat @ A2 @ C2 ) ) ).

% cancel_div_mod_rules(1)
thf(fact_6708_cancel__div__mod__rules_I1_J,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo_int @ A2 @ B2 ) ) @ C2 )
      = ( plus_plus_int @ A2 @ C2 ) ) ).

% cancel_div_mod_rules(1)
thf(fact_6709_cancel__div__mod__rules_I1_J,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ B2 ) @ ( modulo364778990260209775nteger @ A2 @ B2 ) ) @ C2 )
      = ( plus_p5714425477246183910nteger @ A2 @ C2 ) ) ).

% cancel_div_mod_rules(1)
thf(fact_6710_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: nat,A2: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A2 @ B2 ) ) @ ( modulo_modulo_nat @ A2 @ B2 ) ) @ C2 )
      = ( plus_plus_nat @ A2 @ C2 ) ) ).

% cancel_div_mod_rules(2)
thf(fact_6711_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: int,A2: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B2 @ ( divide_divide_int @ A2 @ B2 ) ) @ ( modulo_modulo_int @ A2 @ B2 ) ) @ C2 )
      = ( plus_plus_int @ A2 @ C2 ) ) ).

% cancel_div_mod_rules(2)
thf(fact_6712_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: code_integer,A2: code_integer,C2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A2 @ B2 ) ) @ ( modulo364778990260209775nteger @ A2 @ B2 ) ) @ C2 )
      = ( plus_p5714425477246183910nteger @ A2 @ C2 ) ) ).

% cancel_div_mod_rules(2)
thf(fact_6713_div__mult1__eq,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 )
      = ( plus_plus_nat @ ( times_times_nat @ A2 @ ( divide_divide_nat @ B2 @ C2 ) ) @ ( divide_divide_nat @ ( times_times_nat @ A2 @ ( modulo_modulo_nat @ B2 @ C2 ) ) @ C2 ) ) ) ).

% div_mult1_eq
thf(fact_6714_div__mult1__eq,axiom,
    ! [A2: int,B2: int,C2: int] :
      ( ( divide_divide_int @ ( times_times_int @ A2 @ B2 ) @ C2 )
      = ( plus_plus_int @ ( times_times_int @ A2 @ ( divide_divide_int @ B2 @ C2 ) ) @ ( divide_divide_int @ ( times_times_int @ A2 @ ( modulo_modulo_int @ B2 @ C2 ) ) @ C2 ) ) ) ).

% div_mult1_eq
thf(fact_6715_div__mult1__eq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer] :
      ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) @ C2 )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A2 @ ( divide6298287555418463151nteger @ B2 @ C2 ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A2 @ ( modulo364778990260209775nteger @ B2 @ C2 ) ) @ C2 ) ) ) ).

% div_mult1_eq
thf(fact_6716_zmde,axiom,
    ! [B2: int,A2: int] :
      ( ( times_times_int @ B2 @ ( divide_divide_int @ A2 @ B2 ) )
      = ( minus_minus_int @ A2 @ ( modulo_modulo_int @ A2 @ B2 ) ) ) ).

% zmde
thf(fact_6717_zmde,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A2 @ B2 ) )
      = ( minus_8373710615458151222nteger @ A2 @ ( modulo364778990260209775nteger @ A2 @ B2 ) ) ) ).

% zmde
thf(fact_6718_minus__mult__div__eq__mod,axiom,
    ! [A2: nat,B2: nat] :
      ( ( minus_minus_nat @ A2 @ ( times_times_nat @ B2 @ ( divide_divide_nat @ A2 @ B2 ) ) )
      = ( modulo_modulo_nat @ A2 @ B2 ) ) ).

% minus_mult_div_eq_mod
thf(fact_6719_minus__mult__div__eq__mod,axiom,
    ! [A2: int,B2: int] :
      ( ( minus_minus_int @ A2 @ ( times_times_int @ B2 @ ( divide_divide_int @ A2 @ B2 ) ) )
      = ( modulo_modulo_int @ A2 @ B2 ) ) ).

% minus_mult_div_eq_mod
thf(fact_6720_minus__mult__div__eq__mod,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A2 @ ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A2 @ B2 ) ) )
      = ( modulo364778990260209775nteger @ A2 @ B2 ) ) ).

% minus_mult_div_eq_mod
thf(fact_6721_minus__mod__eq__mult__div,axiom,
    ! [A2: nat,B2: nat] :
      ( ( minus_minus_nat @ A2 @ ( modulo_modulo_nat @ A2 @ B2 ) )
      = ( times_times_nat @ B2 @ ( divide_divide_nat @ A2 @ B2 ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_6722_minus__mod__eq__mult__div,axiom,
    ! [A2: int,B2: int] :
      ( ( minus_minus_int @ A2 @ ( modulo_modulo_int @ A2 @ B2 ) )
      = ( times_times_int @ B2 @ ( divide_divide_int @ A2 @ B2 ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_6723_minus__mod__eq__mult__div,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A2 @ ( modulo364778990260209775nteger @ A2 @ B2 ) )
      = ( times_3573771949741848930nteger @ B2 @ ( divide6298287555418463151nteger @ A2 @ B2 ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_6724_minus__mod__eq__div__mult,axiom,
    ! [A2: nat,B2: nat] :
      ( ( minus_minus_nat @ A2 @ ( modulo_modulo_nat @ A2 @ B2 ) )
      = ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ B2 ) ) ).

% minus_mod_eq_div_mult
thf(fact_6725_minus__mod__eq__div__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( minus_minus_int @ A2 @ ( modulo_modulo_int @ A2 @ B2 ) )
      = ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ B2 ) ) ).

% minus_mod_eq_div_mult
thf(fact_6726_minus__mod__eq__div__mult,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A2 @ ( modulo364778990260209775nteger @ A2 @ B2 ) )
      = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ B2 ) ) ).

% minus_mod_eq_div_mult
thf(fact_6727_minus__div__mult__eq__mod,axiom,
    ! [A2: nat,B2: nat] :
      ( ( minus_minus_nat @ A2 @ ( times_times_nat @ ( divide_divide_nat @ A2 @ B2 ) @ B2 ) )
      = ( modulo_modulo_nat @ A2 @ B2 ) ) ).

% minus_div_mult_eq_mod
thf(fact_6728_minus__div__mult__eq__mod,axiom,
    ! [A2: int,B2: int] :
      ( ( minus_minus_int @ A2 @ ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ B2 ) )
      = ( modulo_modulo_int @ A2 @ B2 ) ) ).

% minus_div_mult_eq_mod
thf(fact_6729_minus__div__mult__eq__mod,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A2 @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ B2 ) )
      = ( modulo364778990260209775nteger @ A2 @ B2 ) ) ).

% minus_div_mult_eq_mod
thf(fact_6730_unit__imp__mod__eq__0,axiom,
    ! [B2: nat,A2: nat] :
      ( ( dvd_dvd_nat @ B2 @ one_one_nat )
     => ( ( modulo_modulo_nat @ A2 @ B2 )
        = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
thf(fact_6731_unit__imp__mod__eq__0,axiom,
    ! [B2: int,A2: int] :
      ( ( dvd_dvd_int @ B2 @ one_one_int )
     => ( ( modulo_modulo_int @ A2 @ B2 )
        = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
thf(fact_6732_unit__imp__mod__eq__0,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
     => ( ( modulo364778990260209775nteger @ A2 @ B2 )
        = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
thf(fact_6733_mod__le__divisor,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N3 ) @ N3 ) ) ).

% mod_le_divisor
thf(fact_6734_div__less__mono,axiom,
    ! [A: nat,B: nat,N3: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( ( modulo_modulo_nat @ A @ N3 )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B @ N3 )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A @ N3 ) @ ( divide_divide_nat @ B @ N3 ) ) ) ) ) ) ).

% div_less_mono
thf(fact_6735_mod__nat__add,axiom,
    ! [X: nat,Z: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Z )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ( ( ord_less_nat @ ( plus_plus_nat @ X @ Y ) @ Z )
           => ( ( modulo_modulo_nat @ ( plus_plus_nat @ X @ Y ) @ Z )
              = ( plus_plus_nat @ X @ Y ) ) )
          & ( ~ ( ord_less_nat @ ( plus_plus_nat @ X @ Y ) @ Z )
           => ( ( modulo_modulo_nat @ ( plus_plus_nat @ X @ Y ) @ Z )
              = ( minus_minus_nat @ ( plus_plus_nat @ X @ Y ) @ Z ) ) ) ) ) ) ).

% mod_nat_add
thf(fact_6736_nat__mod__eq__lemma,axiom,
    ! [X: nat,N3: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N3 )
        = ( modulo_modulo_nat @ Y @ N3 ) )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ? [Q5: nat] :
            ( X
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N3 @ Q5 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_6737_mod__eq__nat2E,axiom,
    ! [M: nat,Q3: nat,N3: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q3 )
        = ( modulo_modulo_nat @ N3 @ Q3 ) )
     => ( ( ord_less_eq_nat @ M @ N3 )
       => ~ ! [S3: nat] :
              ( N3
             != ( plus_plus_nat @ M @ ( times_times_nat @ Q3 @ S3 ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_6738_mod__eq__nat1E,axiom,
    ! [M: nat,Q3: nat,N3: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q3 )
        = ( modulo_modulo_nat @ N3 @ Q3 ) )
     => ( ( ord_less_eq_nat @ N3 @ M )
       => ~ ! [S3: nat] :
              ( M
             != ( plus_plus_nat @ N3 @ ( times_times_nat @ Q3 @ S3 ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_6739_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N3 ) )
      = ( ~ ( dvd_dvd_nat @ N3 @ M ) ) ) ).

% mod_greater_zero_iff_not_dvd
thf(fact_6740_divmod_H__nat__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M5: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_6741_mod__mult2__eq,axiom,
    ! [M: nat,N3: nat,Q3: nat] :
      ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N3 @ Q3 ) )
      = ( plus_plus_nat @ ( times_times_nat @ N3 @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N3 ) @ Q3 ) ) @ ( modulo_modulo_nat @ M @ N3 ) ) ) ).

% mod_mult2_eq
thf(fact_6742_div__mod__decomp,axiom,
    ! [A: nat,N3: nat] :
      ( A
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ N3 ) @ N3 ) @ ( modulo_modulo_nat @ A @ N3 ) ) ) ).

% div_mod_decomp
thf(fact_6743_modulo__nat__def,axiom,
    ( modulo_modulo_nat
    = ( ^ [M5: nat,N2: nat] : ( minus_minus_nat @ M5 @ ( times_times_nat @ ( divide_divide_nat @ M5 @ N2 ) @ N2 ) ) ) ) ).

% modulo_nat_def
thf(fact_6744_mod__eq__dvd__iff__nat,axiom,
    ! [N3: nat,M: nat,Q3: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( ( modulo_modulo_nat @ M @ Q3 )
          = ( modulo_modulo_nat @ N3 @ Q3 ) )
        = ( dvd_dvd_nat @ Q3 @ ( minus_minus_nat @ M @ N3 ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_6745_VEBT__internal_OminNulli_Osimps_I5_J,axiom,
    ! [Uz2: product_prod_nat_nat,Va: nat,Vb2: array_VEBT_VEBTi,Vc2: vEBT_VEBTi] :
      ( ( vEBT_VEBT_minNulli @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va @ Vb2 @ Vc2 ) )
      = ( heap_Time_return_o @ $false ) ) ).

% VEBT_internal.minNulli.simps(5)
thf(fact_6746_VEBT__internal_OminNulli_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux2: array_VEBT_VEBTi,Uy2: vEBT_VEBTi] :
      ( ( vEBT_VEBT_minNulli @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) )
      = ( heap_Time_return_o @ $true ) ) ).

% VEBT_internal.minNulli.simps(4)
thf(fact_6747_vebt__buildupi_Osimps_I1_J,axiom,
    ( ( vEBT_vebt_buildupi @ zero_zero_nat )
    = ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) ).

% vebt_buildupi.simps(1)
thf(fact_6748_VEBT__internal_Ovebt__buildupi_H_Osimps_I1_J,axiom,
    ( ( vEBT_V739175172307565963ildupi @ zero_zero_nat )
    = ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) ).

% VEBT_internal.vebt_buildupi'.simps(1)
thf(fact_6749_star__assoc,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A2 @ B2 ) @ C2 )
      = ( times_times_assn @ A2 @ ( times_times_assn @ B2 @ C2 ) ) ) ).

% star_assoc
thf(fact_6750_star__aci_I2_J,axiom,
    ( times_times_assn
    = ( ^ [A7: assn,B7: assn] : ( times_times_assn @ B7 @ A7 ) ) ) ).

% star_aci(2)
thf(fact_6751_star__aci_I3_J,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( times_times_assn @ A2 @ ( times_times_assn @ B2 @ C2 ) )
      = ( times_times_assn @ B2 @ ( times_times_assn @ A2 @ C2 ) ) ) ).

% star_aci(3)
thf(fact_6752_assn__aci_I10_J,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A2 @ B2 ) @ C2 )
      = ( times_times_assn @ ( times_times_assn @ A2 @ C2 ) @ B2 ) ) ).

% assn_aci(10)
thf(fact_6753_is__entails,axiom,
    ! [P: assn,Q: assn] :
      ( ( entails @ P @ Q )
     => ( entails @ P @ Q ) ) ).

% is_entails
thf(fact_6754_cong__exp__iff__simps_I3_J,axiom,
    ! [N3: num,Q3: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != zero_zero_nat ) ).

% cong_exp_iff_simps(3)
thf(fact_6755_cong__exp__iff__simps_I3_J,axiom,
    ! [N3: num,Q3: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != zero_zero_int ) ).

% cong_exp_iff_simps(3)
thf(fact_6756_cong__exp__iff__simps_I3_J,axiom,
    ! [N3: num,Q3: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(3)
thf(fact_6757_mod__mult2__eq_H,axiom,
    ! [A2: int,M: nat,N3: nat] :
      ( ( modulo_modulo_int @ A2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N3 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A2 @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) @ ( modulo_modulo_int @ A2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_6758_mod__mult2__eq_H,axiom,
    ! [A2: nat,M: nat,N3: nat] :
      ( ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N3 ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A2 @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N3 ) ) ) @ ( modulo_modulo_nat @ A2 @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_6759_mod__mult2__eq_H,axiom,
    ! [A2: code_integer,M: nat,N3: nat] :
      ( ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N3 ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A2 @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N3 ) ) ) @ ( modulo364778990260209775nteger @ A2 @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_6760_even__even__mod__4__iff,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).

% even_even_mod_4_iff
thf(fact_6761_unset__bit__Suc,axiom,
    ! [N3: nat,A2: code_integer] :
      ( ( bit_se8260200283734997820nteger @ ( suc @ N3 ) @ A2 )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N3 @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_6762_unset__bit__Suc,axiom,
    ! [N3: nat,A2: int] :
      ( ( bit_se4203085406695923979it_int @ ( suc @ N3 ) @ A2 )
      = ( plus_plus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N3 @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_6763_unset__bit__Suc,axiom,
    ! [N3: nat,A2: nat] :
      ( ( bit_se4205575877204974255it_nat @ ( suc @ N3 ) @ A2 )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N3 @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_6764_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N3 ) )
      = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N3 ) ) ) @ ( semiri681578069525770553at_rat @ N3 ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_6765_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N3 ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N3 ) ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_6766_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N3: nat] :
      ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N3 ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N3 ) ) ) @ ( semiri8010041392384452111omplex @ N3 ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_6767_mod__lemma,axiom,
    ! [C2: nat,R3: nat,B2: nat,Q3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C2 )
     => ( ( ord_less_nat @ R3 @ B2 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ B2 @ ( modulo_modulo_nat @ Q3 @ C2 ) ) @ R3 ) @ ( times_times_nat @ B2 @ C2 ) ) ) ) ).

% mod_lemma
thf(fact_6768_split__mod,axiom,
    ! [P: nat > $o,M: nat,N3: nat] :
      ( ( P @ ( modulo_modulo_nat @ M @ N3 ) )
      = ( ( ( N3 = zero_zero_nat )
         => ( P @ M ) )
        & ( ( N3 != zero_zero_nat )
         => ! [I2: nat,J: nat] :
              ( ( ord_less_nat @ J @ N3 )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N3 @ I2 ) @ J ) )
               => ( P @ J ) ) ) ) ) ) ).

% split_mod
thf(fact_6769_divmod__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M5: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).

% divmod_def
thf(fact_6770_divmod__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M5: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% divmod_def
thf(fact_6771_divmod__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M5: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).

% divmod_def
thf(fact_6772_diff__mod__le,axiom,
    ! [A2: nat,D2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ D2 )
     => ( ( dvd_dvd_nat @ B2 @ D2 )
       => ( ord_less_eq_nat @ ( minus_minus_nat @ A2 @ ( modulo_modulo_nat @ A2 @ B2 ) ) @ ( minus_minus_nat @ D2 @ B2 ) ) ) ) ).

% diff_mod_le
thf(fact_6773_mod__nat__eqI,axiom,
    ! [R3: nat,N3: nat,M: nat] :
      ( ( ord_less_nat @ R3 @ N3 )
     => ( ( ord_less_eq_nat @ R3 @ M )
       => ( ( dvd_dvd_nat @ N3 @ ( minus_minus_nat @ M @ R3 ) )
         => ( ( modulo_modulo_nat @ M @ N3 )
            = R3 ) ) ) ) ).

% mod_nat_eqI
thf(fact_6774_VEBT__internal_Ovebt__buildupi_H_Osimps_I2_J,axiom,
    ( ( vEBT_V739175172307565963ildupi @ ( suc @ zero_zero_nat ) )
    = ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) ).

% VEBT_internal.vebt_buildupi'.simps(2)
thf(fact_6775_vebt__buildupi_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildupi @ ( suc @ zero_zero_nat ) )
    = ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) ).

% vebt_buildupi.simps(2)
thf(fact_6776_real__of__nat__div__aux,axiom,
    ! [X: nat,D2: nat] :
      ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ D2 ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X @ D2 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X @ D2 ) ) @ ( semiri5074537144036343181t_real @ D2 ) ) ) ) ).

% real_of_nat_div_aux
thf(fact_6777_vebt__maxti_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
      ( ( vEBT_vebt_maxti @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv2 @ Uw ) )
      = ( heap_T3487192422709364219on_nat @ none_nat ) ) ).

% vebt_maxti.simps(2)
thf(fact_6778_vebt__minti_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: array_VEBT_VEBTi,Uw: vEBT_VEBTi] :
      ( ( vEBT_vebt_minti @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu @ Uv2 @ Uw ) )
      = ( heap_T3487192422709364219on_nat @ none_nat ) ) ).

% vebt_minti.simps(2)
thf(fact_6779_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_6780_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
     => ( ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ B2 @ C2 ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B2 @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) @ C2 ) ) @ ( modulo364778990260209775nteger @ A2 @ B2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_6781_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
     => ( ( modulo_modulo_nat @ A2 @ ( times_times_nat @ B2 @ C2 ) )
        = ( plus_plus_nat @ ( times_times_nat @ B2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ A2 @ B2 ) @ C2 ) ) @ ( modulo_modulo_nat @ A2 @ B2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_6782_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C2 )
     => ( ( modulo_modulo_int @ A2 @ ( times_times_int @ B2 @ C2 ) )
        = ( plus_plus_int @ ( times_times_int @ B2 @ ( modulo_modulo_int @ ( divide_divide_int @ A2 @ B2 ) @ C2 ) ) @ ( modulo_modulo_int @ A2 @ B2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_6783_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N3: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N3 ) @ ( numeral_numeral_nat @ Q3 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
thf(fact_6784_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N3: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N3 ) @ ( numeral_numeral_int @ Q3 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
thf(fact_6785_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N3: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N3 ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
thf(fact_6786_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
thf(fact_6787_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
thf(fact_6788_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
thf(fact_6789_even__iff__mod__2__eq__zero,axiom,
    ! [A2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
      = ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_6790_even__iff__mod__2__eq__zero,axiom,
    ! [A2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
      = ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_6791_even__iff__mod__2__eq__zero,axiom,
    ! [A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
      = ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_6792_odd__iff__mod__2__eq__one,axiom,
    ! [A2: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) )
      = ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_6793_odd__iff__mod__2__eq__one,axiom,
    ! [A2: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
      = ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_6794_odd__iff__mod__2__eq__one,axiom,
    ! [A2: code_integer] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) )
      = ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_6795_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N3: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N3 )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N3 ) ) ).

% Suc_mod_eq_add3_mod
thf(fact_6796_Suc__times__mod__eq,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N3 ) ) @ M )
        = one_one_nat ) ) ).

% Suc_times_mod_eq
thf(fact_6797_VEBT__internal_OminNulli_Oelims,axiom,
    ! [X: vEBT_VEBTi,Y: heap_Time_Heap_o] :
      ( ( ( vEBT_VEBT_minNulli @ X )
        = Y )
     => ( ( ( X
            = ( vEBT_Leafi @ $false @ $false ) )
         => ( Y
           != ( heap_Time_return_o @ $true ) ) )
       => ( ( ? [Uv: $o] :
                ( X
                = ( vEBT_Leafi @ $true @ Uv ) )
           => ( Y
             != ( heap_Time_return_o @ $false ) ) )
         => ( ( ? [Uu2: $o] :
                  ( X
                  = ( vEBT_Leafi @ Uu2 @ $true ) )
             => ( Y
               != ( heap_Time_return_o @ $false ) ) )
           => ( ( ? [Uw2: nat,Ux: array_VEBT_VEBTi,Uy: vEBT_VEBTi] :
                    ( X
                    = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uw2 @ Ux @ Uy ) )
               => ( Y
                 != ( heap_Time_return_o @ $true ) ) )
             => ~ ( ? [Uz: product_prod_nat_nat,Va2: nat,Vb: array_VEBT_VEBTi,Vc: vEBT_VEBTi] :
                      ( X
                      = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ Uz ) @ Va2 @ Vb @ Vc ) )
                 => ( Y
                   != ( heap_Time_return_o @ $false ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNulli.elims
thf(fact_6798_vebt__maxti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
      ( ( vEBT_vebt_maxti @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) )
      = ( heap_T3487192422709364219on_nat @ ( some_nat @ Ma ) ) ) ).

% vebt_maxti.simps(3)
thf(fact_6799_vebt__minti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
      ( ( vEBT_vebt_minti @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) )
      = ( heap_T3487192422709364219on_nat @ ( some_nat @ Mi ) ) ) ).

% vebt_minti.simps(3)
thf(fact_6800_divmod__digit__0_I2_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) )
          = ( modulo_modulo_nat @ A2 @ B2 ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_6801_divmod__digit__0_I2_J,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) )
          = ( modulo_modulo_int @ A2 @ B2 ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_6802_divmod__digit__0_I2_J,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) )
          = ( modulo364778990260209775nteger @ A2 @ B2 ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_6803_bits__stable__imp__add__self,axiom,
    ! [A2: nat] :
      ( ( ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A2 )
     => ( ( plus_plus_nat @ A2 @ ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
thf(fact_6804_bits__stable__imp__add__self,axiom,
    ! [A2: int] :
      ( ( ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A2 )
     => ( ( plus_plus_int @ A2 @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
        = zero_zero_int ) ) ).

% bits_stable_imp_add_self
thf(fact_6805_bits__stable__imp__add__self,axiom,
    ! [A2: code_integer] :
      ( ( ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A2 )
     => ( ( plus_p5714425477246183910nteger @ A2 @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
thf(fact_6806_parity__cases,axiom,
    ! [A2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
         != zero_zero_nat ) )
     => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
         => ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
           != one_one_nat ) ) ) ).

% parity_cases
thf(fact_6807_parity__cases,axiom,
    ! [A2: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
         != zero_zero_int ) )
     => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
         => ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
           != one_one_int ) ) ) ).

% parity_cases
thf(fact_6808_parity__cases,axiom,
    ! [A2: code_integer] :
      ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
         != zero_z3403309356797280102nteger ) )
     => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
         => ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
           != one_one_Code_integer ) ) ) ).

% parity_cases
thf(fact_6809_mod2__eq__if,axiom,
    ! [A2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
          = zero_zero_nat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
          = one_one_nat ) ) ) ).

% mod2_eq_if
thf(fact_6810_mod2__eq__if,axiom,
    ! [A2: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
          = zero_zero_int ) )
      & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
          = one_one_int ) ) ) ).

% mod2_eq_if
thf(fact_6811_mod2__eq__if,axiom,
    ! [A2: code_integer] :
      ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          = zero_z3403309356797280102nteger ) )
      & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
       => ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          = one_one_Code_integer ) ) ) ).

% mod2_eq_if
thf(fact_6812_div__exp__mod__exp__eq,axiom,
    ! [A2: nat,N3: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( divide_divide_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
      = ( divide_divide_nat @ ( modulo_modulo_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N3 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_6813_div__exp__mod__exp__eq,axiom,
    ! [A2: int,N3: nat,M: nat] :
      ( ( modulo_modulo_int @ ( divide_divide_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
      = ( divide_divide_int @ ( modulo_modulo_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N3 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_6814_div__exp__mod__exp__eq,axiom,
    ! [A2: code_integer,N3: nat,M: nat] :
      ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
      = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N3 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_6815_power__mod__div,axiom,
    ! [X: nat,N3: nat,M: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
      = ( modulo_modulo_nat @ ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) ) ) ) ).

% power_mod_div
thf(fact_6816_verit__le__mono__div,axiom,
    ! [A: nat,B: nat,N3: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A @ N3 )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B @ N3 )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B @ N3 ) ) ) ) ).

% verit_le_mono_div
thf(fact_6817_vebt__maxti_Osimps_I1_J,axiom,
    ! [B2: $o,A2: $o] :
      ( ( B2
       => ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A2 @ B2 ) )
          = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
      & ( ~ B2
       => ( ( A2
           => ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A2 @ B2 ) )
              = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
          & ( ~ A2
           => ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A2 @ B2 ) )
              = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) ) ).

% vebt_maxti.simps(1)
thf(fact_6818_vebt__minti_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      ( ( A2
       => ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A2 @ B2 ) )
          = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
      & ( ~ A2
       => ( ( B2
           => ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A2 @ B2 ) )
              = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
          & ( ~ B2
           => ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A2 @ B2 ) )
              = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) ) ).

% vebt_minti.simps(1)
thf(fact_6819_divmod__digit__0_I1_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B2 )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
          = ( divide_divide_nat @ A2 @ B2 ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_6820_divmod__digit__0_I1_J,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
          = ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_6821_divmod__digit__0_I1_J,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
       => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) )
          = ( divide6298287555418463151nteger @ A2 @ B2 ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_6822_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N3: nat,A2: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( modulo_modulo_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
        = ( times_times_nat @ ( modulo_modulo_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_6823_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N3: nat,A2: int] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( modulo_modulo_int @ ( times_times_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
        = ( times_times_int @ ( modulo_modulo_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_6824_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N3: nat,A2: code_integer] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) )
        = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_6825_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_6826_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_6827_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_6828_divmod__digit__1_I2_J,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ( ord_le3102999989581377725nteger @ B2 @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
            = ( modulo364778990260209775nteger @ A2 @ B2 ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_6829_divmod__digit__1_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ( ord_less_eq_nat @ B2 @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
            = ( modulo_modulo_nat @ A2 @ B2 ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_6830_divmod__digit__1_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ( ord_less_eq_int @ B2 @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( minus_minus_int @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ B2 )
            = ( modulo_modulo_int @ A2 @ B2 ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_6831_ent__frame__fwd,axiom,
    ! [P: assn,R: assn,Ps: assn,F: assn,Q: assn] :
      ( ( entails @ P @ R )
     => ( ( entails @ Ps @ ( times_times_assn @ P @ F ) )
       => ( ( entails @ ( times_times_assn @ R @ F ) @ Q )
         => ( entails @ Ps @ Q ) ) ) ) ).

% ent_frame_fwd
thf(fact_6832_fr__rot__rhs,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( entails @ A @ ( times_times_assn @ B @ C ) )
     => ( entails @ A @ ( times_times_assn @ C @ B ) ) ) ).

% fr_rot_rhs
thf(fact_6833_fr__refl,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( entails @ A @ B )
     => ( entails @ ( times_times_assn @ A @ C ) @ ( times_times_assn @ B @ C ) ) ) ).

% fr_refl
thf(fact_6834_fr__rot,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( ( entails @ ( times_times_assn @ A @ B ) @ C )
     => ( entails @ ( times_times_assn @ B @ A ) @ C ) ) ).

% fr_rot
thf(fact_6835_eq__diff__eq_H,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( X
        = ( minus_minus_real @ Y @ Z ) )
      = ( Y
        = ( plus_plus_real @ X @ Z ) ) ) ).

% eq_diff_eq'
thf(fact_6836_set__bit__Suc,axiom,
    ! [N3: nat,A2: code_integer] :
      ( ( bit_se2793503036327961859nteger @ ( suc @ N3 ) @ A2 )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N3 @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_6837_set__bit__Suc,axiom,
    ! [N3: nat,A2: int] :
      ( ( bit_se7879613467334960850it_int @ ( suc @ N3 ) @ A2 )
      = ( plus_plus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N3 @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_6838_set__bit__Suc,axiom,
    ! [N3: nat,A2: nat] :
      ( ( bit_se7882103937844011126it_nat @ ( suc @ N3 ) @ A2 )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N3 @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_6839_norm__assertion__simps_I1_J,axiom,
    ! [A2: assn] :
      ( ( times_times_assn @ one_one_assn @ A2 )
      = A2 ) ).

% norm_assertion_simps(1)
thf(fact_6840_norm__assertion__simps_I2_J,axiom,
    ! [A2: assn] :
      ( ( times_times_assn @ A2 @ one_one_assn )
      = A2 ) ).

% norm_assertion_simps(2)
thf(fact_6841_even__mod__4__div__2,axiom,
    ! [N3: nat] :
      ( ( ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( suc @ zero_zero_nat ) )
     => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_mod_4_div_2
thf(fact_6842_even__unset__bit__iff,axiom,
    ! [M: nat,A2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A2 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_6843_even__unset__bit__iff,axiom,
    ! [M: nat,A2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_6844_odd__mod__4__div__2,axiom,
    ! [N3: nat] :
      ( ( ( modulo_modulo_nat @ N3 @ ( 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 @ N3 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% odd_mod_4_div_2
thf(fact_6845_frame__rule__left,axiom,
    ! [P: assn,C2: heap_Time_Heap_o,Q: $o > assn,R: assn] :
      ( ( hoare_hoare_triple_o @ P @ C2 @ Q )
     => ( hoare_hoare_triple_o @ ( times_times_assn @ R @ P ) @ C2
        @ ^ [X3: $o] : ( times_times_assn @ R @ ( Q @ X3 ) ) ) ) ).

% frame_rule_left
thf(fact_6846_frame__rule__left,axiom,
    ! [P: assn,C2: heap_T8145700208782473153_VEBTi,Q: vEBT_VEBTi > assn,R: assn] :
      ( ( hoare_1429296392585015714_VEBTi @ P @ C2 @ Q )
     => ( hoare_1429296392585015714_VEBTi @ ( times_times_assn @ R @ P ) @ C2
        @ ^ [X3: vEBT_VEBTi] : ( times_times_assn @ R @ ( Q @ X3 ) ) ) ) ).

% frame_rule_left
thf(fact_6847_frame__rule__left,axiom,
    ! [P: assn,C2: heap_T2636463487746394924on_nat,Q: option_nat > assn,R: assn] :
      ( ( hoare_7629718768684598413on_nat @ P @ C2 @ Q )
     => ( hoare_7629718768684598413on_nat @ ( times_times_assn @ R @ P ) @ C2
        @ ^ [X3: option_nat] : ( times_times_assn @ R @ ( Q @ X3 ) ) ) ) ).

% frame_rule_left
thf(fact_6848_frame__rule__left,axiom,
    ! [P: assn,C2: heap_Time_Heap_nat,Q: nat > assn,R: assn] :
      ( ( hoare_3067605981109127869le_nat @ P @ C2 @ Q )
     => ( hoare_3067605981109127869le_nat @ ( times_times_assn @ R @ P ) @ C2
        @ ^ [X3: nat] : ( times_times_assn @ R @ ( Q @ X3 ) ) ) ) ).

% frame_rule_left
thf(fact_6849_divmod__digit__1_I1_J,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B2 )
       => ( ( ord_le3102999989581377725nteger @ B2 @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_Code_integer )
            = ( divide6298287555418463151nteger @ A2 @ B2 ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_6850_divmod__digit__1_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ( ord_less_eq_nat @ B2 @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_nat )
            = ( divide_divide_nat @ A2 @ B2 ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_6851_divmod__digit__1_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ( ord_less_eq_int @ B2 @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) )
         => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) @ one_one_int )
            = ( divide_divide_int @ A2 @ B2 ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_6852_vebt__maxti_Oelims,axiom,
    ! [X: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( ( vEBT_vebt_maxti @ X )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leafi @ A3 @ B3 ) )
           => ~ ( ( B3
                 => ( Y
                    = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
                & ( ~ B3
                 => ( ( A3
                     => ( Y
                        = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
                    & ( ~ A3
                     => ( Y
                        = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
                ( X
                = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) )
           => ( Y
             != ( heap_T3487192422709364219on_nat @ none_nat ) ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] :
                    ( X
                    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
               => ( Y
                 != ( heap_T3487192422709364219on_nat @ ( some_nat @ Ma2 ) ) ) ) ) ) ) ).

% vebt_maxti.elims
thf(fact_6853_vebt__minti_Oelims,axiom,
    ! [X: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( ( vEBT_vebt_minti @ X )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leafi @ A3 @ B3 ) )
           => ~ ( ( A3
                 => ( Y
                    = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
                & ( ~ A3
                 => ( ( B3
                     => ( Y
                        = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
                    & ( ~ B3
                     => ( Y
                        = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
                ( X
                = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) )
           => ( Y
             != ( heap_T3487192422709364219on_nat @ none_nat ) ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] :
                    ( X
                    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
               => ( Y
                 != ( heap_T3487192422709364219on_nat @ ( some_nat @ Mi2 ) ) ) ) ) ) ) ).

% vebt_minti.elims
thf(fact_6854_mod__frame__fwd,axiom,
    ! [Ps: assn,H2: produc3658429121746597890et_nat,P: assn,R: assn,F: assn] :
      ( ( rep_assn @ Ps @ H2 )
     => ( ( entails @ P @ R )
       => ( ( entails @ Ps @ ( times_times_assn @ P @ F ) )
         => ( rep_assn @ ( times_times_assn @ R @ F ) @ H2 ) ) ) ) ).

% mod_frame_fwd
thf(fact_6855_div__half__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero_nat )
     => ( ( product_Pair_nat_nat @ ( divide_divide_nat @ X @ Y ) @ ( modulo_modulo_nat @ X @ Y ) )
        = ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ Y @ ( minus_minus_nat @ X @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) @ Y ) ) ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( minus_minus_nat @ X @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) @ Y ) ) @ Y ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) @ ( minus_minus_nat @ X @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) @ Y ) ) ) ) ) ) ).

% div_half_nat
thf(fact_6856_flip__bit__Suc,axiom,
    ! [N3: nat,A2: code_integer] :
      ( ( bit_se1345352211410354436nteger @ ( suc @ N3 ) @ A2 )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N3 @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_6857_flip__bit__Suc,axiom,
    ! [N3: nat,A2: int] :
      ( ( bit_se2159334234014336723it_int @ ( suc @ N3 ) @ A2 )
      = ( plus_plus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N3 @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_6858_flip__bit__Suc,axiom,
    ! [N3: nat,A2: nat] :
      ( ( bit_se2161824704523386999it_nat @ ( suc @ N3 ) @ A2 )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N3 @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_6859_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_6860_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q3: int,R3: int] :
      ( ( ord_less_eq_int @ B2 @ zero_zero_int )
     => ( ( eucl_rel_int @ ( plus_plus_int @ A2 @ one_one_int ) @ B2 @ ( product_Pair_int_int @ Q3 @ R3 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ ( product_Pair_int_int @ Q3 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R3 ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_6861_product__nth,axiom,
    ! [N3: nat,Xs2: list_Code_integer,Ys: list_Code_integer] :
      ( ( ord_less_nat @ N3 @ ( times_times_nat @ ( size_s3445333598471063425nteger @ Xs2 ) @ ( size_s3445333598471063425nteger @ Ys ) ) )
     => ( ( nth_Pr2304437835452373666nteger @ ( produc8792966785426426881nteger @ Xs2 @ Ys ) @ N3 )
        = ( produc1086072967326762835nteger @ ( nth_Code_integer @ Xs2 @ ( divide_divide_nat @ N3 @ ( size_s3445333598471063425nteger @ Ys ) ) ) @ ( nth_Code_integer @ Ys @ ( modulo_modulo_nat @ N3 @ ( size_s3445333598471063425nteger @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6862_product__nth,axiom,
    ! [N3: nat,Xs2: list_num,Ys: list_num] :
      ( ( ord_less_nat @ N3 @ ( times_times_nat @ ( size_size_list_num @ Xs2 ) @ ( size_size_list_num @ Ys ) ) )
     => ( ( nth_Pr6456567536196504476um_num @ ( product_num_num @ Xs2 @ Ys ) @ N3 )
        = ( product_Pair_num_num @ ( nth_num @ Xs2 @ ( divide_divide_nat @ N3 @ ( size_size_list_num @ Ys ) ) ) @ ( nth_num @ Ys @ ( modulo_modulo_nat @ N3 @ ( size_size_list_num @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6863_product__nth,axiom,
    ! [N3: nat,Xs2: list_int,Ys: list_int] :
      ( ( ord_less_nat @ N3 @ ( times_times_nat @ ( size_size_list_int @ Xs2 ) @ ( size_size_list_int @ Ys ) ) )
     => ( ( nth_Pr4439495888332055232nt_int @ ( product_int_int @ Xs2 @ Ys ) @ N3 )
        = ( product_Pair_int_int @ ( nth_int @ Xs2 @ ( divide_divide_nat @ N3 @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N3 @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6864_product__nth,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N3 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys ) @ N3 )
        = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6865_product__nth,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT,Ys: list_real] :
      ( ( ord_less_nat @ N3 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_real @ Ys ) ) )
     => ( ( nth_Pr6842391030413306568T_real @ ( produc4908677263432625371T_real @ Xs2 @ Ys ) @ N3 )
        = ( produc8117437818029410057T_real @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N3 @ ( size_size_list_real @ Ys ) ) ) @ ( nth_real @ Ys @ ( modulo_modulo_nat @ N3 @ ( size_size_list_real @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6866_product__nth,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT,Ys: list_o] :
      ( ( ord_less_nat @ N3 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
     => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys ) @ N3 )
        = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N3 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N3 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6867_product__nth,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT,Ys: list_nat] :
      ( ( ord_less_nat @ N3 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys ) @ N3 )
        = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N3 @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N3 @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6868_product__nth,axiom,
    ! [N3: nat,Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ N3 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s7982070591426661849_VEBTi @ Ys ) ) )
     => ( ( nth_Pr316670251186196177_VEBTi @ ( produc316462671093861988_VEBTi @ Xs2 @ Ys ) @ N3 )
        = ( produc6084888613844515218_VEBTi @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N3 @ ( size_s7982070591426661849_VEBTi @ Ys ) ) ) @ ( nth_VEBT_VEBTi @ Ys @ ( modulo_modulo_nat @ N3 @ ( size_s7982070591426661849_VEBTi @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6869_product__nth,axiom,
    ! [N3: nat,Xs2: list_real,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N3 @ ( times_times_nat @ ( size_size_list_real @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr5429231388385915574T_VEBT @ ( produc3722688996059531265T_VEBT @ Xs2 @ Ys ) @ N3 )
        = ( produc6931449550656315951T_VEBT @ ( nth_real @ Xs2 @ ( divide_divide_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N3 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6870_product__nth,axiom,
    ! [N3: nat,Xs2: list_real,Ys: list_real] :
      ( ( ord_less_nat @ N3 @ ( times_times_nat @ ( size_size_list_real @ Xs2 ) @ ( size_size_list_real @ Ys ) ) )
     => ( ( nth_Pr7489471911767062336l_real @ ( product_real_real @ Xs2 @ Ys ) @ N3 )
        = ( produc4511245868158468465l_real @ ( nth_real @ Xs2 @ ( divide_divide_nat @ N3 @ ( size_size_list_real @ Ys ) ) ) @ ( nth_real @ Ys @ ( modulo_modulo_nat @ N3 @ ( size_size_list_real @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6871_vebt__assn__raw_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa: vEBT_VEBTi,Y: assn] :
      ( ( ( vEBT_vebt_assn_raw @ X @ Xa )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ! [Ai: $o,Bi: $o] :
                ( ( Xa
                  = ( vEBT_Leafi @ Ai @ Bi ) )
               => ( Y
                 != ( pure_assn
                    @ ( ( Ai = A3 )
                      & ( Bi = B3 ) ) ) ) ) )
       => ( ! [Mmo: option4927543243414619207at_nat,Deg2: nat,Tree_list: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mmo @ Deg2 @ Tree_list @ Summary2 ) )
             => ! [Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] :
                  ( ( Xa
                    = ( vEBT_Nodei @ Mmoi @ Degi @ Tree_array @ Summaryi ) )
                 => ( Y
                   != ( times_times_assn
                      @ ( times_times_assn
                        @ ( pure_assn
                          @ ( ( Mmoi = Mmo )
                            & ( Degi = Deg2 ) ) )
                        @ ( vEBT_vebt_assn_raw @ Summary2 @ Summaryi ) )
                      @ ( ex_ass463751140784270563_VEBTi
                        @ ^ [Tree_is2: list_VEBT_VEBTi] : ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Tree_array @ Tree_is2 ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Tree_list @ Tree_is2 ) ) ) ) ) ) )
         => ( ( ? [V2: option4927543243414619207at_nat,Va3: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ V2 @ Va3 @ Vb3 @ Vc3 ) )
             => ( ? [Vd3: $o,Ve3: $o] :
                    ( Xa
                    = ( vEBT_Leafi @ Vd3 @ Ve3 ) )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ? [Vd3: $o,Ve3: $o] :
                    ( X
                    = ( vEBT_Leaf @ Vd3 @ Ve3 ) )
               => ( ? [V2: option4927543243414619207at_nat,Va3: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] :
                      ( Xa
                      = ( vEBT_Nodei @ V2 @ Va3 @ Vb3 @ Vc3 ) )
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% vebt_assn_raw.elims
thf(fact_6872_ex__assn__const,axiom,
    ! [C2: assn] :
      ( ( ex_ass463751140784270563_VEBTi
        @ ^ [X3: list_VEBT_VEBTi] : C2 )
      = C2 ) ).

% ex_assn_const
thf(fact_6873_flip__bit__nonnegative__int__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N3 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% flip_bit_nonnegative_int_iff
thf(fact_6874_flip__bit__negative__int__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N3 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% flip_bit_negative_int_iff
thf(fact_6875_tanh__0,axiom,
    ( ( tanh_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% tanh_0
thf(fact_6876_tanh__0,axiom,
    ( ( tanh_real @ zero_zero_real )
    = zero_zero_real ) ).

% tanh_0
thf(fact_6877_tanh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( ( tanh_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% tanh_real_zero_iff
thf(fact_6878_tanh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% tanh_real_less_iff
thf(fact_6879_tanh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% tanh_real_le_iff
thf(fact_6880_norm__assertion__simps_I17_J,axiom,
    ! [R: assn,Q: list_VEBT_VEBTi > assn] :
      ( ( times_times_assn @ R @ ( ex_ass463751140784270563_VEBTi @ Q ) )
      = ( ex_ass463751140784270563_VEBTi
        @ ^ [X3: list_VEBT_VEBTi] : ( times_times_assn @ R @ ( Q @ X3 ) ) ) ) ).

% norm_assertion_simps(17)
thf(fact_6881_norm__assertion__simps_I16_J,axiom,
    ! [Q: list_VEBT_VEBTi > assn,R: assn] :
      ( ( times_times_assn @ ( ex_ass463751140784270563_VEBTi @ Q ) @ R )
      = ( ex_ass463751140784270563_VEBTi
        @ ^ [X3: list_VEBT_VEBTi] : ( times_times_assn @ ( Q @ X3 ) @ R ) ) ) ).

% norm_assertion_simps(16)
thf(fact_6882_triv__exI,axiom,
    ! [Q: list_VEBT_VEBTi > assn,X: list_VEBT_VEBTi] : ( entails @ ( Q @ X ) @ ( ex_ass463751140784270563_VEBTi @ Q ) ) ).

% triv_exI
thf(fact_6883_mod__ex__dist,axiom,
    ! [P: list_VEBT_VEBTi > assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( ex_ass463751140784270563_VEBTi @ P ) @ H2 )
      = ( ? [X3: list_VEBT_VEBTi] : ( rep_assn @ ( P @ X3 ) @ H2 ) ) ) ).

% mod_ex_dist
thf(fact_6884_mod__neg__neg__trivial,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L2 @ K )
       => ( ( modulo_modulo_int @ K @ L2 )
          = K ) ) ) ).

% mod_neg_neg_trivial
thf(fact_6885_mod__pos__pos__trivial,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L2 )
       => ( ( modulo_modulo_int @ K @ L2 )
          = K ) ) ) ).

% mod_pos_pos_trivial
thf(fact_6886_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_6887_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_6888_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_6889_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_6890_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_6891_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_real] :
      ( ( size_s5035110155006384947T_real @ ( produc4908677263432625371T_real @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_real @ Ys ) ) ) ).

% length_product
thf(fact_6892_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_o] :
      ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_6893_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_nat] :
      ( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_6894_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBTi] :
      ( ( size_s3387956428969716412_VEBTi @ ( produc316462671093861988_VEBTi @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s7982070591426661849_VEBTi @ Ys ) ) ) ).

% length_product
thf(fact_6895_length__product,axiom,
    ! [Xs2: list_real,Ys: list_VEBT_VEBT] :
      ( ( size_s3289364478449617953T_VEBT @ ( produc3722688996059531265T_VEBT @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_real @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_6896_length__product,axiom,
    ! [Xs2: list_real,Ys: list_real] :
      ( ( size_s3932428310213730859l_real @ ( product_real_real @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_real @ Xs2 ) @ ( size_size_list_real @ Ys ) ) ) ).

% length_product
thf(fact_6897_length__product,axiom,
    ! [Xs2: list_real,Ys: list_o] :
      ( ( size_s987546567493390085real_o @ ( product_real_o @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_real @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_6898_length__product,axiom,
    ! [Xs2: list_real,Ys: list_nat] :
      ( ( size_s1877336372972134351al_nat @ ( product_real_nat @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_real @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_6899_length__product,axiom,
    ! [Xs2: list_real,Ys: list_VEBT_VEBTi] :
      ( ( size_s6963394564282275252_VEBTi @ ( produc5599769701989005224_VEBTi @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_real @ Xs2 ) @ ( size_s7982070591426661849_VEBTi @ Ys ) ) ) ).

% length_product
thf(fact_6900_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_6901_mod__h__bot__iff_I8_J,axiom,
    ! [R: list_VEBT_VEBTi > assn,H2: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( ex_ass463751140784270563_VEBTi @ R ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) )
      = ( ? [X3: list_VEBT_VEBTi] : ( rep_assn @ ( R @ X3 ) @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(8)
thf(fact_6902_one__mod__exp__eq__one,axiom,
    ! [N3: nat] :
      ( ( modulo_modulo_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) )
      = one_one_int ) ).

% one_mod_exp_eq_one
thf(fact_6903_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_6904_eucl__rel__int,axiom,
    ! [K: int,L2: int] : ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L2 ) @ ( modulo_modulo_int @ K @ L2 ) ) ) ).

% eucl_rel_int
thf(fact_6905_zmod__helper,axiom,
    ! [N3: int,M: int,K: int,A2: int] :
      ( ( ( modulo_modulo_int @ N3 @ M )
        = K )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ N3 @ A2 ) @ M )
        = ( modulo_modulo_int @ ( plus_plus_int @ K @ A2 ) @ M ) ) ) ).

% zmod_helper
thf(fact_6906_Word_Omod__minus__cong,axiom,
    ! [B2: int,B4: int,X: int,X6: int,Y: int,Y6: int,Z6: int] :
      ( ( B2 = B4 )
     => ( ( ( modulo_modulo_int @ X @ B4 )
          = ( modulo_modulo_int @ X6 @ B4 ) )
       => ( ( ( modulo_modulo_int @ Y @ B4 )
            = ( modulo_modulo_int @ Y6 @ B4 ) )
         => ( ( ( minus_minus_int @ X6 @ Y6 )
              = Z6 )
           => ( ( modulo_modulo_int @ ( minus_minus_int @ X @ Y ) @ B2 )
              = ( modulo_modulo_int @ Z6 @ B4 ) ) ) ) ) ) ).

% Word.mod_minus_cong
thf(fact_6907_div__int__unique,axiom,
    ! [K: int,L2: int,Q3: int,R3: int] :
      ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q3 @ R3 ) )
     => ( ( divide_divide_int @ K @ L2 )
        = Q3 ) ) ).

% div_int_unique
thf(fact_6908_ex__distrib__star,axiom,
    ! [P: list_VEBT_VEBTi > assn,Q: assn] :
      ( ( ex_ass463751140784270563_VEBTi
        @ ^ [X3: list_VEBT_VEBTi] : ( times_times_assn @ ( P @ X3 ) @ Q ) )
      = ( times_times_assn @ ( ex_ass463751140784270563_VEBTi @ P ) @ Q ) ) ).

% ex_distrib_star
thf(fact_6909_ent__ex__preI,axiom,
    ! [P: list_VEBT_VEBTi > assn,Q: assn] :
      ( ! [X4: list_VEBT_VEBTi] : ( entails @ ( P @ X4 ) @ Q )
     => ( entails @ ( ex_ass463751140784270563_VEBTi @ P ) @ Q ) ) ).

% ent_ex_preI
thf(fact_6910_ent__ex__postI,axiom,
    ! [P: assn,Q: list_VEBT_VEBTi > assn,X: list_VEBT_VEBTi] :
      ( ( entails @ P @ ( Q @ X ) )
     => ( entails @ P @ ( ex_ass463751140784270563_VEBTi @ Q ) ) ) ).

% ent_ex_postI
thf(fact_6911_enorm__exI_H,axiom,
    ! [Z7: list_VEBT_VEBTi > $o,P: assn,Q: list_VEBT_VEBTi > assn] :
      ( ! [X4: list_VEBT_VEBTi] :
          ( ( Z7 @ X4 )
         => ( entails @ P @ ( Q @ X4 ) ) )
     => ( ? [X_12: list_VEBT_VEBTi] : ( Z7 @ X_12 )
       => ( entails @ P @ ( ex_ass463751140784270563_VEBTi @ Q ) ) ) ) ).

% enorm_exI'
thf(fact_6912_ex__one__point__gen,axiom,
    ! [P: list_VEBT_VEBTi > assn,V: list_VEBT_VEBTi] :
      ( ! [H4: produc3658429121746597890et_nat,X4: list_VEBT_VEBTi] :
          ( ( rep_assn @ ( P @ X4 ) @ H4 )
         => ( X4 = V ) )
     => ( ( ex_ass463751140784270563_VEBTi @ P )
        = ( P @ V ) ) ) ).

% ex_one_point_gen
thf(fact_6913_mod__exI,axiom,
    ! [P: list_VEBT_VEBTi > assn,H2: produc3658429121746597890et_nat] :
      ( ? [X5: list_VEBT_VEBTi] : ( rep_assn @ ( P @ X5 ) @ H2 )
     => ( rep_assn @ ( ex_ass463751140784270563_VEBTi @ P ) @ H2 ) ) ).

% mod_exI
thf(fact_6914_mod__exE,axiom,
    ! [P: list_VEBT_VEBTi > assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( ex_ass463751140784270563_VEBTi @ P ) @ H2 )
     => ~ ! [X4: list_VEBT_VEBTi] :
            ~ ( rep_assn @ ( P @ X4 ) @ H2 ) ) ).

% mod_exE
thf(fact_6915_neg__mod__bound,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ L2 @ zero_zero_int )
     => ( ord_less_int @ L2 @ ( modulo_modulo_int @ K @ L2 ) ) ) ).

% neg_mod_bound
thf(fact_6916_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L2 )
     => ( ord_less_int @ ( modulo_modulo_int @ K @ L2 ) @ L2 ) ) ).

% Euclidean_Division.pos_mod_bound
thf(fact_6917_divmod__int__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M5: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% divmod_int_def
thf(fact_6918_tanh__real__lt__1,axiom,
    ! [X: real] : ( ord_less_real @ ( tanh_real @ X ) @ one_one_real ) ).

% tanh_real_lt_1
thf(fact_6919_int__mod__ge,axiom,
    ! [A2: int,N3: int] :
      ( ( ord_less_int @ A2 @ N3 )
     => ( ( ord_less_int @ zero_zero_int @ N3 )
       => ( ord_less_eq_int @ A2 @ ( modulo_modulo_int @ A2 @ N3 ) ) ) ) ).

% int_mod_ge
thf(fact_6920_neg__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ B2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A2 @ B2 ) @ zero_zero_int )
        & ( ord_less_int @ B2 @ ( modulo_modulo_int @ A2 @ B2 ) ) ) ) ).

% neg_mod_conj
thf(fact_6921_pos__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A2 @ B2 ) )
        & ( ord_less_int @ ( modulo_modulo_int @ A2 @ B2 ) @ B2 ) ) ) ).

% pos_mod_conj
thf(fact_6922_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( ( modulo_modulo_int @ I @ K )
        = I )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I )
          & ( ord_less_int @ I @ K ) )
        | ( ( ord_less_eq_int @ I @ zero_zero_int )
          & ( ord_less_int @ K @ I ) ) ) ) ).

% zmod_trivial_iff
thf(fact_6923_int__mod__eq,axiom,
    ! [B2: int,N3: int,A2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B2 )
     => ( ( ord_less_int @ B2 @ N3 )
       => ( ( ( modulo_modulo_int @ A2 @ N3 )
            = ( modulo_modulo_int @ B2 @ N3 ) )
         => ( ( modulo_modulo_int @ A2 @ N3 )
            = B2 ) ) ) ) ).

% int_mod_eq
thf(fact_6924_int__mod__lem,axiom,
    ! [N3: int,B2: int] :
      ( ( ord_less_int @ zero_zero_int @ N3 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ B2 )
          & ( ord_less_int @ B2 @ N3 ) )
        = ( ( modulo_modulo_int @ B2 @ N3 )
          = B2 ) ) ) ).

% int_mod_lem
thf(fact_6925_neg__mod__sign,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ L2 @ zero_zero_int )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L2 ) @ zero_zero_int ) ) ).

% neg_mod_sign
thf(fact_6926_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L2 ) ) ) ).

% Euclidean_Division.pos_mod_sign
thf(fact_6927_int__mod__le_H,axiom,
    ! [B2: int,N3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ B2 @ N3 ) )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ B2 @ N3 ) @ ( minus_minus_int @ B2 @ N3 ) ) ) ).

% int_mod_le'
thf(fact_6928_nonneg__mod__div,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A2 @ B2 ) )
          & ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ) ).

% nonneg_mod_div
thf(fact_6929_zdiv__mono__strict,axiom,
    ! [A: int,B: int,N3: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ N3 )
       => ( ( ( modulo_modulo_int @ A @ N3 )
            = zero_zero_int )
         => ( ( ( modulo_modulo_int @ B @ N3 )
              = zero_zero_int )
           => ( ord_less_int @ ( divide_divide_int @ A @ N3 ) @ ( divide_divide_int @ B @ N3 ) ) ) ) ) ) ).

% zdiv_mono_strict
thf(fact_6930_div__mod__decomp__int,axiom,
    ! [A: int,N3: int] :
      ( A
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ N3 ) @ N3 ) @ ( modulo_modulo_int @ A @ N3 ) ) ) ).

% div_mod_decomp_int
thf(fact_6931_mod__div__equality__div__eq,axiom,
    ! [A2: int,B2: int] :
      ( ( times_times_int @ ( divide_divide_int @ A2 @ B2 ) @ B2 )
      = ( minus_minus_int @ A2 @ ( modulo_modulo_int @ A2 @ B2 ) ) ) ).

% mod_div_equality_div_eq
thf(fact_6932_pos__mod__bound2,axiom,
    ! [A2: int] : ( ord_less_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% pos_mod_bound2
thf(fact_6933_int__mod__ge_H,axiom,
    ! [B2: int,N3: int] :
      ( ( ord_less_int @ B2 @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ N3 )
       => ( ord_less_eq_int @ ( plus_plus_int @ B2 @ N3 ) @ ( modulo_modulo_int @ B2 @ N3 ) ) ) ) ).

% int_mod_ge'
thf(fact_6934_mod__pos__neg__trivial,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
       => ( ( modulo_modulo_int @ K @ L2 )
          = ( plus_plus_int @ K @ L2 ) ) ) ) ).

% mod_pos_neg_trivial
thf(fact_6935_mod__pos__geq,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L2 )
     => ( ( ord_less_eq_int @ L2 @ K )
       => ( ( modulo_modulo_int @ K @ L2 )
          = ( modulo_modulo_int @ ( minus_minus_int @ K @ L2 ) @ L2 ) ) ) ) ).

% mod_pos_geq
thf(fact_6936_mod__int__pos__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L2 ) )
      = ( ( dvd_dvd_int @ L2 @ K )
        | ( ( L2 = zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ K ) )
        | ( ord_less_int @ zero_zero_int @ L2 ) ) ) ).

% mod_int_pos_iff
thf(fact_6937_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_6938_real__of__int__div__aux,axiom,
    ! [X: int,D2: int] :
      ( ( divide_divide_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ D2 ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X @ D2 ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X @ D2 ) ) @ ( ring_1_of_int_real @ D2 ) ) ) ) ).

% real_of_int_div_aux
thf(fact_6939_pos__mod__sign2,axiom,
    ! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% pos_mod_sign2
thf(fact_6940_mod__2__neq__1__eq__eq__0,axiom,
    ! [K: int] :
      ( ( ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != one_one_int )
      = ( ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% mod_2_neq_1_eq_eq_0
thf(fact_6941_nmod2,axiom,
    ! [N3: int] :
      ( ( ( modulo_modulo_int @ N3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int )
      | ( ( modulo_modulo_int @ N3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% nmod2
thf(fact_6942_mod__exp__less__eq__exp,axiom,
    ! [A2: int,N3: nat] : ( ord_less_int @ ( modulo_modulo_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ).

% mod_exp_less_eq_exp
thf(fact_6943_mod__power__lem,axiom,
    ! [A2: int,M: nat,N3: nat] :
      ( ( ord_less_int @ one_one_int @ A2 )
     => ( ( ( ord_less_eq_nat @ M @ N3 )
         => ( ( modulo_modulo_int @ ( power_power_int @ A2 @ N3 ) @ ( power_power_int @ A2 @ M ) )
            = zero_zero_int ) )
        & ( ~ ( ord_less_eq_nat @ M @ N3 )
         => ( ( modulo_modulo_int @ ( power_power_int @ A2 @ N3 ) @ ( power_power_int @ A2 @ M ) )
            = ( power_power_int @ A2 @ N3 ) ) ) ) ) ).

% mod_power_lem
thf(fact_6944_split__zmod,axiom,
    ! [P: int > $o,N3: int,K: int] :
      ( ( P @ ( modulo_modulo_int @ N3 @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ N3 ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I2: int,J: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J )
                & ( ord_less_int @ J @ K )
                & ( N3
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J ) ) )
             => ( P @ J ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I2: int,J: int] :
              ( ( ( ord_less_int @ K @ J )
                & ( ord_less_eq_int @ J @ zero_zero_int )
                & ( N3
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J ) ) )
             => ( P @ J ) ) ) ) ) ).

% split_zmod
thf(fact_6945_int__mod__neg__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R3: int] :
      ( ( A2
        = ( plus_plus_int @ ( times_times_int @ B2 @ Q3 ) @ R3 ) )
     => ( ( ord_less_eq_int @ R3 @ zero_zero_int )
       => ( ( ord_less_int @ B2 @ R3 )
         => ( ( modulo_modulo_int @ A2 @ B2 )
            = R3 ) ) ) ) ).

% int_mod_neg_eq
thf(fact_6946_int__mod__pos__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R3: int] :
      ( ( A2
        = ( plus_plus_int @ ( times_times_int @ B2 @ Q3 ) @ R3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
       => ( ( ord_less_int @ R3 @ B2 )
         => ( ( modulo_modulo_int @ A2 @ B2 )
            = R3 ) ) ) ) ).

% int_mod_pos_eq
thf(fact_6947_mod__add__if__z,axiom,
    ! [X: int,Z: int,Y: int] :
      ( ( ord_less_int @ X @ Z )
     => ( ( ord_less_int @ Y @ Z )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( ( ord_less_eq_int @ zero_zero_int @ X )
           => ( ( ord_less_eq_int @ zero_zero_int @ Z )
             => ( ( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ Z )
                 => ( ( modulo_modulo_int @ ( plus_plus_int @ X @ Y ) @ Z )
                    = ( plus_plus_int @ X @ Y ) ) )
                & ( ~ ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ Z )
                 => ( ( modulo_modulo_int @ ( plus_plus_int @ X @ Y ) @ Z )
                    = ( minus_minus_int @ ( plus_plus_int @ X @ Y ) @ Z ) ) ) ) ) ) ) ) ) ).

% mod_add_if_z
thf(fact_6948_mod__sub__if__z,axiom,
    ! [X: int,Z: int,Y: int] :
      ( ( ord_less_int @ X @ Z )
     => ( ( ord_less_int @ Y @ Z )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( ( ord_less_eq_int @ zero_zero_int @ X )
           => ( ( ord_less_eq_int @ zero_zero_int @ Z )
             => ( ( ( ord_less_eq_int @ Y @ X )
                 => ( ( modulo_modulo_int @ ( minus_minus_int @ X @ Y ) @ Z )
                    = ( minus_minus_int @ X @ Y ) ) )
                & ( ~ ( ord_less_eq_int @ Y @ X )
                 => ( ( modulo_modulo_int @ ( minus_minus_int @ X @ Y ) @ Z )
                    = ( plus_plus_int @ ( minus_minus_int @ X @ Y ) @ Z ) ) ) ) ) ) ) ) ) ).

% mod_sub_if_z
thf(fact_6949_zmod__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C2 )
     => ( ( modulo_modulo_int @ A2 @ ( times_times_int @ B2 @ C2 ) )
        = ( plus_plus_int @ ( times_times_int @ B2 @ ( modulo_modulo_int @ ( divide_divide_int @ A2 @ B2 ) @ C2 ) ) @ ( modulo_modulo_int @ A2 @ B2 ) ) ) ) ).

% zmod_zmult2_eq
thf(fact_6950_axxmod2,axiom,
    ! [X: int] :
      ( ( ( modulo_modulo_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ X ) @ X ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int )
      & ( ( modulo_modulo_int @ ( plus_plus_int @ ( plus_plus_int @ zero_zero_int @ X ) @ X ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% axxmod2
thf(fact_6951_z1pmod2,axiom,
    ! [B2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = one_one_int ) ).

% z1pmod2
thf(fact_6952_verit__le__mono__div__int,axiom,
    ! [A: int,B: int,N3: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ N3 )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A @ N3 )
            @ ( if_int
              @ ( ( modulo_modulo_int @ B @ N3 )
                = zero_zero_int )
              @ one_one_int
              @ zero_zero_int ) )
          @ ( divide_divide_int @ B @ N3 ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_6953_split__neg__lemma,axiom,
    ! [K: int,P: int > int > $o,N3: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ( P @ ( divide_divide_int @ N3 @ K ) @ ( modulo_modulo_int @ N3 @ K ) )
        = ( ! [I2: int,J: int] :
              ( ( ( ord_less_int @ K @ J )
                & ( ord_less_eq_int @ J @ zero_zero_int )
                & ( N3
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J ) ) )
             => ( P @ I2 @ J ) ) ) ) ) ).

% split_neg_lemma
thf(fact_6954_split__pos__lemma,axiom,
    ! [K: int,P: int > int > $o,N3: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( P @ ( divide_divide_int @ N3 @ K ) @ ( modulo_modulo_int @ N3 @ K ) )
        = ( ! [I2: int,J: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J )
                & ( ord_less_int @ J @ K )
                & ( N3
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J ) ) )
             => ( P @ I2 @ J ) ) ) ) ) ).

% split_pos_lemma
thf(fact_6955_p1mod22k,axiom,
    ! [B2: int,N3: nat] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ one_one_int ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) @ one_one_int ) ) ).

% p1mod22k
thf(fact_6956_p1mod22k_H,axiom,
    ! [B2: int,N3: nat] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% p1mod22k'
thf(fact_6957_eucl__rel__int__iff,axiom,
    ! [K: int,L2: int,Q3: int,R3: int] :
      ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q3 @ R3 ) )
      = ( ( K
          = ( plus_plus_int @ ( times_times_int @ L2 @ Q3 ) @ R3 ) )
        & ( ( ord_less_int @ zero_zero_int @ L2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
            & ( ord_less_int @ R3 @ L2 ) ) )
        & ( ~ ( ord_less_int @ zero_zero_int @ L2 )
         => ( ( ( ord_less_int @ L2 @ zero_zero_int )
             => ( ( ord_less_int @ L2 @ R3 )
                & ( ord_less_eq_int @ R3 @ zero_zero_int ) ) )
            & ( ~ ( ord_less_int @ L2 @ zero_zero_int )
             => ( Q3 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_6958_pos__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B2 @ A2 ) ) ) ) ) ).

% pos_zmod_mult_2
thf(fact_6959_eme1p,axiom,
    ! [N3: int,D2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ D2 )
         => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ N3 ) @ D2 )
            = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ N3 @ D2 ) ) ) ) ) ) ).

% eme1p
thf(fact_6960_emep1,axiom,
    ! [N3: int,D2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ D2 )
         => ( ( modulo_modulo_int @ ( plus_plus_int @ N3 @ one_one_int ) @ D2 )
            = ( plus_plus_int @ ( modulo_modulo_int @ N3 @ D2 ) @ one_one_int ) ) ) ) ) ).

% emep1
thf(fact_6961_even__flip__bit__iff,axiom,
    ! [M: nat,A2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A2 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_6962_even__flip__bit__iff,axiom,
    ! [M: nat,A2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_6963_sb__inc__lem,axiom,
    ! [A2: int,K: nat] :
      ( ( ord_less_int @ ( plus_plus_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ zero_zero_int )
     => ( ord_less_eq_int @ ( plus_plus_int @ ( plus_plus_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ K ) ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ K ) ) ) ) ) ).

% sb_inc_lem
thf(fact_6964_vebt__assn__raw_Osimps_I2_J,axiom,
    ! [Mmo2: option4927543243414619207at_nat,Deg: nat,Tree_list2: list_VEBT_VEBT,Summary: vEBT_VEBT,Mmoi2: option4927543243414619207at_nat,Degi2: nat,Tree_array2: array_VEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
      ( ( vEBT_vebt_assn_raw @ ( vEBT_Node @ Mmo2 @ Deg @ Tree_list2 @ Summary ) @ ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) )
      = ( times_times_assn
        @ ( times_times_assn
          @ ( pure_assn
            @ ( ( Mmoi2 = Mmo2 )
              & ( Degi2 = Deg ) ) )
          @ ( vEBT_vebt_assn_raw @ Summary @ Summaryi2 ) )
        @ ( ex_ass463751140784270563_VEBTi
          @ ^ [Tree_is2: list_VEBT_VEBTi] : ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Tree_array2 @ Tree_is2 ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Tree_list2 @ Tree_is2 ) ) ) ) ) ).

% vebt_assn_raw.simps(2)
thf(fact_6965_neg__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
        = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B2 @ one_one_int ) @ A2 ) ) @ one_one_int ) ) ) ).

% neg_zmod_mult_2
thf(fact_6966_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q3: int,R3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B2 )
     => ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair_int_int @ Q3 @ R3 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ ( product_Pair_int_int @ Q3 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R3 ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_6967_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_6968_triangle__def,axiom,
    ( nat_triangle
    = ( ^ [N2: nat] : ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% triangle_def
thf(fact_6969_obtain__set__pred,axiom,
    ! [Z: nat,X: nat,A: set_nat] :
      ( ( ord_less_nat @ Z @ X )
     => ( ( vEBT_VEBT_min_in_set @ A @ Z )
       => ( ( finite_finite_nat @ A )
         => ? [X_1: nat] : ( vEBT_is_pred_in_set @ A @ X @ X_1 ) ) ) ) ).

% obtain_set_pred
thf(fact_6970_obtain__set__succ,axiom,
    ! [X: nat,Z: nat,A: set_nat,B: set_nat] :
      ( ( ord_less_nat @ X @ Z )
     => ( ( vEBT_VEBT_max_in_set @ A @ Z )
       => ( ( finite_finite_nat @ B )
         => ( ( A = B )
           => ? [X_1: nat] : ( vEBT_is_succ_in_set @ A @ X @ X_1 ) ) ) ) ) ).

% obtain_set_succ
thf(fact_6971_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ T @ N3 )
     => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_finite
thf(fact_6972_succ__none__empty,axiom,
    ! [Xs2: set_nat,A2: nat] :
      ( ~ ? [X_1: nat] : ( vEBT_is_succ_in_set @ Xs2 @ A2 @ X_1 )
     => ( ( finite_finite_nat @ Xs2 )
       => ~ ? [X5: nat] :
              ( ( member_nat @ X5 @ Xs2 )
              & ( ord_less_nat @ A2 @ X5 ) ) ) ) ).

% succ_none_empty
thf(fact_6973_pred__none__empty,axiom,
    ! [Xs2: set_nat,A2: nat] :
      ( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs2 @ A2 @ X_1 )
     => ( ( finite_finite_nat @ Xs2 )
       => ~ ? [X5: nat] :
              ( ( member_nat @ X5 @ Xs2 )
              & ( ord_less_nat @ X5 @ A2 ) ) ) ) ).

% pred_none_empty
thf(fact_6974_neg__equal__iff__equal,axiom,
    ! [A2: real,B2: real] :
      ( ( ( uminus_uminus_real @ A2 )
        = ( uminus_uminus_real @ B2 ) )
      = ( A2 = B2 ) ) ).

% neg_equal_iff_equal
thf(fact_6975_neg__equal__iff__equal,axiom,
    ! [A2: int,B2: int] :
      ( ( ( uminus_uminus_int @ A2 )
        = ( uminus_uminus_int @ B2 ) )
      = ( A2 = B2 ) ) ).

% neg_equal_iff_equal
thf(fact_6976_neg__equal__iff__equal,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ( uminus1482373934393186551omplex @ A2 )
        = ( uminus1482373934393186551omplex @ B2 ) )
      = ( A2 = B2 ) ) ).

% neg_equal_iff_equal
thf(fact_6977_neg__equal__iff__equal,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A2 )
        = ( uminus1351360451143612070nteger @ B2 ) )
      = ( A2 = B2 ) ) ).

% neg_equal_iff_equal
thf(fact_6978_neg__equal__iff__equal,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( uminus_uminus_rat @ A2 )
        = ( uminus_uminus_rat @ B2 ) )
      = ( A2 = B2 ) ) ).

% neg_equal_iff_equal
thf(fact_6979_add_Oinverse__inverse,axiom,
    ! [A2: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ A2 ) )
      = A2 ) ).

% add.inverse_inverse
thf(fact_6980_add_Oinverse__inverse,axiom,
    ! [A2: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A2 ) )
      = A2 ) ).

% add.inverse_inverse
thf(fact_6981_add_Oinverse__inverse,axiom,
    ! [A2: complex] :
      ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A2 ) )
      = A2 ) ).

% add.inverse_inverse
thf(fact_6982_add_Oinverse__inverse,axiom,
    ! [A2: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A2 ) )
      = A2 ) ).

% add.inverse_inverse
thf(fact_6983_add_Oinverse__inverse,axiom,
    ! [A2: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A2 ) )
      = A2 ) ).

% add.inverse_inverse
thf(fact_6984_neg__equal__zero,axiom,
    ! [A2: real] :
      ( ( ( uminus_uminus_real @ A2 )
        = A2 )
      = ( A2 = zero_zero_real ) ) ).

% neg_equal_zero
thf(fact_6985_neg__equal__zero,axiom,
    ! [A2: int] :
      ( ( ( uminus_uminus_int @ A2 )
        = A2 )
      = ( A2 = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_6986_neg__equal__zero,axiom,
    ! [A2: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A2 )
        = A2 )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
thf(fact_6987_neg__equal__zero,axiom,
    ! [A2: rat] :
      ( ( ( uminus_uminus_rat @ A2 )
        = A2 )
      = ( A2 = zero_zero_rat ) ) ).

% neg_equal_zero
thf(fact_6988_equal__neg__zero,axiom,
    ! [A2: real] :
      ( ( A2
        = ( uminus_uminus_real @ A2 ) )
      = ( A2 = zero_zero_real ) ) ).

% equal_neg_zero
thf(fact_6989_equal__neg__zero,axiom,
    ! [A2: int] :
      ( ( A2
        = ( uminus_uminus_int @ A2 ) )
      = ( A2 = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_6990_equal__neg__zero,axiom,
    ! [A2: code_integer] :
      ( ( A2
        = ( uminus1351360451143612070nteger @ A2 ) )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
thf(fact_6991_equal__neg__zero,axiom,
    ! [A2: rat] :
      ( ( A2
        = ( uminus_uminus_rat @ A2 ) )
      = ( A2 = zero_zero_rat ) ) ).

% equal_neg_zero
thf(fact_6992_neg__equal__0__iff__equal,axiom,
    ! [A2: real] :
      ( ( ( uminus_uminus_real @ A2 )
        = zero_zero_real )
      = ( A2 = zero_zero_real ) ) ).

% neg_equal_0_iff_equal
thf(fact_6993_neg__equal__0__iff__equal,axiom,
    ! [A2: int] :
      ( ( ( uminus_uminus_int @ A2 )
        = zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
thf(fact_6994_neg__equal__0__iff__equal,axiom,
    ! [A2: complex] :
      ( ( ( uminus1482373934393186551omplex @ A2 )
        = zero_zero_complex )
      = ( A2 = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
thf(fact_6995_neg__equal__0__iff__equal,axiom,
    ! [A2: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A2 )
        = zero_z3403309356797280102nteger )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
thf(fact_6996_neg__equal__0__iff__equal,axiom,
    ! [A2: rat] :
      ( ( ( uminus_uminus_rat @ A2 )
        = zero_zero_rat )
      = ( A2 = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
thf(fact_6997_neg__0__equal__iff__equal,axiom,
    ! [A2: real] :
      ( ( zero_zero_real
        = ( uminus_uminus_real @ A2 ) )
      = ( zero_zero_real = A2 ) ) ).

% neg_0_equal_iff_equal
thf(fact_6998_neg__0__equal__iff__equal,axiom,
    ! [A2: int] :
      ( ( zero_zero_int
        = ( uminus_uminus_int @ A2 ) )
      = ( zero_zero_int = A2 ) ) ).

% neg_0_equal_iff_equal
thf(fact_6999_neg__0__equal__iff__equal,axiom,
    ! [A2: complex] :
      ( ( zero_zero_complex
        = ( uminus1482373934393186551omplex @ A2 ) )
      = ( zero_zero_complex = A2 ) ) ).

% neg_0_equal_iff_equal
thf(fact_7000_neg__0__equal__iff__equal,axiom,
    ! [A2: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( uminus1351360451143612070nteger @ A2 ) )
      = ( zero_z3403309356797280102nteger = A2 ) ) ).

% neg_0_equal_iff_equal
thf(fact_7001_neg__0__equal__iff__equal,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat
        = ( uminus_uminus_rat @ A2 ) )
      = ( zero_zero_rat = A2 ) ) ).

% neg_0_equal_iff_equal
thf(fact_7002_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_real @ zero_zero_real )
    = zero_zero_real ) ).

% add.inverse_neutral
thf(fact_7003_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_7004_add_Oinverse__neutral,axiom,
    ( ( uminus1482373934393186551omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% add.inverse_neutral
thf(fact_7005_add_Oinverse__neutral,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% add.inverse_neutral
thf(fact_7006_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% add.inverse_neutral
thf(fact_7007_neg__le__iff__le,axiom,
    ! [B2: real,A2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A2 ) )
      = ( ord_less_eq_real @ A2 @ B2 ) ) ).

% neg_le_iff_le
thf(fact_7008_neg__le__iff__le,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ).

% neg_le_iff_le
thf(fact_7009_neg__le__iff__le,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A2 ) )
      = ( ord_less_eq_rat @ A2 @ B2 ) ) ).

% neg_le_iff_le
thf(fact_7010_neg__le__iff__le,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) )
      = ( ord_less_eq_int @ A2 @ B2 ) ) ).

% neg_le_iff_le
thf(fact_7011_neg__numeral__eq__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
      = ( M = N3 ) ) ).

% neg_numeral_eq_iff
thf(fact_7012_neg__numeral__eq__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( M = N3 ) ) ).

% neg_numeral_eq_iff
thf(fact_7013_neg__numeral__eq__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N3 ) ) )
      = ( M = N3 ) ) ).

% neg_numeral_eq_iff
thf(fact_7014_neg__numeral__eq__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N3 ) ) )
      = ( M = N3 ) ) ).

% neg_numeral_eq_iff
thf(fact_7015_neg__numeral__eq__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) )
      = ( M = N3 ) ) ).

% neg_numeral_eq_iff
thf(fact_7016_neg__less__iff__less,axiom,
    ! [B2: real,A2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A2 ) )
      = ( ord_less_real @ A2 @ B2 ) ) ).

% neg_less_iff_less
thf(fact_7017_neg__less__iff__less,axiom,
    ! [B2: int,A2: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) )
      = ( ord_less_int @ A2 @ B2 ) ) ).

% neg_less_iff_less
thf(fact_7018_neg__less__iff__less,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( ord_le6747313008572928689nteger @ A2 @ B2 ) ) ).

% neg_less_iff_less
thf(fact_7019_neg__less__iff__less,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A2 ) )
      = ( ord_less_rat @ A2 @ B2 ) ) ).

% neg_less_iff_less
thf(fact_7020_minus__add__distrib,axiom,
    ! [A2: real,B2: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A2 @ B2 ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B2 ) ) ) ).

% minus_add_distrib
thf(fact_7021_minus__add__distrib,axiom,
    ! [A2: int,B2: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B2 ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B2 ) ) ) ).

% minus_add_distrib
thf(fact_7022_minus__add__distrib,axiom,
    ! [A2: complex,B2: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A2 @ B2 ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( uminus1482373934393186551omplex @ B2 ) ) ) ).

% minus_add_distrib
thf(fact_7023_minus__add__distrib,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( uminus1351360451143612070nteger @ B2 ) ) ) ).

% minus_add_distrib
thf(fact_7024_minus__add__distrib,axiom,
    ! [A2: rat,B2: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A2 @ B2 ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ ( uminus_uminus_rat @ B2 ) ) ) ).

% minus_add_distrib
thf(fact_7025_minus__add__cancel,axiom,
    ! [A2: real,B2: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ ( plus_plus_real @ A2 @ B2 ) )
      = B2 ) ).

% minus_add_cancel
thf(fact_7026_minus__add__cancel,axiom,
    ! [A2: int,B2: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( plus_plus_int @ A2 @ B2 ) )
      = B2 ) ).

% minus_add_cancel
thf(fact_7027_minus__add__cancel,axiom,
    ! [A2: complex,B2: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( plus_plus_complex @ A2 @ B2 ) )
      = B2 ) ).

% minus_add_cancel
thf(fact_7028_minus__add__cancel,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) )
      = B2 ) ).

% minus_add_cancel
thf(fact_7029_minus__add__cancel,axiom,
    ! [A2: rat,B2: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ ( plus_plus_rat @ A2 @ B2 ) )
      = B2 ) ).

% minus_add_cancel
thf(fact_7030_add__minus__cancel,axiom,
    ! [A2: real,B2: real] :
      ( ( plus_plus_real @ A2 @ ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ B2 ) )
      = B2 ) ).

% add_minus_cancel
thf(fact_7031_add__minus__cancel,axiom,
    ! [A2: int,B2: int] :
      ( ( plus_plus_int @ A2 @ ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B2 ) )
      = B2 ) ).

% add_minus_cancel
thf(fact_7032_add__minus__cancel,axiom,
    ! [A2: complex,B2: complex] :
      ( ( plus_plus_complex @ A2 @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B2 ) )
      = B2 ) ).

% add_minus_cancel
thf(fact_7033_add__minus__cancel,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A2 @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 ) )
      = B2 ) ).

% add_minus_cancel
thf(fact_7034_add__minus__cancel,axiom,
    ! [A2: rat,B2: rat] :
      ( ( plus_plus_rat @ A2 @ ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ B2 ) )
      = B2 ) ).

% add_minus_cancel
thf(fact_7035_mult__minus__right,axiom,
    ! [A2: real,B2: real] :
      ( ( times_times_real @ A2 @ ( uminus_uminus_real @ B2 ) )
      = ( uminus_uminus_real @ ( times_times_real @ A2 @ B2 ) ) ) ).

% mult_minus_right
thf(fact_7036_mult__minus__right,axiom,
    ! [A2: int,B2: int] :
      ( ( times_times_int @ A2 @ ( uminus_uminus_int @ B2 ) )
      = ( uminus_uminus_int @ ( times_times_int @ A2 @ B2 ) ) ) ).

% mult_minus_right
thf(fact_7037_mult__minus__right,axiom,
    ! [A2: complex,B2: complex] :
      ( ( times_times_complex @ A2 @ ( uminus1482373934393186551omplex @ B2 ) )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A2 @ B2 ) ) ) ).

% mult_minus_right
thf(fact_7038_mult__minus__right,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( times_3573771949741848930nteger @ A2 @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) ) ) ).

% mult_minus_right
thf(fact_7039_mult__minus__right,axiom,
    ! [A2: rat,B2: rat] :
      ( ( times_times_rat @ A2 @ ( uminus_uminus_rat @ B2 ) )
      = ( uminus_uminus_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ).

% mult_minus_right
thf(fact_7040_minus__mult__minus,axiom,
    ! [A2: real,B2: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B2 ) )
      = ( times_times_real @ A2 @ B2 ) ) ).

% minus_mult_minus
thf(fact_7041_minus__mult__minus,axiom,
    ! [A2: int,B2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B2 ) )
      = ( times_times_int @ A2 @ B2 ) ) ).

% minus_mult_minus
thf(fact_7042_minus__mult__minus,axiom,
    ! [A2: complex,B2: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( uminus1482373934393186551omplex @ B2 ) )
      = ( times_times_complex @ A2 @ B2 ) ) ).

% minus_mult_minus
thf(fact_7043_minus__mult__minus,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( times_3573771949741848930nteger @ A2 @ B2 ) ) ).

% minus_mult_minus
thf(fact_7044_minus__mult__minus,axiom,
    ! [A2: rat,B2: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A2 ) @ ( uminus_uminus_rat @ B2 ) )
      = ( times_times_rat @ A2 @ B2 ) ) ).

% minus_mult_minus
thf(fact_7045_mult__minus__left,axiom,
    ! [A2: real,B2: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A2 ) @ B2 )
      = ( uminus_uminus_real @ ( times_times_real @ A2 @ B2 ) ) ) ).

% mult_minus_left
thf(fact_7046_mult__minus__left,axiom,
    ! [A2: int,B2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ B2 )
      = ( uminus_uminus_int @ ( times_times_int @ A2 @ B2 ) ) ) ).

% mult_minus_left
thf(fact_7047_mult__minus__left,axiom,
    ! [A2: complex,B2: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B2 )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A2 @ B2 ) ) ) ).

% mult_minus_left
thf(fact_7048_mult__minus__left,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 )
      = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A2 @ B2 ) ) ) ).

% mult_minus_left
thf(fact_7049_mult__minus__left,axiom,
    ! [A2: rat,B2: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A2 ) @ B2 )
      = ( uminus_uminus_rat @ ( times_times_rat @ A2 @ B2 ) ) ) ).

% mult_minus_left
thf(fact_7050_minus__diff__eq,axiom,
    ! [A2: real,B2: real] :
      ( ( uminus_uminus_real @ ( minus_minus_real @ A2 @ B2 ) )
      = ( minus_minus_real @ B2 @ A2 ) ) ).

% minus_diff_eq
thf(fact_7051_minus__diff__eq,axiom,
    ! [A2: int,B2: int] :
      ( ( uminus_uminus_int @ ( minus_minus_int @ A2 @ B2 ) )
      = ( minus_minus_int @ B2 @ A2 ) ) ).

% minus_diff_eq
thf(fact_7052_minus__diff__eq,axiom,
    ! [A2: complex,B2: complex] :
      ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A2 @ B2 ) )
      = ( minus_minus_complex @ B2 @ A2 ) ) ).

% minus_diff_eq
thf(fact_7053_minus__diff__eq,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) )
      = ( minus_8373710615458151222nteger @ B2 @ A2 ) ) ).

% minus_diff_eq
thf(fact_7054_minus__diff__eq,axiom,
    ! [A2: rat,B2: rat] :
      ( ( uminus_uminus_rat @ ( minus_minus_rat @ A2 @ B2 ) )
      = ( minus_minus_rat @ B2 @ A2 ) ) ).

% minus_diff_eq
thf(fact_7055_div__minus__minus,axiom,
    ! [A2: int,B2: int] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B2 ) )
      = ( divide_divide_int @ A2 @ B2 ) ) ).

% div_minus_minus
thf(fact_7056_div__minus__minus,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( divide6298287555418463151nteger @ A2 @ B2 ) ) ).

% div_minus_minus
thf(fact_7057_minus__dvd__iff,axiom,
    ! [X: real,Y: real] :
      ( ( dvd_dvd_real @ ( uminus_uminus_real @ X ) @ Y )
      = ( dvd_dvd_real @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_7058_minus__dvd__iff,axiom,
    ! [X: int,Y: int] :
      ( ( dvd_dvd_int @ ( uminus_uminus_int @ X ) @ Y )
      = ( dvd_dvd_int @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_7059_minus__dvd__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X ) @ Y )
      = ( dvd_dvd_complex @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_7060_minus__dvd__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ Y )
      = ( dvd_dvd_Code_integer @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_7061_minus__dvd__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X ) @ Y )
      = ( dvd_dvd_rat @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_7062_dvd__minus__iff,axiom,
    ! [X: real,Y: real] :
      ( ( dvd_dvd_real @ X @ ( uminus_uminus_real @ Y ) )
      = ( dvd_dvd_real @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_7063_dvd__minus__iff,axiom,
    ! [X: int,Y: int] :
      ( ( dvd_dvd_int @ X @ ( uminus_uminus_int @ Y ) )
      = ( dvd_dvd_int @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_7064_dvd__minus__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( dvd_dvd_complex @ X @ ( uminus1482373934393186551omplex @ Y ) )
      = ( dvd_dvd_complex @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_7065_dvd__minus__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( dvd_dvd_Code_integer @ X @ ( uminus1351360451143612070nteger @ Y ) )
      = ( dvd_dvd_Code_integer @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_7066_dvd__minus__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( dvd_dvd_rat @ X @ ( uminus_uminus_rat @ Y ) )
      = ( dvd_dvd_rat @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_7067_mod__minus__minus,axiom,
    ! [A2: int,B2: int] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B2 ) )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ A2 @ B2 ) ) ) ).

% mod_minus_minus
thf(fact_7068_mod__minus__minus,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A2 @ B2 ) ) ) ).

% mod_minus_minus
thf(fact_7069_List_Ofinite__set,axiom,
    ! [Xs2: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_7070_List_Ofinite__set,axiom,
    ! [Xs2: list_real] : ( finite_finite_real @ ( set_real2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_7071_List_Ofinite__set,axiom,
    ! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_7072_List_Ofinite__set,axiom,
    ! [Xs2: list_int] : ( finite_finite_int @ ( set_int2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_7073_List_Ofinite__set,axiom,
    ! [Xs2: list_Code_integer] : ( finite6017078050557962740nteger @ ( set_Code_integer2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_7074_List_Ofinite__set,axiom,
    ! [Xs2: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_7075_finite__atLeastLessThan,axiom,
    ! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L2 @ U ) ) ).

% finite_atLeastLessThan
thf(fact_7076_finite__atLeastAtMost,axiom,
    ! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L2 @ U ) ) ).

% finite_atLeastAtMost
thf(fact_7077_triangle__0,axiom,
    ( ( nat_triangle @ zero_zero_nat )
    = zero_zero_nat ) ).

% triangle_0
thf(fact_7078_neg__0__le__iff__le,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A2 ) )
      = ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).

% neg_0_le_iff_le
thf(fact_7079_neg__0__le__iff__le,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% neg_0_le_iff_le
thf(fact_7080_neg__0__le__iff__le,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A2 ) )
      = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).

% neg_0_le_iff_le
thf(fact_7081_neg__0__le__iff__le,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
      = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).

% neg_0_le_iff_le
thf(fact_7082_neg__le__0__iff__le,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ zero_zero_real )
      = ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).

% neg_le_0_iff_le
thf(fact_7083_neg__le__0__iff__le,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ zero_z3403309356797280102nteger )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).

% neg_le_0_iff_le
thf(fact_7084_neg__le__0__iff__le,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).

% neg_le_0_iff_le
thf(fact_7085_neg__le__0__iff__le,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
      = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).

% neg_le_0_iff_le
thf(fact_7086_less__eq__neg__nonpos,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ A2 ) )
      = ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).

% less_eq_neg_nonpos
thf(fact_7087_less__eq__neg__nonpos,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% less_eq_neg_nonpos
thf(fact_7088_less__eq__neg__nonpos,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
      = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).

% less_eq_neg_nonpos
thf(fact_7089_less__eq__neg__nonpos,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ A2 ) )
      = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).

% less_eq_neg_nonpos
thf(fact_7090_neg__less__eq__nonneg,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ A2 )
      = ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).

% neg_less_eq_nonneg
thf(fact_7091_neg__less__eq__nonneg,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).

% neg_less_eq_nonneg
thf(fact_7092_neg__less__eq__nonneg,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
      = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).

% neg_less_eq_nonneg
thf(fact_7093_neg__less__eq__nonneg,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ A2 )
      = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).

% neg_less_eq_nonneg
thf(fact_7094_neg__less__0__iff__less,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ zero_zero_real )
      = ( ord_less_real @ zero_zero_real @ A2 ) ) ).

% neg_less_0_iff_less
thf(fact_7095_neg__less__0__iff__less,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
      = ( ord_less_int @ zero_zero_int @ A2 ) ) ).

% neg_less_0_iff_less
thf(fact_7096_neg__less__0__iff__less,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ zero_z3403309356797280102nteger )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).

% neg_less_0_iff_less
thf(fact_7097_neg__less__0__iff__less,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ zero_zero_rat )
      = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).

% neg_less_0_iff_less
thf(fact_7098_neg__0__less__iff__less,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A2 ) )
      = ( ord_less_real @ A2 @ zero_zero_real ) ) ).

% neg_0_less_iff_less
thf(fact_7099_neg__0__less__iff__less,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
      = ( ord_less_int @ A2 @ zero_zero_int ) ) ).

% neg_0_less_iff_less
thf(fact_7100_neg__0__less__iff__less,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% neg_0_less_iff_less
thf(fact_7101_neg__0__less__iff__less,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A2 ) )
      = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).

% neg_0_less_iff_less
thf(fact_7102_neg__less__pos,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ A2 )
      = ( ord_less_real @ zero_zero_real @ A2 ) ) ).

% neg_less_pos
thf(fact_7103_neg__less__pos,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ A2 )
      = ( ord_less_int @ zero_zero_int @ A2 ) ) ).

% neg_less_pos
thf(fact_7104_neg__less__pos,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).

% neg_less_pos
thf(fact_7105_neg__less__pos,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
      = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).

% neg_less_pos
thf(fact_7106_less__neg__neg,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ A2 @ ( uminus_uminus_real @ A2 ) )
      = ( ord_less_real @ A2 @ zero_zero_real ) ) ).

% less_neg_neg
thf(fact_7107_less__neg__neg,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ A2 @ ( uminus_uminus_int @ A2 ) )
      = ( ord_less_int @ A2 @ zero_zero_int ) ) ).

% less_neg_neg
thf(fact_7108_less__neg__neg,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).

% less_neg_neg
thf(fact_7109_less__neg__neg,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
      = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).

% less_neg_neg
thf(fact_7110_ab__left__minus,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ A2 )
      = zero_zero_real ) ).

% ab_left_minus
thf(fact_7111_ab__left__minus,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_7112_ab__left__minus,axiom,
    ! [A2: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ A2 )
      = zero_zero_complex ) ).

% ab_left_minus
thf(fact_7113_ab__left__minus,axiom,
    ! [A2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
      = zero_z3403309356797280102nteger ) ).

% ab_left_minus
thf(fact_7114_ab__left__minus,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_7115_add_Oright__inverse,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ A2 @ ( uminus_uminus_real @ A2 ) )
      = zero_zero_real ) ).

% add.right_inverse
thf(fact_7116_add_Oright__inverse,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ A2 @ ( uminus_uminus_int @ A2 ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_7117_add_Oright__inverse,axiom,
    ! [A2: complex] :
      ( ( plus_plus_complex @ A2 @ ( uminus1482373934393186551omplex @ A2 ) )
      = zero_zero_complex ) ).

% add.right_inverse
thf(fact_7118_add_Oright__inverse,axiom,
    ! [A2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A2 @ ( uminus1351360451143612070nteger @ A2 ) )
      = zero_z3403309356797280102nteger ) ).

% add.right_inverse
thf(fact_7119_add_Oright__inverse,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_7120_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
      = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N3 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_7121_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N3 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_7122_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N3 ) ) )
      = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N3 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_7123_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N3 ) ) )
      = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N3 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_7124_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) )
      = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N3 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_7125_diff__0,axiom,
    ! [A2: real] :
      ( ( minus_minus_real @ zero_zero_real @ A2 )
      = ( uminus_uminus_real @ A2 ) ) ).

% diff_0
thf(fact_7126_diff__0,axiom,
    ! [A2: int] :
      ( ( minus_minus_int @ zero_zero_int @ A2 )
      = ( uminus_uminus_int @ A2 ) ) ).

% diff_0
thf(fact_7127_diff__0,axiom,
    ! [A2: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ A2 )
      = ( uminus1482373934393186551omplex @ A2 ) ) ).

% diff_0
thf(fact_7128_diff__0,axiom,
    ! [A2: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A2 )
      = ( uminus1351360451143612070nteger @ A2 ) ) ).

% diff_0
thf(fact_7129_diff__0,axiom,
    ! [A2: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ A2 )
      = ( uminus_uminus_rat @ A2 ) ) ).

% diff_0
thf(fact_7130_verit__minus__simplify_I3_J,axiom,
    ! [B2: real] :
      ( ( minus_minus_real @ zero_zero_real @ B2 )
      = ( uminus_uminus_real @ B2 ) ) ).

% verit_minus_simplify(3)
thf(fact_7131_verit__minus__simplify_I3_J,axiom,
    ! [B2: int] :
      ( ( minus_minus_int @ zero_zero_int @ B2 )
      = ( uminus_uminus_int @ B2 ) ) ).

% verit_minus_simplify(3)
thf(fact_7132_verit__minus__simplify_I3_J,axiom,
    ! [B2: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ B2 )
      = ( uminus1482373934393186551omplex @ B2 ) ) ).

% verit_minus_simplify(3)
thf(fact_7133_verit__minus__simplify_I3_J,axiom,
    ! [B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B2 )
      = ( uminus1351360451143612070nteger @ B2 ) ) ).

% verit_minus_simplify(3)
thf(fact_7134_verit__minus__simplify_I3_J,axiom,
    ! [B2: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ B2 )
      = ( uminus_uminus_rat @ B2 ) ) ).

% verit_minus_simplify(3)
thf(fact_7135_mult__minus1,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
      = ( uminus_uminus_real @ Z ) ) ).

% mult_minus1
thf(fact_7136_mult__minus1,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1
thf(fact_7137_mult__minus1,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1
thf(fact_7138_mult__minus1,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1
thf(fact_7139_mult__minus1,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1
thf(fact_7140_mult__minus1__right,axiom,
    ! [Z: real] :
      ( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ Z ) ) ).

% mult_minus1_right
thf(fact_7141_mult__minus1__right,axiom,
    ! [Z: int] :
      ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1_right
thf(fact_7142_mult__minus1__right,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1_right
thf(fact_7143_mult__minus1__right,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1_right
thf(fact_7144_mult__minus1__right,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1_right
thf(fact_7145_uminus__add__conv__diff,axiom,
    ! [A2: real,B2: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ B2 )
      = ( minus_minus_real @ B2 @ A2 ) ) ).

% uminus_add_conv_diff
thf(fact_7146_uminus__add__conv__diff,axiom,
    ! [A2: int,B2: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B2 )
      = ( minus_minus_int @ B2 @ A2 ) ) ).

% uminus_add_conv_diff
thf(fact_7147_uminus__add__conv__diff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B2 )
      = ( minus_minus_complex @ B2 @ A2 ) ) ).

% uminus_add_conv_diff
thf(fact_7148_uminus__add__conv__diff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 )
      = ( minus_8373710615458151222nteger @ B2 @ A2 ) ) ).

% uminus_add_conv_diff
thf(fact_7149_uminus__add__conv__diff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ B2 )
      = ( minus_minus_rat @ B2 @ A2 ) ) ).

% uminus_add_conv_diff
thf(fact_7150_diff__minus__eq__add,axiom,
    ! [A2: real,B2: real] :
      ( ( minus_minus_real @ A2 @ ( uminus_uminus_real @ B2 ) )
      = ( plus_plus_real @ A2 @ B2 ) ) ).

% diff_minus_eq_add
thf(fact_7151_diff__minus__eq__add,axiom,
    ! [A2: int,B2: int] :
      ( ( minus_minus_int @ A2 @ ( uminus_uminus_int @ B2 ) )
      = ( plus_plus_int @ A2 @ B2 ) ) ).

% diff_minus_eq_add
thf(fact_7152_diff__minus__eq__add,axiom,
    ! [A2: complex,B2: complex] :
      ( ( minus_minus_complex @ A2 @ ( uminus1482373934393186551omplex @ B2 ) )
      = ( plus_plus_complex @ A2 @ B2 ) ) ).

% diff_minus_eq_add
thf(fact_7153_diff__minus__eq__add,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ A2 @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( plus_p5714425477246183910nteger @ A2 @ B2 ) ) ).

% diff_minus_eq_add
thf(fact_7154_diff__minus__eq__add,axiom,
    ! [A2: rat,B2: rat] :
      ( ( minus_minus_rat @ A2 @ ( uminus_uminus_rat @ B2 ) )
      = ( plus_plus_rat @ A2 @ B2 ) ) ).

% diff_minus_eq_add
thf(fact_7155_divide__minus1,axiom,
    ! [X: real] :
      ( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ X ) ) ).

% divide_minus1
thf(fact_7156_divide__minus1,axiom,
    ! [X: complex] :
      ( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ X ) ) ).

% divide_minus1
thf(fact_7157_divide__minus1,axiom,
    ! [X: rat] :
      ( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ X ) ) ).

% divide_minus1
thf(fact_7158_div__minus1__right,axiom,
    ! [A2: int] :
      ( ( divide_divide_int @ A2 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ A2 ) ) ).

% div_minus1_right
thf(fact_7159_div__minus1__right,axiom,
    ! [A2: code_integer] :
      ( ( divide6298287555418463151nteger @ A2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ A2 ) ) ).

% div_minus1_right
thf(fact_7160_minus__mod__self1,axiom,
    ! [B2: int,A2: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ B2 @ A2 ) @ B2 )
      = ( modulo_modulo_int @ ( uminus_uminus_int @ A2 ) @ B2 ) ) ).

% minus_mod_self1
thf(fact_7161_minus__mod__self1,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B2 @ A2 ) @ B2 )
      = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 ) ) ).

% minus_mod_self1
thf(fact_7162_infinite__Icc__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A2 @ B2 ) ) )
      = ( ord_less_rat @ A2 @ B2 ) ) ).

% infinite_Icc_iff
thf(fact_7163_infinite__Icc__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) ) )
      = ( ord_less_real @ A2 @ B2 ) ) ).

% infinite_Icc_iff
thf(fact_7164_infinite__Ico__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ~ ( finite_finite_real @ ( set_or66887138388493659n_real @ A2 @ B2 ) ) )
      = ( ord_less_real @ A2 @ B2 ) ) ).

% infinite_Ico_iff
thf(fact_7165_infinite__Ico__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ~ ( finite_finite_rat @ ( set_or4029947393144176647an_rat @ A2 @ B2 ) ) )
      = ( ord_less_rat @ A2 @ B2 ) ) ).

% infinite_Ico_iff
thf(fact_7166_real__add__minus__iff,axiom,
    ! [X: real,A2: real] :
      ( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A2 ) )
        = zero_zero_real )
      = ( X = A2 ) ) ).

% real_add_minus_iff
thf(fact_7167_triangle__Suc,axiom,
    ! [N3: nat] :
      ( ( nat_triangle @ ( suc @ N3 ) )
      = ( plus_plus_nat @ ( nat_triangle @ N3 ) @ ( suc @ N3 ) ) ) ).

% triangle_Suc
thf(fact_7168_listI__assn__finite,axiom,
    ! [I3: set_nat,A: vEBT_VEBT > vEBT_VEBTi > assn,Xs2: list_VEBT_VEBT,Xsi: list_VEBT_VEBTi] :
      ( ~ ( finite_finite_nat @ I3 )
     => ( ( vEBT_L1528199826722428489_VEBTi @ I3 @ A @ Xs2 @ Xsi )
        = bot_bot_assn ) ) ).

% listI_assn_finite
thf(fact_7169_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_7170_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_7171_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_7172_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_7173_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_7174_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_7175_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_7176_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_7177_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_7178_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_7179_numeral__eq__neg__one__iff,axiom,
    ! [N3: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( N3 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_7180_numeral__eq__neg__one__iff,axiom,
    ! [N3: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( N3 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_7181_numeral__eq__neg__one__iff,axiom,
    ! [N3: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N3 ) )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( N3 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_7182_numeral__eq__neg__one__iff,axiom,
    ! [N3: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N3 ) )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( N3 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_7183_numeral__eq__neg__one__iff,axiom,
    ! [N3: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( N3 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_7184_neg__one__eq__numeral__iff,axiom,
    ! [N3: num] :
      ( ( ( uminus_uminus_real @ one_one_real )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
      = ( N3 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_7185_neg__one__eq__numeral__iff,axiom,
    ! [N3: num] :
      ( ( ( uminus_uminus_int @ one_one_int )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( N3 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_7186_neg__one__eq__numeral__iff,axiom,
    ! [N3: num] :
      ( ( ( uminus1482373934393186551omplex @ one_one_complex )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N3 ) ) )
      = ( N3 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_7187_neg__one__eq__numeral__iff,axiom,
    ! [N3: num] :
      ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N3 ) ) )
      = ( N3 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_7188_neg__one__eq__numeral__iff,axiom,
    ! [N3: num] :
      ( ( ( uminus_uminus_rat @ one_one_rat )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) )
      = ( N3 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_7189_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_7190_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_7191_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_7192_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_7193_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_7194_left__minus__one__mult__self,axiom,
    ! [N3: nat,A2: real] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ A2 ) )
      = A2 ) ).

% left_minus_one_mult_self
thf(fact_7195_left__minus__one__mult__self,axiom,
    ! [N3: nat,A2: int] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N3 ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N3 ) @ A2 ) )
      = A2 ) ).

% left_minus_one_mult_self
thf(fact_7196_left__minus__one__mult__self,axiom,
    ! [N3: nat,A2: complex] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N3 ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N3 ) @ A2 ) )
      = A2 ) ).

% left_minus_one_mult_self
thf(fact_7197_left__minus__one__mult__self,axiom,
    ! [N3: nat,A2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N3 ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N3 ) @ A2 ) )
      = A2 ) ).

% left_minus_one_mult_self
thf(fact_7198_left__minus__one__mult__self,axiom,
    ! [N3: nat,A2: rat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N3 ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N3 ) @ A2 ) )
      = A2 ) ).

% left_minus_one_mult_self
thf(fact_7199_minus__one__mult__self,axiom,
    ! [N3: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) )
      = one_one_real ) ).

% minus_one_mult_self
thf(fact_7200_minus__one__mult__self,axiom,
    ! [N3: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N3 ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N3 ) )
      = one_one_int ) ).

% minus_one_mult_self
thf(fact_7201_minus__one__mult__self,axiom,
    ! [N3: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N3 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N3 ) )
      = one_one_complex ) ).

% minus_one_mult_self
thf(fact_7202_minus__one__mult__self,axiom,
    ! [N3: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N3 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N3 ) )
      = one_one_Code_integer ) ).

% minus_one_mult_self
thf(fact_7203_minus__one__mult__self,axiom,
    ! [N3: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N3 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N3 ) )
      = one_one_rat ) ).

% minus_one_mult_self
thf(fact_7204_mod__minus1__right,axiom,
    ! [A2: int] :
      ( ( modulo_modulo_int @ A2 @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_7205_mod__minus1__right,axiom,
    ! [A2: code_integer] :
      ( ( modulo364778990260209775nteger @ A2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = zero_z3403309356797280102nteger ) ).

% mod_minus1_right
thf(fact_7206_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_7207_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_7208_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_7209_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_7210_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_7211_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_7212_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_7213_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_7214_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_7215_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_7216_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_7217_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_7218_norm__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_neg_numeral
thf(fact_7219_norm__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_neg_numeral
thf(fact_7220_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_7221_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_7222_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_7223_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_7224_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_7225_ceiling__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_neg_numeral
thf(fact_7226_ceiling__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_neg_numeral
thf(fact_7227_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N3: num] :
      ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M @ N3 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_7228_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N3: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N3 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_7229_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N3: num] :
      ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N3 ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N3 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_7230_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N3: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N3 ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N3 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_7231_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N3: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N3 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_7232_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N3 ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N3 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_7233_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N3 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N3 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_7234_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N3 ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N3 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_7235_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N3 ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N3 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_7236_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N3 ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N3 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_7237_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
      = ( numeral_numeral_real @ ( times_times_num @ M @ N3 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_7238_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N3 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_7239_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N3 ) ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N3 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_7240_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N3 ) ) )
      = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N3 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_7241_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M @ N3 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_7242_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N3 ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N3 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_7243_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N3 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N3 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_7244_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N3 ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N3 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_7245_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N3 ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N3 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_7246_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N3 ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N3 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_7247_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N3 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_7248_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N3 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_7249_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N3 ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N3 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_7250_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N3 ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N3 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_7251_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N3 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_7252_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_7253_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_7254_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_7255_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_7256_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_7257_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_7258_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_7259_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_7260_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_7261_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_7262_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_7263_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_7264_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_7265_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_7266_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_7267_neg__numeral__le__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
      = ( ord_less_eq_num @ N3 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_7268_neg__numeral__le__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N3 ) ) )
      = ( ord_less_eq_num @ N3 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_7269_neg__numeral__le__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) )
      = ( ord_less_eq_num @ N3 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_7270_neg__numeral__le__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( ord_less_eq_num @ N3 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_7271_neg__numeral__less__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
      = ( ord_less_num @ N3 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_7272_neg__numeral__less__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( ord_less_num @ N3 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_7273_neg__numeral__less__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N3 ) ) )
      = ( ord_less_num @ N3 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_7274_neg__numeral__less__iff,axiom,
    ! [M: num,N3: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) )
      = ( ord_less_num @ N3 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_7275_round__neg__numeral,axiom,
    ! [N3: num] :
      ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) ) ).

% round_neg_numeral
thf(fact_7276_round__neg__numeral,axiom,
    ! [N3: num] :
      ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) ) ).

% round_neg_numeral
thf(fact_7277_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_7278_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_7279_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_7280_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_7281_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_7282_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_7283_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_7284_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_7285_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( A2
        = ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
           != zero_zero_real )
         => ( ( times_times_real @ A2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
            = B2 ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_7286_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A2: complex,B2: complex,W: num] :
      ( ( A2
        = ( divide1717551699836669952omplex @ B2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
           != zero_zero_complex )
         => ( ( times_times_complex @ A2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
            = B2 ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_7287_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( ( A2
        = ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
           != zero_zero_rat )
         => ( ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
            = B2 ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_7288_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: real,W: num,A2: real] :
      ( ( ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
        = A2 )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
           != zero_zero_real )
         => ( B2
            = ( times_times_real @ A2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_7289_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: complex,W: num,A2: complex] :
      ( ( ( divide1717551699836669952omplex @ B2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
        = A2 )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
           != zero_zero_complex )
         => ( B2
            = ( times_times_complex @ A2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_7290_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( ( ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
        = A2 )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
           != zero_zero_rat )
         => ( B2
            = ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_7291_le__divide__eq__numeral1_I2_J,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( ord_less_eq_real @ A2 @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ord_less_eq_real @ B2 @ ( times_times_real @ A2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_7292_le__divide__eq__numeral1_I2_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ord_less_eq_rat @ B2 @ ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_7293_divide__le__eq__numeral1_I2_J,axiom,
    ! [B2: real,W: num,A2: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A2 )
      = ( ord_less_eq_real @ ( times_times_real @ A2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B2 ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_7294_divide__le__eq__numeral1_I2_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A2 )
      = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B2 ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_7295_less__divide__eq__numeral1_I2_J,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( ord_less_real @ A2 @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ord_less_real @ B2 @ ( times_times_real @ A2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_7296_less__divide__eq__numeral1_I2_J,axiom,
    ! [A2: rat,B2: rat,W: num] :
      ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ord_less_rat @ B2 @ ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_7297_divide__less__eq__numeral1_I2_J,axiom,
    ! [B2: real,W: num,A2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A2 )
      = ( ord_less_real @ ( times_times_real @ A2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B2 ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_7298_divide__less__eq__numeral1_I2_J,axiom,
    ! [B2: rat,W: num,A2: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A2 )
      = ( ord_less_rat @ ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B2 ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_7299_power2__minus,axiom,
    ! [A2: real] :
      ( ( power_power_real @ ( uminus_uminus_real @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_7300_power2__minus,axiom,
    ! [A2: int] :
      ( ( power_power_int @ ( uminus_uminus_int @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_7301_power2__minus,axiom,
    ! [A2: complex] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_complex @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_7302_power2__minus,axiom,
    ! [A2: code_integer] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_7303_power2__minus,axiom,
    ! [A2: rat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_7304_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_7305_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_7306_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_7307_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_7308_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_7309_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_7310_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_7311_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_7312_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_7313_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_7314_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_7315_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_7316_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_7317_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_7318_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_7319_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_7320_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_7321_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_7322_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_7323_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_7324_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_7325_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A2: real,N3: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( power_power_real @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_7326_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A2: int,N3: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( power_power_int @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_7327_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A2: complex,N3: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( power_power_complex @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_7328_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( power_8256067586552552935nteger @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_7329_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A2: rat,N3: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( power_power_rat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_7330_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N3: nat,A2: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_real @ ( uminus_uminus_real @ A2 ) @ N3 )
        = ( power_power_real @ A2 @ N3 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_7331_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N3: nat,A2: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_int @ ( uminus_uminus_int @ A2 ) @ N3 )
        = ( power_power_int @ A2 @ N3 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_7332_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N3: nat,A2: complex] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A2 ) @ N3 )
        = ( power_power_complex @ A2 @ N3 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_7333_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N3: nat,A2: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ N3 )
        = ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_7334_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N3: nat,A2: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ N3 )
        = ( power_power_rat @ A2 @ N3 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_7335_power__minus__odd,axiom,
    ! [N3: nat,A2: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_real @ ( uminus_uminus_real @ A2 ) @ N3 )
        = ( uminus_uminus_real @ ( power_power_real @ A2 @ N3 ) ) ) ) ).

% power_minus_odd
thf(fact_7336_power__minus__odd,axiom,
    ! [N3: nat,A2: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_int @ ( uminus_uminus_int @ A2 ) @ N3 )
        = ( uminus_uminus_int @ ( power_power_int @ A2 @ N3 ) ) ) ) ).

% power_minus_odd
thf(fact_7337_power__minus__odd,axiom,
    ! [N3: nat,A2: complex] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A2 ) @ N3 )
        = ( uminus1482373934393186551omplex @ ( power_power_complex @ A2 @ N3 ) ) ) ) ).

% power_minus_odd
thf(fact_7338_power__minus__odd,axiom,
    ! [N3: nat,A2: code_integer] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ N3 )
        = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ) ).

% power_minus_odd
thf(fact_7339_power__minus__odd,axiom,
    ! [N3: nat,A2: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ N3 )
        = ( uminus_uminus_rat @ ( power_power_rat @ A2 @ N3 ) ) ) ) ).

% power_minus_odd
thf(fact_7340_diff__numeral__special_I3_J,axiom,
    ! [N3: num] :
      ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N3 ) ) ) ).

% diff_numeral_special(3)
thf(fact_7341_diff__numeral__special_I3_J,axiom,
    ! [N3: num] :
      ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N3 ) ) ) ).

% diff_numeral_special(3)
thf(fact_7342_diff__numeral__special_I3_J,axiom,
    ! [N3: num] :
      ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N3 ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N3 ) ) ) ).

% diff_numeral_special(3)
thf(fact_7343_diff__numeral__special_I3_J,axiom,
    ! [N3: num] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N3 ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N3 ) ) ) ).

% diff_numeral_special(3)
thf(fact_7344_diff__numeral__special_I3_J,axiom,
    ! [N3: num] :
      ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N3 ) ) ) ).

% diff_numeral_special(3)
thf(fact_7345_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_7346_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_7347_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_7348_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_7349_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_7350_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N3: nat] :
      ( ( ( ring_1_of_int_real @ Y )
        = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N3 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7351_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N3: nat] :
      ( ( ( ring_1_of_int_int @ Y )
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7352_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N3: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N3 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7353_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N3: nat] :
      ( ( ( ring_18347121197199848620nteger @ Y )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N3 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7354_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N3: nat] :
      ( ( ( ring_1_of_int_rat @ Y )
        = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N3 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7355_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: int] :
      ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N3 )
        = ( ring_1_of_int_real @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7356_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: int] :
      ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 )
        = ( ring_1_of_int_int @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7357_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: int] :
      ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N3 )
        = ( ring_17405671764205052669omplex @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7358_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: int] :
      ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N3 )
        = ( ring_18347121197199848620nteger @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7359_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,Y: int] :
      ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N3 )
        = ( ring_1_of_int_rat @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7360_ceiling__le__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).

% ceiling_le_neg_numeral
thf(fact_7361_ceiling__le__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).

% ceiling_le_neg_numeral
thf(fact_7362_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X ) ) ).

% neg_numeral_less_ceiling
thf(fact_7363_neg__numeral__less__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X ) ) ).

% neg_numeral_less_ceiling
thf(fact_7364_ceiling__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ zero_zero_int )
      = ( ord_less_eq_real @ X @ ( uminus_uminus_real @ one_one_real ) ) ) ).

% ceiling_less_zero
thf(fact_7365_ceiling__less__zero,axiom,
    ! [X: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% ceiling_less_zero
thf(fact_7366_zero__le__ceiling,axiom,
    ! [X: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X ) ) ).

% zero_le_ceiling
thf(fact_7367_zero__le__ceiling,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X ) ) ).

% zero_le_ceiling
thf(fact_7368_power__minus1__even,axiom,
    ! [N3: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = one_one_real ) ).

% power_minus1_even
thf(fact_7369_power__minus1__even,axiom,
    ! [N3: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = one_one_int ) ).

% power_minus1_even
thf(fact_7370_power__minus1__even,axiom,
    ! [N3: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = one_one_complex ) ).

% power_minus1_even
thf(fact_7371_power__minus1__even,axiom,
    ! [N3: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = one_one_Code_integer ) ).

% power_minus1_even
thf(fact_7372_power__minus1__even,axiom,
    ! [N3: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = one_one_rat ) ).

% power_minus1_even
thf(fact_7373_neg__one__even__power,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 )
        = one_one_real ) ) ).

% neg_one_even_power
thf(fact_7374_neg__one__even__power,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N3 )
        = one_one_int ) ) ).

% neg_one_even_power
thf(fact_7375_neg__one__even__power,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N3 )
        = one_one_complex ) ) ).

% neg_one_even_power
thf(fact_7376_neg__one__even__power,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N3 )
        = one_one_Code_integer ) ) ).

% neg_one_even_power
thf(fact_7377_neg__one__even__power,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N3 )
        = one_one_rat ) ) ).

% neg_one_even_power
thf(fact_7378_neg__one__odd__power,axiom,
    ! [N3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 )
        = ( uminus_uminus_real @ one_one_real ) ) ) ).

% neg_one_odd_power
thf(fact_7379_neg__one__odd__power,axiom,
    ! [N3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N3 )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% neg_one_odd_power
thf(fact_7380_neg__one__odd__power,axiom,
    ! [N3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N3 )
        = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).

% neg_one_odd_power
thf(fact_7381_neg__one__odd__power,axiom,
    ! [N3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N3 )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).

% neg_one_odd_power
thf(fact_7382_neg__one__odd__power,axiom,
    ! [N3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N3 )
        = ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% neg_one_odd_power
thf(fact_7383_ceiling__less__neg__numeral,axiom,
    ! [X: real,V: num] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_real @ X @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_7384_ceiling__less__neg__numeral,axiom,
    ! [X: rat,V: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_rat @ X @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_7385_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: real] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X ) )
      = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X ) ) ).

% neg_numeral_le_ceiling
thf(fact_7386_neg__numeral__le__ceiling,axiom,
    ! [V: num,X: rat] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X ) ) ).

% neg_numeral_le_ceiling
thf(fact_7387_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A2: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N3 ) @ ( ring_1_of_int_real @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) @ A2 ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_7388_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A2: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N3 ) @ ( ring_18347121197199848620nteger @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) @ A2 ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_7389_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A2: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N3 ) @ ( ring_1_of_int_rat @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) @ A2 ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_7390_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) @ ( ring_1_of_int_int @ A2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) @ A2 ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_7391_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N3: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A2 ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N3 ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_7392_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N3: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N3 ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_7393_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N3: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N3 ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_7394_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N3: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A2 ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) )
      = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_7395_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N3: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A2 ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N3 ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_7396_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N3: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A2 ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_7397_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N3 ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_7398_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N3: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N3 ) )
      = ( ord_less_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_7399_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A2: int] :
      ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N3 ) @ ( ring_1_of_int_real @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) @ A2 ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_7400_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A2: int] :
      ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) @ ( ring_1_of_int_int @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) @ A2 ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_7401_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A2: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N3 ) @ ( ring_18347121197199848620nteger @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) @ A2 ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_7402_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N3: nat,A2: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N3 ) @ ( ring_1_of_int_rat @ A2 ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N3 ) @ A2 ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_7403_le__imp__neg__le,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A2 ) ) ) ).

% le_imp_neg_le
thf(fact_7404_le__imp__neg__le,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% le_imp_neg_le
thf(fact_7405_le__imp__neg__le,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A2 ) ) ) ).

% le_imp_neg_le
thf(fact_7406_le__imp__neg__le,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) ) ) ).

% le_imp_neg_le
thf(fact_7407_minus__le__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ B2 )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ A2 ) ) ).

% minus_le_iff
thf(fact_7408_minus__le__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 )
      = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B2 ) @ A2 ) ) ).

% minus_le_iff
thf(fact_7409_minus__le__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ B2 )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ A2 ) ) ).

% minus_le_iff
thf(fact_7410_minus__le__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B2 )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ A2 ) ) ).

% minus_le_iff
thf(fact_7411_le__minus__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ B2 ) )
      = ( ord_less_eq_real @ B2 @ ( uminus_uminus_real @ A2 ) ) ) ).

% le_minus_iff
thf(fact_7412_le__minus__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( ord_le3102999989581377725nteger @ B2 @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% le_minus_iff
thf(fact_7413_le__minus__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ B2 ) )
      = ( ord_less_eq_rat @ B2 @ ( uminus_uminus_rat @ A2 ) ) ) ).

% le_minus_iff
thf(fact_7414_le__minus__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ B2 ) )
      = ( ord_less_eq_int @ B2 @ ( uminus_uminus_int @ A2 ) ) ) ).

% le_minus_iff
thf(fact_7415_finite__if__eq__beyond__finite,axiom,
    ! [S: set_int,S4: set_int] :
      ( ( finite_finite_int @ S )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [S5: set_int] :
              ( ( minus_minus_set_int @ S5 @ S )
              = ( minus_minus_set_int @ S4 @ S ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_7416_finite__if__eq__beyond__finite,axiom,
    ! [S: set_Code_integer,S4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ S )
     => ( finite6931041176100689706nteger
        @ ( collec574505750873337192nteger
          @ ^ [S5: set_Code_integer] :
              ( ( minus_2355218937544613996nteger @ S5 @ S )
              = ( minus_2355218937544613996nteger @ S4 @ S ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_7417_finite__if__eq__beyond__finite,axiom,
    ! [S: set_complex,S4: set_complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( finite6551019134538273531omplex
        @ ( collect_set_complex
          @ ^ [S5: set_complex] :
              ( ( minus_811609699411566653omplex @ S5 @ S )
              = ( minus_811609699411566653omplex @ S4 @ S ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_7418_finite__if__eq__beyond__finite,axiom,
    ! [S: set_nat,S4: set_nat] :
      ( ( finite_finite_nat @ S )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [S5: set_nat] :
              ( ( minus_minus_set_nat @ S5 @ S )
              = ( minus_minus_set_nat @ S4 @ S ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_7419_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L2: code_integer,U: code_integer] :
      ( ( set_or8404916559141939852nteger @ L2 @ ( plus_p5714425477246183910nteger @ U @ one_one_Code_integer ) )
      = ( set_or189985376899183464nteger @ L2 @ U ) ) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
thf(fact_7420_minus__equation__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ( uminus_uminus_real @ A2 )
        = B2 )
      = ( ( uminus_uminus_real @ B2 )
        = A2 ) ) ).

% minus_equation_iff
thf(fact_7421_minus__equation__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ( uminus_uminus_int @ A2 )
        = B2 )
      = ( ( uminus_uminus_int @ B2 )
        = A2 ) ) ).

% minus_equation_iff
thf(fact_7422_minus__equation__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ( uminus1482373934393186551omplex @ A2 )
        = B2 )
      = ( ( uminus1482373934393186551omplex @ B2 )
        = A2 ) ) ).

% minus_equation_iff
thf(fact_7423_minus__equation__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A2 )
        = B2 )
      = ( ( uminus1351360451143612070nteger @ B2 )
        = A2 ) ) ).

% minus_equation_iff
thf(fact_7424_minus__equation__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( uminus_uminus_rat @ A2 )
        = B2 )
      = ( ( uminus_uminus_rat @ B2 )
        = A2 ) ) ).

% minus_equation_iff
thf(fact_7425_equation__minus__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( A2
        = ( uminus_uminus_real @ B2 ) )
      = ( B2
        = ( uminus_uminus_real @ A2 ) ) ) ).

% equation_minus_iff
thf(fact_7426_equation__minus__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( A2
        = ( uminus_uminus_int @ B2 ) )
      = ( B2
        = ( uminus_uminus_int @ A2 ) ) ) ).

% equation_minus_iff
thf(fact_7427_equation__minus__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( A2
        = ( uminus1482373934393186551omplex @ B2 ) )
      = ( B2
        = ( uminus1482373934393186551omplex @ A2 ) ) ) ).

% equation_minus_iff
thf(fact_7428_equation__minus__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2
        = ( uminus1351360451143612070nteger @ B2 ) )
      = ( B2
        = ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% equation_minus_iff
thf(fact_7429_equation__minus__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2
        = ( uminus_uminus_rat @ B2 ) )
      = ( B2
        = ( uminus_uminus_rat @ A2 ) ) ) ).

% equation_minus_iff
thf(fact_7430_numeral__neq__neg__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( numeral_numeral_real @ M )
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_7431_numeral__neq__neg__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( numeral_numeral_int @ M )
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_7432_numeral__neq__neg__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( numera6690914467698888265omplex @ M )
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N3 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_7433_numeral__neq__neg__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( numera6620942414471956472nteger @ M )
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N3 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_7434_numeral__neq__neg__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( numeral_numeral_rat @ M )
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_7435_neg__numeral__neq__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
     != ( numeral_numeral_real @ N3 ) ) ).

% neg_numeral_neq_numeral
thf(fact_7436_neg__numeral__neq__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
     != ( numeral_numeral_int @ N3 ) ) ).

% neg_numeral_neq_numeral
thf(fact_7437_neg__numeral__neq__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
     != ( numera6690914467698888265omplex @ N3 ) ) ).

% neg_numeral_neq_numeral
thf(fact_7438_neg__numeral__neq__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
     != ( numera6620942414471956472nteger @ N3 ) ) ).

% neg_numeral_neq_numeral
thf(fact_7439_neg__numeral__neq__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
     != ( numeral_numeral_rat @ N3 ) ) ).

% neg_numeral_neq_numeral
thf(fact_7440_minus__less__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ B2 )
      = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ A2 ) ) ).

% minus_less_iff
thf(fact_7441_minus__less__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B2 )
      = ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ A2 ) ) ).

% minus_less_iff
thf(fact_7442_minus__less__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 )
      = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B2 ) @ A2 ) ) ).

% minus_less_iff
thf(fact_7443_minus__less__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ B2 )
      = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ A2 ) ) ).

% minus_less_iff
thf(fact_7444_less__minus__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ ( uminus_uminus_real @ B2 ) )
      = ( ord_less_real @ B2 @ ( uminus_uminus_real @ A2 ) ) ) ).

% less_minus_iff
thf(fact_7445_less__minus__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ ( uminus_uminus_int @ B2 ) )
      = ( ord_less_int @ B2 @ ( uminus_uminus_int @ A2 ) ) ) ).

% less_minus_iff
thf(fact_7446_less__minus__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( ord_le6747313008572928689nteger @ B2 @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% less_minus_iff
thf(fact_7447_less__minus__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ B2 ) )
      = ( ord_less_rat @ B2 @ ( uminus_uminus_rat @ A2 ) ) ) ).

% less_minus_iff
thf(fact_7448_verit__negate__coefficient_I2_J,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A2 ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_7449_verit__negate__coefficient_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_7450_verit__negate__coefficient_I2_J,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
     => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_7451_verit__negate__coefficient_I2_J,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A2 ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_7452_is__num__normalize_I8_J,axiom,
    ! [A2: real,B2: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A2 @ B2 ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A2 ) ) ) ).

% is_num_normalize(8)
thf(fact_7453_is__num__normalize_I8_J,axiom,
    ! [A2: int,B2: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B2 ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) ) ) ).

% is_num_normalize(8)
thf(fact_7454_is__num__normalize_I8_J,axiom,
    ! [A2: complex,B2: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A2 @ B2 ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B2 ) @ ( uminus1482373934393186551omplex @ A2 ) ) ) ).

% is_num_normalize(8)
thf(fact_7455_is__num__normalize_I8_J,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% is_num_normalize(8)
thf(fact_7456_is__num__normalize_I8_J,axiom,
    ! [A2: rat,B2: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A2 @ B2 ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A2 ) ) ) ).

% is_num_normalize(8)
thf(fact_7457_add_Oinverse__distrib__swap,axiom,
    ! [A2: real,B2: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A2 @ B2 ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A2 ) ) ) ).

% add.inverse_distrib_swap
thf(fact_7458_add_Oinverse__distrib__swap,axiom,
    ! [A2: int,B2: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B2 ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) ) ) ).

% add.inverse_distrib_swap
thf(fact_7459_add_Oinverse__distrib__swap,axiom,
    ! [A2: complex,B2: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A2 @ B2 ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B2 ) @ ( uminus1482373934393186551omplex @ A2 ) ) ) ).

% add.inverse_distrib_swap
thf(fact_7460_add_Oinverse__distrib__swap,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B2 ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% add.inverse_distrib_swap
thf(fact_7461_add_Oinverse__distrib__swap,axiom,
    ! [A2: rat,B2: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A2 @ B2 ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B2 ) @ ( uminus_uminus_rat @ A2 ) ) ) ).

% add.inverse_distrib_swap
thf(fact_7462_group__cancel_Oneg1,axiom,
    ! [A: real,K: real,A2: real] :
      ( ( A
        = ( plus_plus_real @ K @ A2 ) )
     => ( ( uminus_uminus_real @ A )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A2 ) ) ) ) ).

% group_cancel.neg1
thf(fact_7463_group__cancel_Oneg1,axiom,
    ! [A: int,K: int,A2: int] :
      ( ( A
        = ( plus_plus_int @ K @ A2 ) )
     => ( ( uminus_uminus_int @ A )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A2 ) ) ) ) ).

% group_cancel.neg1
thf(fact_7464_group__cancel_Oneg1,axiom,
    ! [A: complex,K: complex,A2: complex] :
      ( ( A
        = ( plus_plus_complex @ K @ A2 ) )
     => ( ( uminus1482373934393186551omplex @ A )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A2 ) ) ) ) ).

% group_cancel.neg1
thf(fact_7465_group__cancel_Oneg1,axiom,
    ! [A: code_integer,K: code_integer,A2: code_integer] :
      ( ( A
        = ( plus_p5714425477246183910nteger @ K @ A2 ) )
     => ( ( uminus1351360451143612070nteger @ A )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ) ).

% group_cancel.neg1
thf(fact_7466_group__cancel_Oneg1,axiom,
    ! [A: rat,K: rat,A2: rat] :
      ( ( A
        = ( plus_plus_rat @ K @ A2 ) )
     => ( ( uminus_uminus_rat @ A )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A2 ) ) ) ) ).

% group_cancel.neg1
thf(fact_7467_one__neq__neg__one,axiom,
    ( one_one_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% one_neq_neg_one
thf(fact_7468_one__neq__neg__one,axiom,
    ( one_one_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% one_neq_neg_one
thf(fact_7469_one__neq__neg__one,axiom,
    ( one_one_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% one_neq_neg_one
thf(fact_7470_one__neq__neg__one,axiom,
    ( one_one_Code_integer
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% one_neq_neg_one
thf(fact_7471_one__neq__neg__one,axiom,
    ( one_one_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% one_neq_neg_one
thf(fact_7472_minus__mult__commute,axiom,
    ! [A2: real,B2: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A2 ) @ B2 )
      = ( times_times_real @ A2 @ ( uminus_uminus_real @ B2 ) ) ) ).

% minus_mult_commute
thf(fact_7473_minus__mult__commute,axiom,
    ! [A2: int,B2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A2 ) @ B2 )
      = ( times_times_int @ A2 @ ( uminus_uminus_int @ B2 ) ) ) ).

% minus_mult_commute
thf(fact_7474_minus__mult__commute,axiom,
    ! [A2: complex,B2: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B2 )
      = ( times_times_complex @ A2 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ).

% minus_mult_commute
thf(fact_7475_minus__mult__commute,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 )
      = ( times_3573771949741848930nteger @ A2 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ).

% minus_mult_commute
thf(fact_7476_minus__mult__commute,axiom,
    ! [A2: rat,B2: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A2 ) @ B2 )
      = ( times_times_rat @ A2 @ ( uminus_uminus_rat @ B2 ) ) ) ).

% minus_mult_commute
thf(fact_7477_square__eq__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ( times_times_real @ A2 @ A2 )
        = ( times_times_real @ B2 @ B2 ) )
      = ( ( A2 = B2 )
        | ( A2
          = ( uminus_uminus_real @ B2 ) ) ) ) ).

% square_eq_iff
thf(fact_7478_square__eq__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ( times_times_int @ A2 @ A2 )
        = ( times_times_int @ B2 @ B2 ) )
      = ( ( A2 = B2 )
        | ( A2
          = ( uminus_uminus_int @ B2 ) ) ) ) ).

% square_eq_iff
thf(fact_7479_square__eq__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ( times_times_complex @ A2 @ A2 )
        = ( times_times_complex @ B2 @ B2 ) )
      = ( ( A2 = B2 )
        | ( A2
          = ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).

% square_eq_iff
thf(fact_7480_square__eq__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A2 @ A2 )
        = ( times_3573771949741848930nteger @ B2 @ B2 ) )
      = ( ( A2 = B2 )
        | ( A2
          = ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).

% square_eq_iff
thf(fact_7481_square__eq__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( times_times_rat @ A2 @ A2 )
        = ( times_times_rat @ B2 @ B2 ) )
      = ( ( A2 = B2 )
        | ( A2
          = ( uminus_uminus_rat @ B2 ) ) ) ) ).

% square_eq_iff
thf(fact_7482_minus__diff__minus,axiom,
    ! [A2: real,B2: real] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B2 ) )
      = ( uminus_uminus_real @ ( minus_minus_real @ A2 @ B2 ) ) ) ).

% minus_diff_minus
thf(fact_7483_minus__diff__minus,axiom,
    ! [A2: int,B2: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B2 ) )
      = ( uminus_uminus_int @ ( minus_minus_int @ A2 @ B2 ) ) ) ).

% minus_diff_minus
thf(fact_7484_minus__diff__minus,axiom,
    ! [A2: complex,B2: complex] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( uminus1482373934393186551omplex @ B2 ) )
      = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A2 @ B2 ) ) ) ).

% minus_diff_minus
thf(fact_7485_minus__diff__minus,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A2 @ B2 ) ) ) ).

% minus_diff_minus
thf(fact_7486_minus__diff__minus,axiom,
    ! [A2: rat,B2: rat] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ A2 ) @ ( uminus_uminus_rat @ B2 ) )
      = ( uminus_uminus_rat @ ( minus_minus_rat @ A2 @ B2 ) ) ) ).

% minus_diff_minus
thf(fact_7487_minus__diff__commute,axiom,
    ! [B2: real,A2: real] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ B2 ) @ A2 )
      = ( minus_minus_real @ ( uminus_uminus_real @ A2 ) @ B2 ) ) ).

% minus_diff_commute
thf(fact_7488_minus__diff__commute,axiom,
    ! [B2: int,A2: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ B2 ) @ A2 )
      = ( minus_minus_int @ ( uminus_uminus_int @ A2 ) @ B2 ) ) ).

% minus_diff_commute
thf(fact_7489_minus__diff__commute,axiom,
    ! [B2: complex,A2: complex] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B2 ) @ A2 )
      = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B2 ) ) ).

% minus_diff_commute
thf(fact_7490_minus__diff__commute,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B2 ) @ A2 )
      = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 ) ) ).

% minus_diff_commute
thf(fact_7491_minus__diff__commute,axiom,
    ! [B2: rat,A2: rat] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ B2 ) @ A2 )
      = ( minus_minus_rat @ ( uminus_uminus_rat @ A2 ) @ B2 ) ) ).

% minus_diff_commute
thf(fact_7492_minus__divide__left,axiom,
    ! [A2: real,B2: real] :
      ( ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B2 ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ A2 ) @ B2 ) ) ).

% minus_divide_left
thf(fact_7493_minus__divide__left,axiom,
    ! [A2: complex,B2: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ B2 ) )
      = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A2 ) @ B2 ) ) ).

% minus_divide_left
thf(fact_7494_minus__divide__left,axiom,
    ! [A2: rat,B2: rat] :
      ( ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ B2 ) )
      = ( divide_divide_rat @ ( uminus_uminus_rat @ A2 ) @ B2 ) ) ).

% minus_divide_left
thf(fact_7495_minus__divide__divide,axiom,
    ! [A2: real,B2: real] :
      ( ( divide_divide_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B2 ) )
      = ( divide_divide_real @ A2 @ B2 ) ) ).

% minus_divide_divide
thf(fact_7496_minus__divide__divide,axiom,
    ! [A2: complex,B2: complex] :
      ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A2 ) @ ( uminus1482373934393186551omplex @ B2 ) )
      = ( divide1717551699836669952omplex @ A2 @ B2 ) ) ).

% minus_divide_divide
thf(fact_7497_minus__divide__divide,axiom,
    ! [A2: rat,B2: rat] :
      ( ( divide_divide_rat @ ( uminus_uminus_rat @ A2 ) @ ( uminus_uminus_rat @ B2 ) )
      = ( divide_divide_rat @ A2 @ B2 ) ) ).

% minus_divide_divide
thf(fact_7498_minus__divide__right,axiom,
    ! [A2: real,B2: real] :
      ( ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B2 ) )
      = ( divide_divide_real @ A2 @ ( uminus_uminus_real @ B2 ) ) ) ).

% minus_divide_right
thf(fact_7499_minus__divide__right,axiom,
    ! [A2: complex,B2: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ B2 ) )
      = ( divide1717551699836669952omplex @ A2 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ).

% minus_divide_right
thf(fact_7500_minus__divide__right,axiom,
    ! [A2: rat,B2: rat] :
      ( ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ B2 ) )
      = ( divide_divide_rat @ A2 @ ( uminus_uminus_rat @ B2 ) ) ) ).

% minus_divide_right
thf(fact_7501_div__minus__right,axiom,
    ! [A2: int,B2: int] :
      ( ( divide_divide_int @ A2 @ ( uminus_uminus_int @ B2 ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ A2 ) @ B2 ) ) ).

% div_minus_right
thf(fact_7502_div__minus__right,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( divide6298287555418463151nteger @ A2 @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 ) ) ).

% div_minus_right
thf(fact_7503_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N8: set_nat] :
        ? [M5: nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ N8 )
         => ( ord_less_nat @ X3 @ M5 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_7504_bounded__nat__set__is__finite,axiom,
    ! [N7: set_nat,N3: nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ N7 )
         => ( ord_less_nat @ X4 @ N3 ) )
     => ( finite_finite_nat @ N7 ) ) ).

% bounded_nat_set_is_finite
thf(fact_7505_mod__minus__right,axiom,
    ! [A2: int,B2: int] :
      ( ( modulo_modulo_int @ A2 @ ( uminus_uminus_int @ B2 ) )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A2 ) @ B2 ) ) ) ).

% mod_minus_right
thf(fact_7506_mod__minus__right,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ A2 @ ( uminus1351360451143612070nteger @ B2 ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 ) ) ) ).

% mod_minus_right
thf(fact_7507_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
    ! [A2: int,B2: int,A4: int] :
      ( ( ( modulo_modulo_int @ A2 @ B2 )
        = ( modulo_modulo_int @ A4 @ B2 ) )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A2 ) @ B2 )
        = ( modulo_modulo_int @ ( uminus_uminus_int @ A4 ) @ B2 ) ) ) ).

% euclidean_ring_cancel_class.mod_minus_cong
thf(fact_7508_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
    ! [A2: code_integer,B2: code_integer,A4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A2 @ B2 )
        = ( modulo364778990260209775nteger @ A4 @ B2 ) )
     => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 )
        = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A4 ) @ B2 ) ) ) ).

% euclidean_ring_cancel_class.mod_minus_cong
thf(fact_7509_mod__minus__eq,axiom,
    ! [A2: int,B2: int] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A2 @ B2 ) ) @ B2 )
      = ( modulo_modulo_int @ ( uminus_uminus_int @ A2 ) @ B2 ) ) ).

% mod_minus_eq
thf(fact_7510_mod__minus__eq,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A2 @ B2 ) ) @ B2 )
      = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 ) ) ).

% mod_minus_eq
thf(fact_7511_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N8: set_nat] :
        ? [M5: nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ N8 )
         => ( ord_less_eq_nat @ X3 @ M5 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_7512_finite__list,axiom,
    ! [A: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A )
     => ? [Xs3: list_VEBT_VEBT] :
          ( ( set_VEBT_VEBT2 @ Xs3 )
          = A ) ) ).

% finite_list
thf(fact_7513_finite__list,axiom,
    ! [A: set_real] :
      ( ( finite_finite_real @ A )
     => ? [Xs3: list_real] :
          ( ( set_real2 @ Xs3 )
          = A ) ) ).

% finite_list
thf(fact_7514_finite__list,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ? [Xs3: list_nat] :
          ( ( set_nat2 @ Xs3 )
          = A ) ) ).

% finite_list
thf(fact_7515_finite__list,axiom,
    ! [A: set_int] :
      ( ( finite_finite_int @ A )
     => ? [Xs3: list_int] :
          ( ( set_int2 @ Xs3 )
          = A ) ) ).

% finite_list
thf(fact_7516_finite__list,axiom,
    ! [A: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A )
     => ? [Xs3: list_Code_integer] :
          ( ( set_Code_integer2 @ Xs3 )
          = A ) ) ).

% finite_list
thf(fact_7517_finite__list,axiom,
    ! [A: set_complex] :
      ( ( finite3207457112153483333omplex @ A )
     => ? [Xs3: list_complex] :
          ( ( set_complex2 @ Xs3 )
          = A ) ) ).

% finite_list
thf(fact_7518_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_7519_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_7520_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_7521_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_7522_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_7523_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K3: nat] :
            ( ( P @ K3 )
            & ( ord_less_nat @ K3 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_7524_finite__less__ub,axiom,
    ! [F2: nat > nat,U: nat] :
      ( ! [N: nat] : ( ord_less_eq_nat @ N @ ( F2 @ N ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N2: nat] : ( ord_less_eq_nat @ ( F2 @ N2 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_7525_finite__lists__length__eq,axiom,
    ! [A: set_int,N3: nat] :
      ( ( finite_finite_int @ A )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A )
              & ( ( size_size_list_int @ Xs )
                = N3 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7526_finite__lists__length__eq,axiom,
    ! [A: set_Code_integer,N3: nat] :
      ( ( finite6017078050557962740nteger @ A )
     => ( finite1283093830868386564nteger
        @ ( collec3483841146883906114nteger
          @ ^ [Xs: list_Code_integer] :
              ( ( ord_le7084787975880047091nteger @ ( set_Code_integer2 @ Xs ) @ A )
              & ( ( size_s3445333598471063425nteger @ Xs )
                = N3 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7527_finite__lists__length__eq,axiom,
    ! [A: set_complex,N3: nat] :
      ( ( finite3207457112153483333omplex @ A )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A )
              & ( ( size_s3451745648224563538omplex @ Xs )
                = N3 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7528_finite__lists__length__eq,axiom,
    ! [A: set_VEBT_VEBT,N3: nat] :
      ( ( finite5795047828879050333T_VEBT @ A )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A )
              & ( ( size_s6755466524823107622T_VEBT @ Xs )
                = N3 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7529_finite__lists__length__eq,axiom,
    ! [A: set_real,N3: nat] :
      ( ( finite_finite_real @ A )
     => ( finite306553202115118035t_real
        @ ( collect_list_real
          @ ^ [Xs: list_real] :
              ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A )
              & ( ( size_size_list_real @ Xs )
                = N3 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7530_finite__lists__length__eq,axiom,
    ! [A: set_o,N3: nat] :
      ( ( finite_finite_o @ A )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A )
              & ( ( size_size_list_o @ Xs )
                = N3 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7531_finite__lists__length__eq,axiom,
    ! [A: set_VEBT_VEBTi,N3: nat] :
      ( ( finite6580341952239402572_VEBTi @ A )
     => ( finite5722435864697464412_VEBTi
        @ ( collec1288327022334394266_VEBTi
          @ ^ [Xs: list_VEBT_VEBTi] :
              ( ( ord_le6592769550269828683_VEBTi @ ( set_VEBT_VEBTi2 @ Xs ) @ A )
              & ( ( size_s7982070591426661849_VEBTi @ Xs )
                = N3 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7532_finite__lists__length__eq,axiom,
    ! [A: set_nat,N3: nat] :
      ( ( finite_finite_nat @ A )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A )
              & ( ( size_size_list_nat @ Xs )
                = N3 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_7533_zero__neq__neg__numeral,axiom,
    ! [N3: num] :
      ( zero_zero_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_7534_zero__neq__neg__numeral,axiom,
    ! [N3: num] :
      ( zero_zero_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_7535_zero__neq__neg__numeral,axiom,
    ! [N3: num] :
      ( zero_zero_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N3 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_7536_zero__neq__neg__numeral,axiom,
    ! [N3: num] :
      ( zero_z3403309356797280102nteger
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N3 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_7537_zero__neq__neg__numeral,axiom,
    ! [N3: num] :
      ( zero_zero_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_7538_neg__numeral__le__numeral,axiom,
    ! [M: num,N3: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N3 ) ) ).

% neg_numeral_le_numeral
thf(fact_7539_neg__numeral__le__numeral,axiom,
    ! [M: num,N3: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N3 ) ) ).

% neg_numeral_le_numeral
thf(fact_7540_neg__numeral__le__numeral,axiom,
    ! [M: num,N3: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N3 ) ) ).

% neg_numeral_le_numeral
thf(fact_7541_neg__numeral__le__numeral,axiom,
    ! [M: num,N3: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N3 ) ) ).

% neg_numeral_le_numeral
thf(fact_7542_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N3: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_7543_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N3: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N3 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_7544_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N3: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_7545_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N3: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_7546_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N3: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_7547_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N3: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_7548_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N3: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N3 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_7549_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N3: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_7550_neg__numeral__less__numeral,axiom,
    ! [M: num,N3: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N3 ) ) ).

% neg_numeral_less_numeral
thf(fact_7551_neg__numeral__less__numeral,axiom,
    ! [M: num,N3: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N3 ) ) ).

% neg_numeral_less_numeral
thf(fact_7552_neg__numeral__less__numeral,axiom,
    ! [M: num,N3: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N3 ) ) ).

% neg_numeral_less_numeral
thf(fact_7553_neg__numeral__less__numeral,axiom,
    ! [M: num,N3: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N3 ) ) ).

% neg_numeral_less_numeral
thf(fact_7554_neg__eq__iff__add__eq__0,axiom,
    ! [A2: real,B2: real] :
      ( ( ( uminus_uminus_real @ A2 )
        = B2 )
      = ( ( plus_plus_real @ A2 @ B2 )
        = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_7555_neg__eq__iff__add__eq__0,axiom,
    ! [A2: int,B2: int] :
      ( ( ( uminus_uminus_int @ A2 )
        = B2 )
      = ( ( plus_plus_int @ A2 @ B2 )
        = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_7556_neg__eq__iff__add__eq__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ( uminus1482373934393186551omplex @ A2 )
        = B2 )
      = ( ( plus_plus_complex @ A2 @ B2 )
        = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_7557_neg__eq__iff__add__eq__0,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A2 )
        = B2 )
      = ( ( plus_p5714425477246183910nteger @ A2 @ B2 )
        = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_7558_neg__eq__iff__add__eq__0,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( uminus_uminus_rat @ A2 )
        = B2 )
      = ( ( plus_plus_rat @ A2 @ B2 )
        = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_7559_eq__neg__iff__add__eq__0,axiom,
    ! [A2: real,B2: real] :
      ( ( A2
        = ( uminus_uminus_real @ B2 ) )
      = ( ( plus_plus_real @ A2 @ B2 )
        = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_7560_eq__neg__iff__add__eq__0,axiom,
    ! [A2: int,B2: int] :
      ( ( A2
        = ( uminus_uminus_int @ B2 ) )
      = ( ( plus_plus_int @ A2 @ B2 )
        = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_7561_eq__neg__iff__add__eq__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( A2
        = ( uminus1482373934393186551omplex @ B2 ) )
      = ( ( plus_plus_complex @ A2 @ B2 )
        = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_7562_eq__neg__iff__add__eq__0,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2
        = ( uminus1351360451143612070nteger @ B2 ) )
      = ( ( plus_p5714425477246183910nteger @ A2 @ B2 )
        = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_7563_eq__neg__iff__add__eq__0,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2
        = ( uminus_uminus_rat @ B2 ) )
      = ( ( plus_plus_rat @ A2 @ B2 )
        = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_7564_add_Oinverse__unique,axiom,
    ! [A2: real,B2: real] :
      ( ( ( plus_plus_real @ A2 @ B2 )
        = zero_zero_real )
     => ( ( uminus_uminus_real @ A2 )
        = B2 ) ) ).

% add.inverse_unique
thf(fact_7565_add_Oinverse__unique,axiom,
    ! [A2: int,B2: int] :
      ( ( ( plus_plus_int @ A2 @ B2 )
        = zero_zero_int )
     => ( ( uminus_uminus_int @ A2 )
        = B2 ) ) ).

% add.inverse_unique
thf(fact_7566_add_Oinverse__unique,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ( plus_plus_complex @ A2 @ B2 )
        = zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ A2 )
        = B2 ) ) ).

% add.inverse_unique
thf(fact_7567_add_Oinverse__unique,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A2 @ B2 )
        = zero_z3403309356797280102nteger )
     => ( ( uminus1351360451143612070nteger @ A2 )
        = B2 ) ) ).

% add.inverse_unique
thf(fact_7568_add_Oinverse__unique,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( plus_plus_rat @ A2 @ B2 )
        = zero_zero_rat )
     => ( ( uminus_uminus_rat @ A2 )
        = B2 ) ) ).

% add.inverse_unique
thf(fact_7569_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ A2 )
      = zero_zero_real ) ).

% ab_group_add_class.ab_left_minus
thf(fact_7570_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
      = zero_zero_int ) ).

% ab_group_add_class.ab_left_minus
thf(fact_7571_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ A2 )
      = zero_zero_complex ) ).

% ab_group_add_class.ab_left_minus
thf(fact_7572_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
      = zero_z3403309356797280102nteger ) ).

% ab_group_add_class.ab_left_minus
thf(fact_7573_ab__group__add__class_Oab__left__minus,axiom,
    ! [A2: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
      = zero_zero_rat ) ).

% ab_group_add_class.ab_left_minus
thf(fact_7574_add__eq__0__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ( plus_plus_real @ A2 @ B2 )
        = zero_zero_real )
      = ( B2
        = ( uminus_uminus_real @ A2 ) ) ) ).

% add_eq_0_iff
thf(fact_7575_add__eq__0__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ( plus_plus_int @ A2 @ B2 )
        = zero_zero_int )
      = ( B2
        = ( uminus_uminus_int @ A2 ) ) ) ).

% add_eq_0_iff
thf(fact_7576_add__eq__0__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ( plus_plus_complex @ A2 @ B2 )
        = zero_zero_complex )
      = ( B2
        = ( uminus1482373934393186551omplex @ A2 ) ) ) ).

% add_eq_0_iff
thf(fact_7577_add__eq__0__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A2 @ B2 )
        = zero_z3403309356797280102nteger )
      = ( B2
        = ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% add_eq_0_iff
thf(fact_7578_add__eq__0__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( plus_plus_rat @ A2 @ B2 )
        = zero_zero_rat )
      = ( B2
        = ( uminus_uminus_rat @ A2 ) ) ) ).

% add_eq_0_iff
thf(fact_7579_zero__neq__neg__one,axiom,
    ( zero_zero_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% zero_neq_neg_one
thf(fact_7580_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_7581_zero__neq__neg__one,axiom,
    ( zero_zero_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% zero_neq_neg_one
thf(fact_7582_zero__neq__neg__one,axiom,
    ( zero_z3403309356797280102nteger
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% zero_neq_neg_one
thf(fact_7583_zero__neq__neg__one,axiom,
    ( zero_zero_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% zero_neq_neg_one
thf(fact_7584_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_7585_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_7586_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_7587_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_7588_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_7589_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_7590_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_7591_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_7592_numeral__neq__neg__one,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_real @ N3 )
     != ( uminus_uminus_real @ one_one_real ) ) ).

% numeral_neq_neg_one
thf(fact_7593_numeral__neq__neg__one,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_int @ N3 )
     != ( uminus_uminus_int @ one_one_int ) ) ).

% numeral_neq_neg_one
thf(fact_7594_numeral__neq__neg__one,axiom,
    ! [N3: num] :
      ( ( numera6690914467698888265omplex @ N3 )
     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% numeral_neq_neg_one
thf(fact_7595_numeral__neq__neg__one,axiom,
    ! [N3: num] :
      ( ( numera6620942414471956472nteger @ N3 )
     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% numeral_neq_neg_one
thf(fact_7596_numeral__neq__neg__one,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_rat @ N3 )
     != ( uminus_uminus_rat @ one_one_rat ) ) ).

% numeral_neq_neg_one
thf(fact_7597_one__neq__neg__numeral,axiom,
    ! [N3: num] :
      ( one_one_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) ) ).

% one_neq_neg_numeral
thf(fact_7598_one__neq__neg__numeral,axiom,
    ! [N3: num] :
      ( one_one_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) ) ).

% one_neq_neg_numeral
thf(fact_7599_one__neq__neg__numeral,axiom,
    ! [N3: num] :
      ( one_one_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N3 ) ) ) ).

% one_neq_neg_numeral
thf(fact_7600_one__neq__neg__numeral,axiom,
    ! [N3: num] :
      ( one_one_Code_integer
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N3 ) ) ) ).

% one_neq_neg_numeral
thf(fact_7601_one__neq__neg__numeral,axiom,
    ! [N3: num] :
      ( one_one_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) ) ).

% one_neq_neg_numeral
thf(fact_7602_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_7603_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_7604_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_7605_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_7606_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_7607_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_7608_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_7609_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_7610_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_7611_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_7612_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_7613_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_7614_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_7615_nonzero__minus__divide__divide,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( ( divide_divide_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B2 ) )
        = ( divide_divide_real @ A2 @ B2 ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_7616_nonzero__minus__divide__divide,axiom,
    ! [B2: complex,A2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A2 ) @ ( uminus1482373934393186551omplex @ B2 ) )
        = ( divide1717551699836669952omplex @ A2 @ B2 ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_7617_nonzero__minus__divide__divide,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A2 ) @ ( uminus_uminus_rat @ B2 ) )
        = ( divide_divide_rat @ A2 @ B2 ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_7618_nonzero__minus__divide__right,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B2 ) )
        = ( divide_divide_real @ A2 @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_7619_nonzero__minus__divide__right,axiom,
    ! [B2: complex,A2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ B2 ) )
        = ( divide1717551699836669952omplex @ A2 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_7620_nonzero__minus__divide__right,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ B2 ) )
        = ( divide_divide_rat @ A2 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_7621_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_7622_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_7623_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_7624_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_7625_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_7626_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A7: real,B7: real] : ( plus_plus_real @ A7 @ ( uminus_uminus_real @ B7 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_7627_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A7: int,B7: int] : ( plus_plus_int @ A7 @ ( uminus_uminus_int @ B7 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_7628_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A7: complex,B7: complex] : ( plus_plus_complex @ A7 @ ( uminus1482373934393186551omplex @ B7 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_7629_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A7: code_integer,B7: code_integer] : ( plus_p5714425477246183910nteger @ A7 @ ( uminus1351360451143612070nteger @ B7 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_7630_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A7: rat,B7: rat] : ( plus_plus_rat @ A7 @ ( uminus_uminus_rat @ B7 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_7631_diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A7: real,B7: real] : ( plus_plus_real @ A7 @ ( uminus_uminus_real @ B7 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_7632_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A7: int,B7: int] : ( plus_plus_int @ A7 @ ( uminus_uminus_int @ B7 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_7633_diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A7: complex,B7: complex] : ( plus_plus_complex @ A7 @ ( uminus1482373934393186551omplex @ B7 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_7634_diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A7: code_integer,B7: code_integer] : ( plus_p5714425477246183910nteger @ A7 @ ( uminus1351360451143612070nteger @ B7 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_7635_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A7: rat,B7: rat] : ( plus_plus_rat @ A7 @ ( uminus_uminus_rat @ B7 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_7636_group__cancel_Osub2,axiom,
    ! [B: real,K: real,B2: real,A2: real] :
      ( ( B
        = ( plus_plus_real @ K @ B2 ) )
     => ( ( minus_minus_real @ A2 @ B )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A2 @ B2 ) ) ) ) ).

% group_cancel.sub2
thf(fact_7637_group__cancel_Osub2,axiom,
    ! [B: int,K: int,B2: int,A2: int] :
      ( ( B
        = ( plus_plus_int @ K @ B2 ) )
     => ( ( minus_minus_int @ A2 @ B )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A2 @ B2 ) ) ) ) ).

% group_cancel.sub2
thf(fact_7638_group__cancel_Osub2,axiom,
    ! [B: complex,K: complex,B2: complex,A2: complex] :
      ( ( B
        = ( plus_plus_complex @ K @ B2 ) )
     => ( ( minus_minus_complex @ A2 @ B )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A2 @ B2 ) ) ) ) ).

% group_cancel.sub2
thf(fact_7639_group__cancel_Osub2,axiom,
    ! [B: code_integer,K: code_integer,B2: code_integer,A2: code_integer] :
      ( ( B
        = ( plus_p5714425477246183910nteger @ K @ B2 ) )
     => ( ( minus_8373710615458151222nteger @ A2 @ B )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A2 @ B2 ) ) ) ) ).

% group_cancel.sub2
thf(fact_7640_group__cancel_Osub2,axiom,
    ! [B: rat,K: rat,B2: rat,A2: rat] :
      ( ( B
        = ( plus_plus_rat @ K @ B2 ) )
     => ( ( minus_minus_rat @ A2 @ B )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A2 @ B2 ) ) ) ) ).

% group_cancel.sub2
thf(fact_7641_dvd__neg__div,axiom,
    ! [B2: real,A2: real] :
      ( ( dvd_dvd_real @ B2 @ A2 )
     => ( ( divide_divide_real @ ( uminus_uminus_real @ A2 ) @ B2 )
        = ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B2 ) ) ) ) ).

% dvd_neg_div
thf(fact_7642_dvd__neg__div,axiom,
    ! [B2: int,A2: int] :
      ( ( dvd_dvd_int @ B2 @ A2 )
     => ( ( divide_divide_int @ ( uminus_uminus_int @ A2 ) @ B2 )
        = ( uminus_uminus_int @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% dvd_neg_div
thf(fact_7643_dvd__neg__div,axiom,
    ! [B2: complex,A2: complex] :
      ( ( dvd_dvd_complex @ B2 @ A2 )
     => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A2 ) @ B2 )
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ B2 ) ) ) ) ).

% dvd_neg_div
thf(fact_7644_dvd__neg__div,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ A2 )
     => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 )
        = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) ) ) ) ).

% dvd_neg_div
thf(fact_7645_dvd__neg__div,axiom,
    ! [B2: rat,A2: rat] :
      ( ( dvd_dvd_rat @ B2 @ A2 )
     => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A2 ) @ B2 )
        = ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ B2 ) ) ) ) ).

% dvd_neg_div
thf(fact_7646_dvd__div__neg,axiom,
    ! [B2: real,A2: real] :
      ( ( dvd_dvd_real @ B2 @ A2 )
     => ( ( divide_divide_real @ A2 @ ( uminus_uminus_real @ B2 ) )
        = ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B2 ) ) ) ) ).

% dvd_div_neg
thf(fact_7647_dvd__div__neg,axiom,
    ! [B2: int,A2: int] :
      ( ( dvd_dvd_int @ B2 @ A2 )
     => ( ( divide_divide_int @ A2 @ ( uminus_uminus_int @ B2 ) )
        = ( uminus_uminus_int @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).

% dvd_div_neg
thf(fact_7648_dvd__div__neg,axiom,
    ! [B2: complex,A2: complex] :
      ( ( dvd_dvd_complex @ B2 @ A2 )
     => ( ( divide1717551699836669952omplex @ A2 @ ( uminus1482373934393186551omplex @ B2 ) )
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ B2 ) ) ) ) ).

% dvd_div_neg
thf(fact_7649_dvd__div__neg,axiom,
    ! [B2: code_integer,A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B2 @ A2 )
     => ( ( divide6298287555418463151nteger @ A2 @ ( uminus1351360451143612070nteger @ B2 ) )
        = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A2 @ B2 ) ) ) ) ).

% dvd_div_neg
thf(fact_7650_dvd__div__neg,axiom,
    ! [B2: rat,A2: rat] :
      ( ( dvd_dvd_rat @ B2 @ A2 )
     => ( ( divide_divide_rat @ A2 @ ( uminus_uminus_rat @ B2 ) )
        = ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ B2 ) ) ) ) ).

% dvd_div_neg
thf(fact_7651_infinite__Icc,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A2 @ B2 ) ) ) ).

% infinite_Icc
thf(fact_7652_infinite__Icc,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) ) ) ).

% infinite_Icc
thf(fact_7653_infinite__Ico,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ~ ( finite_finite_real @ ( set_or66887138388493659n_real @ A2 @ B2 ) ) ) ).

% infinite_Ico
thf(fact_7654_infinite__Ico,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ A2 @ B2 )
     => ~ ( finite_finite_rat @ ( set_or4029947393144176647an_rat @ A2 @ B2 ) ) ) ).

% infinite_Ico
thf(fact_7655_finite__lists__length__le,axiom,
    ! [A: set_int,N3: nat] :
      ( ( finite_finite_int @ A )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A )
              & ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N3 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7656_finite__lists__length__le,axiom,
    ! [A: set_Code_integer,N3: nat] :
      ( ( finite6017078050557962740nteger @ A )
     => ( finite1283093830868386564nteger
        @ ( collec3483841146883906114nteger
          @ ^ [Xs: list_Code_integer] :
              ( ( ord_le7084787975880047091nteger @ ( set_Code_integer2 @ Xs ) @ A )
              & ( ord_less_eq_nat @ ( size_s3445333598471063425nteger @ Xs ) @ N3 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7657_finite__lists__length__le,axiom,
    ! [A: set_complex,N3: nat] :
      ( ( finite3207457112153483333omplex @ A )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A )
              & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ N3 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7658_finite__lists__length__le,axiom,
    ! [A: set_VEBT_VEBT,N3: nat] :
      ( ( finite5795047828879050333T_VEBT @ A )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A )
              & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ N3 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7659_finite__lists__length__le,axiom,
    ! [A: set_real,N3: nat] :
      ( ( finite_finite_real @ A )
     => ( finite306553202115118035t_real
        @ ( collect_list_real
          @ ^ [Xs: list_real] :
              ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A )
              & ( ord_less_eq_nat @ ( size_size_list_real @ Xs ) @ N3 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7660_finite__lists__length__le,axiom,
    ! [A: set_o,N3: nat] :
      ( ( finite_finite_o @ A )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A )
              & ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N3 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7661_finite__lists__length__le,axiom,
    ! [A: set_VEBT_VEBTi,N3: nat] :
      ( ( finite6580341952239402572_VEBTi @ A )
     => ( finite5722435864697464412_VEBTi
        @ ( collec1288327022334394266_VEBTi
          @ ^ [Xs: list_VEBT_VEBTi] :
              ( ( ord_le6592769550269828683_VEBTi @ ( set_VEBT_VEBTi2 @ Xs ) @ A )
              & ( ord_less_eq_nat @ ( size_s7982070591426661849_VEBTi @ Xs ) @ N3 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7662_finite__lists__length__le,axiom,
    ! [A: set_nat,N3: nat] :
      ( ( finite_finite_nat @ A )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A )
              & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N3 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_7663_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_7664_minus__real__def,axiom,
    ( minus_minus_real
    = ( ^ [X3: real,Y2: real] : ( plus_plus_real @ X3 @ ( uminus_uminus_real @ Y2 ) ) ) ) ).

% minus_real_def
thf(fact_7665_of__int__of__nat,axiom,
    ( ring_1_of_int_rat
    = ( ^ [K3: int] : ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri681578069525770553at_rat @ ( nat2 @ K3 ) ) ) ) ) ).

% of_int_of_nat
thf(fact_7666_of__int__of__nat,axiom,
    ( ring_1_of_int_real
    = ( ^ [K3: int] : ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri5074537144036343181t_real @ ( nat2 @ K3 ) ) ) ) ) ).

% of_int_of_nat
thf(fact_7667_of__int__of__nat,axiom,
    ( ring_1_of_int_int
    = ( ^ [K3: int] : ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K3 ) ) ) ) ) ).

% of_int_of_nat
thf(fact_7668_of__int__of__nat,axiom,
    ( ring_17405671764205052669omplex
    = ( ^ [K3: int] : ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri8010041392384452111omplex @ ( nat2 @ K3 ) ) ) ) ) ).

% of_int_of_nat
thf(fact_7669_of__int__of__nat,axiom,
    ( ring_18347121197199848620nteger
    = ( ^ [K3: int] : ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri4939895301339042750nteger @ ( nat2 @ K3 ) ) ) ) ) ).

% of_int_of_nat
thf(fact_7670_not__zero__le__neg__numeral,axiom,
    ! [N3: num] :
      ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_7671_not__zero__le__neg__numeral,axiom,
    ! [N3: num] :
      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N3 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_7672_not__zero__le__neg__numeral,axiom,
    ! [N3: num] :
      ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_7673_not__zero__le__neg__numeral,axiom,
    ! [N3: num] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_7674_neg__numeral__le__zero,axiom,
    ! [N3: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) @ zero_zero_real ) ).

% neg_numeral_le_zero
thf(fact_7675_neg__numeral__le__zero,axiom,
    ! [N3: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N3 ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_le_zero
thf(fact_7676_neg__numeral__le__zero,axiom,
    ! [N3: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) @ zero_zero_rat ) ).

% neg_numeral_le_zero
thf(fact_7677_neg__numeral__le__zero,axiom,
    ! [N3: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) @ zero_zero_int ) ).

% neg_numeral_le_zero
thf(fact_7678_not__zero__less__neg__numeral,axiom,
    ! [N3: num] :
      ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_7679_not__zero__less__neg__numeral,axiom,
    ! [N3: num] :
      ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_7680_not__zero__less__neg__numeral,axiom,
    ! [N3: num] :
      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N3 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_7681_not__zero__less__neg__numeral,axiom,
    ! [N3: num] :
      ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_7682_neg__numeral__less__zero,axiom,
    ! [N3: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) @ zero_zero_real ) ).

% neg_numeral_less_zero
thf(fact_7683_neg__numeral__less__zero,axiom,
    ! [N3: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) @ zero_zero_int ) ).

% neg_numeral_less_zero
thf(fact_7684_neg__numeral__less__zero,axiom,
    ! [N3: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N3 ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_less_zero
thf(fact_7685_neg__numeral__less__zero,axiom,
    ! [N3: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) @ zero_zero_rat ) ).

% neg_numeral_less_zero
thf(fact_7686_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_7687_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(3)
thf(fact_7688_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_7689_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_7690_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_7691_le__minus__one__simps_I1_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% le_minus_one_simps(1)
thf(fact_7692_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_7693_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_7694_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_7695_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_7696_less__minus__one__simps_I1_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% less_minus_one_simps(1)
thf(fact_7697_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_7698_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_7699_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_7700_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(3)
thf(fact_7701_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_7702_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_7703_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_le_one
thf(fact_7704_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_7705_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_7706_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_7707_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_le_numeral
thf(fact_7708_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_7709_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_7710_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_7711_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_7712_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_7713_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_7714_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_7715_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_le_neg_one
thf(fact_7716_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_7717_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_7718_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_7719_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_7720_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_7721_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_7722_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_7723_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_7724_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_7725_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_7726_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_7727_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_7728_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_7729_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_7730_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_7731_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_7732_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_less_neg_one
thf(fact_7733_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_7734_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_7735_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_7736_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_less_numeral
thf(fact_7737_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_7738_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_7739_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_7740_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_less_one
thf(fact_7741_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_7742_uminus__numeral__One,axiom,
    ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% uminus_numeral_One
thf(fact_7743_uminus__numeral__One,axiom,
    ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% uminus_numeral_One
thf(fact_7744_uminus__numeral__One,axiom,
    ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% uminus_numeral_One
thf(fact_7745_uminus__numeral__One,axiom,
    ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% uminus_numeral_One
thf(fact_7746_uminus__numeral__One,axiom,
    ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% uminus_numeral_One
thf(fact_7747_mult__1s__ring__1_I1_J,axiom,
    ! [B2: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B2 )
      = ( uminus_uminus_real @ B2 ) ) ).

% mult_1s_ring_1(1)
thf(fact_7748_mult__1s__ring__1_I1_J,axiom,
    ! [B2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B2 )
      = ( uminus_uminus_int @ B2 ) ) ).

% mult_1s_ring_1(1)
thf(fact_7749_mult__1s__ring__1_I1_J,axiom,
    ! [B2: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B2 )
      = ( uminus1482373934393186551omplex @ B2 ) ) ).

% mult_1s_ring_1(1)
thf(fact_7750_mult__1s__ring__1_I1_J,axiom,
    ! [B2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B2 )
      = ( uminus1351360451143612070nteger @ B2 ) ) ).

% mult_1s_ring_1(1)
thf(fact_7751_mult__1s__ring__1_I1_J,axiom,
    ! [B2: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B2 )
      = ( uminus_uminus_rat @ B2 ) ) ).

% mult_1s_ring_1(1)
thf(fact_7752_mult__1s__ring__1_I2_J,axiom,
    ! [B2: real] :
      ( ( times_times_real @ B2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
      = ( uminus_uminus_real @ B2 ) ) ).

% mult_1s_ring_1(2)
thf(fact_7753_mult__1s__ring__1_I2_J,axiom,
    ! [B2: int] :
      ( ( times_times_int @ B2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
      = ( uminus_uminus_int @ B2 ) ) ).

% mult_1s_ring_1(2)
thf(fact_7754_mult__1s__ring__1_I2_J,axiom,
    ! [B2: complex] :
      ( ( times_times_complex @ B2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
      = ( uminus1482373934393186551omplex @ B2 ) ) ).

% mult_1s_ring_1(2)
thf(fact_7755_mult__1s__ring__1_I2_J,axiom,
    ! [B2: code_integer] :
      ( ( times_3573771949741848930nteger @ B2 @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
      = ( uminus1351360451143612070nteger @ B2 ) ) ).

% mult_1s_ring_1(2)
thf(fact_7756_mult__1s__ring__1_I2_J,axiom,
    ! [B2: rat] :
      ( ( times_times_rat @ B2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
      = ( uminus_uminus_rat @ B2 ) ) ).

% mult_1s_ring_1(2)
thf(fact_7757_divide__eq__minus__1__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ( divide_divide_real @ A2 @ B2 )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( ( B2 != zero_zero_real )
        & ( A2
          = ( uminus_uminus_real @ B2 ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_7758_divide__eq__minus__1__iff,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ( divide1717551699836669952omplex @ A2 @ B2 )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( ( B2 != zero_zero_complex )
        & ( A2
          = ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_7759_divide__eq__minus__1__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( divide_divide_rat @ A2 @ B2 )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( ( B2 != zero_zero_rat )
        & ( A2
          = ( uminus_uminus_rat @ B2 ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_7760_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: real,C2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( ( C2
          = ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B2 ) ) )
        = ( ( times_times_real @ C2 @ B2 )
          = ( uminus_uminus_real @ A2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_7761_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: complex,C2: complex,A2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( C2
          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ B2 ) ) )
        = ( ( times_times_complex @ C2 @ B2 )
          = ( uminus1482373934393186551omplex @ A2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_7762_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( C2
          = ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ B2 ) ) )
        = ( ( times_times_rat @ C2 @ B2 )
          = ( uminus_uminus_rat @ A2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_7763_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: real,A2: real,C2: real] :
      ( ( B2 != zero_zero_real )
     => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B2 ) )
          = C2 )
        = ( ( uminus_uminus_real @ A2 )
          = ( times_times_real @ C2 @ B2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_7764_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: complex,A2: complex,C2: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ B2 ) )
          = C2 )
        = ( ( uminus1482373934393186551omplex @ A2 )
          = ( times_times_complex @ C2 @ B2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_7765_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: rat,A2: rat,C2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ B2 ) )
          = C2 )
        = ( ( uminus_uminus_rat @ A2 )
          = ( times_times_rat @ C2 @ B2 ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_7766_minus__divide__eq__eq,axiom,
    ! [B2: real,C2: real,A2: real] :
      ( ( ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C2 ) )
        = A2 )
      = ( ( ( C2 != zero_zero_real )
         => ( ( uminus_uminus_real @ B2 )
            = ( times_times_real @ A2 @ C2 ) ) )
        & ( ( C2 = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_7767_minus__divide__eq__eq,axiom,
    ! [B2: complex,C2: complex,A2: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B2 @ C2 ) )
        = A2 )
      = ( ( ( C2 != zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ B2 )
            = ( times_times_complex @ A2 @ C2 ) ) )
        & ( ( C2 = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_7768_minus__divide__eq__eq,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) )
        = A2 )
      = ( ( ( C2 != zero_zero_rat )
         => ( ( uminus_uminus_rat @ B2 )
            = ( times_times_rat @ A2 @ C2 ) ) )
        & ( ( C2 = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_7769_eq__minus__divide__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( A2
        = ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C2 ) ) )
      = ( ( ( C2 != zero_zero_real )
         => ( ( times_times_real @ A2 @ C2 )
            = ( uminus_uminus_real @ B2 ) ) )
        & ( ( C2 = zero_zero_real )
         => ( A2 = zero_zero_real ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_7770_eq__minus__divide__eq,axiom,
    ! [A2: complex,B2: complex,C2: complex] :
      ( ( A2
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B2 @ C2 ) ) )
      = ( ( ( C2 != zero_zero_complex )
         => ( ( times_times_complex @ A2 @ C2 )
            = ( uminus1482373934393186551omplex @ B2 ) ) )
        & ( ( C2 = zero_zero_complex )
         => ( A2 = zero_zero_complex ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_7771_eq__minus__divide__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( A2
        = ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) )
      = ( ( ( C2 != zero_zero_rat )
         => ( ( times_times_rat @ A2 @ C2 )
            = ( uminus_uminus_rat @ B2 ) ) )
        & ( ( C2 = zero_zero_rat )
         => ( A2 = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_7772_power__minus,axiom,
    ! [A2: real,N3: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A2 ) @ N3 )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( power_power_real @ A2 @ N3 ) ) ) ).

% power_minus
thf(fact_7773_power__minus,axiom,
    ! [A2: int,N3: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A2 ) @ N3 )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N3 ) @ ( power_power_int @ A2 @ N3 ) ) ) ).

% power_minus
thf(fact_7774_power__minus,axiom,
    ! [A2: complex,N3: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A2 ) @ N3 )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N3 ) @ ( power_power_complex @ A2 @ N3 ) ) ) ).

% power_minus
thf(fact_7775_power__minus,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ N3 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N3 ) @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ).

% power_minus
thf(fact_7776_power__minus,axiom,
    ! [A2: rat,N3: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ N3 )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N3 ) @ ( power_power_rat @ A2 @ N3 ) ) ) ).

% power_minus
thf(fact_7777_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_7778_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_7779_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_7780_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_7781_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_7782_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_7783_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_7784_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_7785_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_7786_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_7787_norm__uminus__minus,axiom,
    ! [X: real,Y: real] :
      ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ Y ) )
      = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) ) ).

% norm_uminus_minus
thf(fact_7788_norm__uminus__minus,axiom,
    ! [X: complex,Y: complex] :
      ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ Y ) )
      = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) ) ).

% norm_uminus_minus
thf(fact_7789_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_7790_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_7791_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_7792_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_7793_tanh__real__gt__neg1,axiom,
    ! [X: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X ) ) ).

% tanh_real_gt_neg1
thf(fact_7794_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [D3: nat] : ( dvd_dvd_nat @ D3 @ M ) ) ) ) ).

% finite_divisors_nat
thf(fact_7795_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N7: set_nat,N3: nat] :
      ( ( ord_less_eq_set_nat @ N7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) )
     => ( finite_finite_nat @ N7 ) ) ).

% subset_eq_atLeast0_lessThan_finite
thf(fact_7796_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N7: set_nat,N3: nat] :
      ( ( ord_less_eq_set_nat @ N7 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
     => ( finite_finite_nat @ N7 ) ) ).

% subset_eq_atLeast0_atMost_finite
thf(fact_7797_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: real,C2: real] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
        = ( divide_divide_real @ B2 @ C2 ) )
      = ( ( ( C2 != zero_zero_real )
         => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C2 )
            = B2 ) )
        & ( ( C2 = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_7798_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: complex,C2: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
        = ( divide1717551699836669952omplex @ B2 @ C2 ) )
      = ( ( ( C2 != zero_zero_complex )
         => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C2 )
            = B2 ) )
        & ( ( C2 = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_7799_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: rat,C2: rat] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
        = ( divide_divide_rat @ B2 @ C2 ) )
      = ( ( ( C2 != zero_zero_rat )
         => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C2 )
            = B2 ) )
        & ( ( C2 = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_7800_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: real,C2: real,W: num] :
      ( ( ( divide_divide_real @ B2 @ C2 )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( C2 != zero_zero_real )
         => ( B2
            = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C2 ) ) )
        & ( ( C2 = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_7801_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: complex,C2: complex,W: num] :
      ( ( ( divide1717551699836669952omplex @ B2 @ C2 )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( ( C2 != zero_zero_complex )
         => ( B2
            = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C2 ) ) )
        & ( ( C2 = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_7802_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: rat,C2: rat,W: num] :
      ( ( ( divide_divide_rat @ B2 @ C2 )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( C2 != zero_zero_rat )
         => ( B2
            = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C2 ) ) )
        & ( ( C2 = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_7803_pos__minus__divide__less__eq,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( ( ord_less_real @ zero_zero_real @ C2 )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C2 ) ) @ A2 )
        = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A2 @ C2 ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_7804_pos__minus__divide__less__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) @ A2 )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_7805_pos__less__minus__divide__eq,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C2 )
     => ( ( ord_less_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C2 ) ) )
        = ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_7806_pos__less__minus__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) )
        = ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_7807_neg__minus__divide__less__eq,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( ( ord_less_real @ C2 @ zero_zero_real )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C2 ) ) @ A2 )
        = ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_7808_neg__minus__divide__less__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) @ A2 )
        = ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_7809_neg__less__minus__divide__eq,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ord_less_real @ C2 @ zero_zero_real )
     => ( ( ord_less_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C2 ) ) )
        = ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A2 @ C2 ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_7810_neg__less__minus__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_7811_minus__divide__less__eq,axiom,
    ! [B2: real,C2: real,A2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C2 ) ) @ A2 )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A2 @ C2 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( uminus_uminus_real @ B2 ) ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A2 ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_7812_minus__divide__less__eq,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) @ A2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( uminus_uminus_rat @ B2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A2 ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_7813_less__minus__divide__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C2 ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ ( times_times_real @ A2 @ C2 ) @ ( uminus_uminus_real @ B2 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A2 @ C2 ) ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ A2 @ zero_zero_real ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_7814_less__minus__divide__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ ( times_times_rat @ A2 @ C2 ) @ ( uminus_uminus_rat @ B2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ A2 @ zero_zero_rat ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_7815_minus__divide__add__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_7816_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_7817_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_7818_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: real,A2: real,B2: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A2 @ Z ) ) @ B2 )
          = B2 ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A2 @ Z ) ) @ B2 )
          = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_7819_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: complex,A2: complex,B2: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ Z ) ) @ B2 )
          = B2 ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ Z ) ) @ B2 )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_7820_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: rat,A2: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ Z ) ) @ B2 )
          = B2 ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ Z ) ) @ B2 )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_7821_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
        = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_7822_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_7823_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_7824_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: real,A2: real,B2: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A2 @ Z ) @ B2 )
          = ( uminus_uminus_real @ B2 ) ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A2 @ Z ) @ B2 )
          = ( divide_divide_real @ ( minus_minus_real @ A2 @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_7825_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: complex,A2: complex,B2: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A2 @ Z ) @ B2 )
          = ( uminus1482373934393186551omplex @ B2 ) ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A2 @ Z ) @ B2 )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A2 @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_7826_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: rat,A2: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A2 @ Z ) @ B2 )
          = ( uminus_uminus_rat @ B2 ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A2 @ Z ) @ B2 )
          = ( divide_divide_rat @ ( minus_minus_rat @ A2 @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_7827_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: real,A2: real,B2: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A2 @ Z ) ) @ B2 )
          = ( uminus_uminus_real @ B2 ) ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A2 @ Z ) ) @ B2 )
          = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A2 ) @ ( times_times_real @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_7828_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: complex,A2: complex,B2: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ Z ) ) @ B2 )
          = ( uminus1482373934393186551omplex @ B2 ) ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ Z ) ) @ B2 )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( times_times_complex @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_7829_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: rat,A2: rat,B2: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ Z ) ) @ B2 )
          = ( uminus_uminus_rat @ B2 ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ Z ) ) @ B2 )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A2 ) @ ( times_times_rat @ B2 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_7830_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_7831_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_7832_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_7833_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_7834_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_7835_even__minus,axiom,
    ! [A2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A2 ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) ).

% even_minus
thf(fact_7836_even__minus,axiom,
    ! [A2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) ) ).

% even_minus
thf(fact_7837_finite__roots__unity,axiom,
    ! [N3: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N3 )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Z5: real] :
              ( ( power_power_real @ Z5 @ N3 )
              = one_one_real ) ) ) ) ).

% finite_roots_unity
thf(fact_7838_finite__roots__unity,axiom,
    ! [N3: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N3 )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z5: complex] :
              ( ( power_power_complex @ Z5 @ N3 )
              = one_one_complex ) ) ) ) ).

% finite_roots_unity
thf(fact_7839_pos__minus__divide__le__eq,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( ( ord_less_real @ zero_zero_real @ C2 )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C2 ) ) @ A2 )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A2 @ C2 ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_7840_pos__minus__divide__le__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) @ A2 )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_7841_pos__le__minus__divide__eq,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ C2 )
     => ( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C2 ) ) )
        = ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_7842_pos__le__minus__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_7843_neg__minus__divide__le__eq,axiom,
    ! [C2: real,B2: real,A2: real] :
      ( ( ord_less_real @ C2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C2 ) ) @ A2 )
        = ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_7844_neg__minus__divide__le__eq,axiom,
    ! [C2: rat,B2: rat,A2: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) @ A2 )
        = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_7845_neg__le__minus__divide__eq,axiom,
    ! [C2: real,A2: real,B2: real] :
      ( ( ord_less_real @ C2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C2 ) ) )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A2 @ C2 ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_7846_neg__le__minus__divide__eq,axiom,
    ! [C2: rat,A2: rat,B2: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_7847_minus__divide__le__eq,axiom,
    ! [B2: real,C2: real,A2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C2 ) ) @ A2 )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A2 @ C2 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( uminus_uminus_real @ B2 ) ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_7848_minus__divide__le__eq,axiom,
    ! [B2: rat,C2: rat,A2: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) @ A2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ ( uminus_uminus_rat @ B2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_7849_le__minus__divide__eq,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B2 @ C2 ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ ( times_times_real @ A2 @ C2 ) @ ( uminus_uminus_real @ B2 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( times_times_real @ A2 @ C2 ) ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_7850_le__minus__divide__eq,axiom,
    ! [A2: rat,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B2 @ C2 ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C2 ) @ ( uminus_uminus_rat @ B2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B2 ) @ ( times_times_rat @ A2 @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_7851_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: real,C2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B2 @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C2 ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C2 ) ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_7852_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: rat,C2: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B2 @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C2 ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_7853_divide__less__eq__numeral_I2_J,axiom,
    ! [B2: real,C2: real,W: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B2 @ C2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C2 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C2 ) @ B2 ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_7854_divide__less__eq__numeral_I2_J,axiom,
    ! [B2: rat,C2: rat,W: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B2 @ C2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C2 ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_7855_power2__eq__1__iff,axiom,
    ! [A2: real] :
      ( ( ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
      = ( ( A2 = one_one_real )
        | ( A2
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% power2_eq_1_iff
thf(fact_7856_power2__eq__1__iff,axiom,
    ! [A2: int] :
      ( ( ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_int )
      = ( ( A2 = one_one_int )
        | ( A2
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% power2_eq_1_iff
thf(fact_7857_power2__eq__1__iff,axiom,
    ! [A2: complex] :
      ( ( ( power_power_complex @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_complex )
      = ( ( A2 = one_one_complex )
        | ( A2
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% power2_eq_1_iff
thf(fact_7858_power2__eq__1__iff,axiom,
    ! [A2: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_Code_integer )
      = ( ( A2 = one_one_Code_integer )
        | ( A2
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% power2_eq_1_iff
thf(fact_7859_power2__eq__1__iff,axiom,
    ! [A2: rat] :
      ( ( ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_rat )
      = ( ( A2 = one_one_rat )
        | ( A2
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% power2_eq_1_iff
thf(fact_7860_uminus__power__if,axiom,
    ! [N3: nat,A2: real] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_real @ ( uminus_uminus_real @ A2 ) @ N3 )
          = ( power_power_real @ A2 @ N3 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_real @ ( uminus_uminus_real @ A2 ) @ N3 )
          = ( uminus_uminus_real @ ( power_power_real @ A2 @ N3 ) ) ) ) ) ).

% uminus_power_if
thf(fact_7861_uminus__power__if,axiom,
    ! [N3: nat,A2: int] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_int @ ( uminus_uminus_int @ A2 ) @ N3 )
          = ( power_power_int @ A2 @ N3 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_int @ ( uminus_uminus_int @ A2 ) @ N3 )
          = ( uminus_uminus_int @ ( power_power_int @ A2 @ N3 ) ) ) ) ) ).

% uminus_power_if
thf(fact_7862_uminus__power__if,axiom,
    ! [N3: nat,A2: complex] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A2 ) @ N3 )
          = ( power_power_complex @ A2 @ N3 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A2 ) @ N3 )
          = ( uminus1482373934393186551omplex @ ( power_power_complex @ A2 @ N3 ) ) ) ) ) ).

% uminus_power_if
thf(fact_7863_uminus__power__if,axiom,
    ! [N3: nat,A2: code_integer] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ N3 )
          = ( power_8256067586552552935nteger @ A2 @ N3 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ N3 )
          = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ) ) ).

% uminus_power_if
thf(fact_7864_uminus__power__if,axiom,
    ! [N3: nat,A2: rat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ N3 )
          = ( power_power_rat @ A2 @ N3 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ N3 )
          = ( uminus_uminus_rat @ ( power_power_rat @ A2 @ N3 ) ) ) ) ) ).

% uminus_power_if
thf(fact_7865_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N3 @ K ) )
        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N3 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_7866_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N3 @ K ) )
        = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N3 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_7867_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N3 @ K ) )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N3 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_7868_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N3 @ K ) )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N3 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_7869_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N3 @ K ) )
        = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N3 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_7870_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_7871_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).

% ln_add_one_self_le_self2
thf(fact_7872_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: real,C2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B2 @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C2 ) @ B2 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C2 ) ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_7873_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: rat,C2: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B2 @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C2 ) @ B2 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_7874_divide__le__eq__numeral_I2_J,axiom,
    ! [B2: real,C2: real,W: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ B2 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C2 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C2 ) @ B2 ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_7875_divide__le__eq__numeral_I2_J,axiom,
    ! [B2: rat,C2: rat,W: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B2 @ C2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ B2 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C2 ) @ B2 ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_7876_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_7877_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_7878_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_7879_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_7880_minus__power__mult__self,axiom,
    ! [A2: real,N3: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A2 ) @ N3 ) @ ( power_power_real @ ( uminus_uminus_real @ A2 ) @ N3 ) )
      = ( power_power_real @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% minus_power_mult_self
thf(fact_7881_minus__power__mult__self,axiom,
    ! [A2: int,N3: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A2 ) @ N3 ) @ ( power_power_int @ ( uminus_uminus_int @ A2 ) @ N3 ) )
      = ( power_power_int @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% minus_power_mult_self
thf(fact_7882_minus__power__mult__self,axiom,
    ! [A2: complex,N3: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A2 ) @ N3 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A2 ) @ N3 ) )
      = ( power_power_complex @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% minus_power_mult_self
thf(fact_7883_minus__power__mult__self,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ N3 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ N3 ) )
      = ( power_8256067586552552935nteger @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% minus_power_mult_self
thf(fact_7884_minus__power__mult__self,axiom,
    ! [A2: rat,N3: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ N3 ) @ ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ N3 ) )
      = ( power_power_rat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% minus_power_mult_self
thf(fact_7885_minus__one__power__iff,axiom,
    ! [N3: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 )
          = one_one_real ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 )
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% minus_one_power_iff
thf(fact_7886_minus__one__power__iff,axiom,
    ! [N3: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N3 )
          = one_one_int ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N3 )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% minus_one_power_iff
thf(fact_7887_minus__one__power__iff,axiom,
    ! [N3: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N3 )
          = one_one_complex ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N3 )
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% minus_one_power_iff
thf(fact_7888_minus__one__power__iff,axiom,
    ! [N3: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N3 )
          = one_one_Code_integer ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N3 )
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% minus_one_power_iff
thf(fact_7889_minus__one__power__iff,axiom,
    ! [N3: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N3 )
          = one_one_rat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N3 )
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% minus_one_power_iff
thf(fact_7890_Bernoulli__inequality,axiom,
    ! [X: real,N3: 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 @ N3 ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N3 ) ) ) ).

% Bernoulli_inequality
thf(fact_7891_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_7892_power__minus1__odd,axiom,
    ! [N3: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( uminus_uminus_real @ one_one_real ) ) ).

% power_minus1_odd
thf(fact_7893_power__minus1__odd,axiom,
    ! [N3: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% power_minus1_odd
thf(fact_7894_power__minus1__odd,axiom,
    ! [N3: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power_minus1_odd
thf(fact_7895_power__minus1__odd,axiom,
    ! [N3: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% power_minus1_odd
thf(fact_7896_power__minus1__odd,axiom,
    ! [N3: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( uminus_uminus_rat @ one_one_rat ) ) ).

% power_minus1_odd
thf(fact_7897_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_7898_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_7899_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_7900_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_7901_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_7902_finite__Diff__insert,axiom,
    ! [A: set_VEBT_VEBT,A2: vEBT_VEBT,B: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ ( minus_5127226145743854075T_VEBT @ A @ ( insert_VEBT_VEBT @ A2 @ B ) ) )
      = ( finite5795047828879050333T_VEBT @ ( minus_5127226145743854075T_VEBT @ A @ B ) ) ) ).

% finite_Diff_insert
thf(fact_7903_finite__Diff__insert,axiom,
    ! [A: set_real,A2: real,B: set_real] :
      ( ( finite_finite_real @ ( minus_minus_set_real @ A @ ( insert_real @ A2 @ B ) ) )
      = ( finite_finite_real @ ( minus_minus_set_real @ A @ B ) ) ) ).

% finite_Diff_insert
thf(fact_7904_finite__Diff__insert,axiom,
    ! [A: set_o,A2: $o,B: set_o] :
      ( ( finite_finite_o @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ B ) ) )
      = ( finite_finite_o @ ( minus_minus_set_o @ A @ B ) ) ) ).

% finite_Diff_insert
thf(fact_7905_finite__Diff__insert,axiom,
    ! [A: set_int,A2: int,B: set_int] :
      ( ( finite_finite_int @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ B ) ) )
      = ( finite_finite_int @ ( minus_minus_set_int @ A @ B ) ) ) ).

% finite_Diff_insert
thf(fact_7906_finite__Diff__insert,axiom,
    ! [A: set_Code_integer,A2: code_integer,B: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A @ ( insert_Code_integer @ A2 @ B ) ) )
      = ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A @ B ) ) ) ).

% finite_Diff_insert
thf(fact_7907_finite__Diff__insert,axiom,
    ! [A: set_complex,A2: complex,B: set_complex] :
      ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A @ ( insert_complex @ A2 @ B ) ) )
      = ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A @ B ) ) ) ).

% finite_Diff_insert
thf(fact_7908_finite__Diff__insert,axiom,
    ! [A: set_nat,A2: nat,B: set_nat] :
      ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) ) )
      = ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).

% finite_Diff_insert
thf(fact_7909_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_7910_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_7911_finite__Collect__subsets,axiom,
    ! [A: set_int] :
      ( ( finite_finite_int @ A )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [B5: set_int] : ( ord_less_eq_set_int @ B5 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_7912_finite__Collect__subsets,axiom,
    ! [A: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A )
     => ( finite6931041176100689706nteger
        @ ( collec574505750873337192nteger
          @ ^ [B5: set_Code_integer] : ( ord_le7084787975880047091nteger @ B5 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_7913_finite__Collect__subsets,axiom,
    ! [A: set_complex] :
      ( ( finite3207457112153483333omplex @ A )
     => ( finite6551019134538273531omplex
        @ ( collect_set_complex
          @ ^ [B5: set_complex] : ( ord_le211207098394363844omplex @ B5 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_7914_finite__Collect__subsets,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B5: set_nat] : ( ord_less_eq_set_nat @ B5 @ A ) ) ) ) ).

% finite_Collect_subsets
thf(fact_7915_Compl__subset__Compl__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( uminus5710092332889474511et_nat @ B ) )
      = ( ord_less_eq_set_nat @ B @ A ) ) ).

% Compl_subset_Compl_iff
thf(fact_7916_Compl__anti__mono,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B ) @ ( uminus5710092332889474511et_nat @ A ) ) ) ).

% Compl_anti_mono
thf(fact_7917_finite__atLeastAtMost__int,axiom,
    ! [L2: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L2 @ U ) ) ).

% finite_atLeastAtMost_int
thf(fact_7918_finite__atLeastLessThan__int,axiom,
    ! [L2: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L2 @ U ) ) ).

% finite_atLeastLessThan_int
thf(fact_7919_finite__interval__int4,axiom,
    ! [A2: int,B2: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I2: int] :
            ( ( ord_less_int @ A2 @ I2 )
            & ( ord_less_int @ I2 @ B2 ) ) ) ) ).

% finite_interval_int4
thf(fact_7920_finite__atLeastLessThan__integer,axiom,
    ! [L2: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ L2 @ U ) ) ).

% finite_atLeastLessThan_integer
thf(fact_7921_finite__atLeastAtMost__integer,axiom,
    ! [L2: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or189985376899183464nteger @ L2 @ U ) ) ).

% finite_atLeastAtMost_integer
thf(fact_7922_finite__insert,axiom,
    ! [A2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ ( insert_VEBT_VEBT @ A2 @ A ) )
      = ( finite5795047828879050333T_VEBT @ A ) ) ).

% finite_insert
thf(fact_7923_finite__insert,axiom,
    ! [A2: real,A: set_real] :
      ( ( finite_finite_real @ ( insert_real @ A2 @ A ) )
      = ( finite_finite_real @ A ) ) ).

% finite_insert
thf(fact_7924_finite__insert,axiom,
    ! [A2: $o,A: set_o] :
      ( ( finite_finite_o @ ( insert_o @ A2 @ A ) )
      = ( finite_finite_o @ A ) ) ).

% finite_insert
thf(fact_7925_finite__insert,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( finite_finite_nat @ ( insert_nat @ A2 @ A ) )
      = ( finite_finite_nat @ A ) ) ).

% finite_insert
thf(fact_7926_finite__insert,axiom,
    ! [A2: int,A: set_int] :
      ( ( finite_finite_int @ ( insert_int @ A2 @ A ) )
      = ( finite_finite_int @ A ) ) ).

% finite_insert
thf(fact_7927_finite__insert,axiom,
    ! [A2: code_integer,A: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ ( insert_Code_integer @ A2 @ A ) )
      = ( finite6017078050557962740nteger @ A ) ) ).

% finite_insert
thf(fact_7928_finite__insert,axiom,
    ! [A2: complex,A: set_complex] :
      ( ( finite3207457112153483333omplex @ ( insert_complex @ A2 @ A ) )
      = ( finite3207457112153483333omplex @ A ) ) ).

% finite_insert
thf(fact_7929_finite__Diff2,axiom,
    ! [B: set_int,A: set_int] :
      ( ( finite_finite_int @ B )
     => ( ( finite_finite_int @ ( minus_minus_set_int @ A @ B ) )
        = ( finite_finite_int @ A ) ) ) ).

% finite_Diff2
thf(fact_7930_finite__Diff2,axiom,
    ! [B: set_Code_integer,A: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B )
     => ( ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A @ B ) )
        = ( finite6017078050557962740nteger @ A ) ) ) ).

% finite_Diff2
thf(fact_7931_finite__Diff2,axiom,
    ! [B: set_complex,A: set_complex] :
      ( ( finite3207457112153483333omplex @ B )
     => ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A @ B ) )
        = ( finite3207457112153483333omplex @ A ) ) ) ).

% finite_Diff2
thf(fact_7932_finite__Diff2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) )
        = ( finite_finite_nat @ A ) ) ) ).

% finite_Diff2
thf(fact_7933_finite__Diff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( finite_finite_int @ A )
     => ( finite_finite_int @ ( minus_minus_set_int @ A @ B ) ) ) ).

% finite_Diff
thf(fact_7934_finite__Diff,axiom,
    ! [A: set_Code_integer,B: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A )
     => ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A @ B ) ) ) ).

% finite_Diff
thf(fact_7935_finite__Diff,axiom,
    ! [A: set_complex,B: set_complex] :
      ( ( finite3207457112153483333omplex @ A )
     => ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A @ B ) ) ) ).

% finite_Diff
thf(fact_7936_finite__Diff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).

% finite_Diff
thf(fact_7937_finite__interval__int3,axiom,
    ! [A2: int,B2: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I2: int] :
            ( ( ord_less_int @ A2 @ I2 )
            & ( ord_less_eq_int @ I2 @ B2 ) ) ) ) ).

% finite_interval_int3
thf(fact_7938_finite__interval__int2,axiom,
    ! [A2: int,B2: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I2: int] :
            ( ( ord_less_eq_int @ A2 @ I2 )
            & ( ord_less_int @ I2 @ B2 ) ) ) ) ).

% finite_interval_int2
thf(fact_7939_subset__Compl__singleton,axiom,
    ! [A: set_VEBT_VEBT,B2: vEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ A @ ( uminus8041839845116263051T_VEBT @ ( insert_VEBT_VEBT @ B2 @ bot_bo8194388402131092736T_VEBT ) ) )
      = ( ~ ( member_VEBT_VEBT @ B2 @ A ) ) ) ).

% subset_Compl_singleton
thf(fact_7940_subset__Compl__singleton,axiom,
    ! [A: set_complex,B2: complex] :
      ( ( ord_le211207098394363844omplex @ A @ ( uminus8566677241136511917omplex @ ( insert_complex @ B2 @ bot_bot_set_complex ) ) )
      = ( ~ ( member_complex @ B2 @ A ) ) ) ).

% subset_Compl_singleton
thf(fact_7941_subset__Compl__singleton,axiom,
    ! [A: set_real,B2: real] :
      ( ( ord_less_eq_set_real @ A @ ( uminus612125837232591019t_real @ ( insert_real @ B2 @ bot_bot_set_real ) ) )
      = ( ~ ( member_real @ B2 @ A ) ) ) ).

% subset_Compl_singleton
thf(fact_7942_subset__Compl__singleton,axiom,
    ! [A: set_o,B2: $o] :
      ( ( ord_less_eq_set_o @ A @ ( uminus_uminus_set_o @ ( insert_o @ B2 @ bot_bot_set_o ) ) )
      = ( ~ ( member_o @ B2 @ A ) ) ) ).

% subset_Compl_singleton
thf(fact_7943_subset__Compl__singleton,axiom,
    ! [A: set_int,B2: int] :
      ( ( ord_less_eq_set_int @ A @ ( uminus1532241313380277803et_int @ ( insert_int @ B2 @ bot_bot_set_int ) ) )
      = ( ~ ( member_int @ B2 @ A ) ) ) ).

% subset_Compl_singleton
thf(fact_7944_subset__Compl__singleton,axiom,
    ! [A: set_nat,B2: nat] :
      ( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) )
      = ( ~ ( member_nat @ B2 @ A ) ) ) ).

% subset_Compl_singleton
thf(fact_7945_negative__eq__positive,axiom,
    ! [N3: nat,M: nat] :
      ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) )
        = ( semiri1314217659103216013at_int @ M ) )
      = ( ( N3 = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
thf(fact_7946_negative__zless,axiom,
    ! [N3: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zless
thf(fact_7947_nat__neg__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = zero_zero_nat ) ).

% nat_neg_numeral
thf(fact_7948_nat__zminus__int,axiom,
    ! [N3: nat] :
      ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
      = zero_zero_nat ) ).

% nat_zminus_int
thf(fact_7949_int__div__minus__is__minus1,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( ( divide_divide_int @ A2 @ B2 )
          = ( uminus_uminus_int @ A2 ) )
        = ( B2
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% int_div_minus_is_minus1
thf(fact_7950_ceiling__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] :
      ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A2 ) @ ( numeral_numeral_real @ B2 ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A2 ) ) @ ( numeral_numeral_int @ B2 ) ) ) ) ).

% ceiling_divide_eq_div_numeral
thf(fact_7951_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] :
      ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A2 ) @ ( numeral_numeral_real @ B2 ) ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A2 ) @ ( numeral_numeral_int @ B2 ) ) ) ) ).

% ceiling_minus_divide_eq_div_numeral
thf(fact_7952_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ U ) ) ).

% finite_atLeastZeroLessThan_integer
thf(fact_7953_subset__Compl__self__eq,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ A @ ( uminus612125837232591019t_real @ A ) )
      = ( A = bot_bot_set_real ) ) ).

% subset_Compl_self_eq
thf(fact_7954_subset__Compl__self__eq,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ A @ ( uminus_uminus_set_o @ A ) )
      = ( A = bot_bot_set_o ) ) ).

% subset_Compl_self_eq
thf(fact_7955_subset__Compl__self__eq,axiom,
    ! [A: set_int] :
      ( ( ord_less_eq_set_int @ A @ ( uminus1532241313380277803et_int @ A ) )
      = ( A = bot_bot_set_int ) ) ).

% subset_Compl_self_eq
thf(fact_7956_subset__Compl__self__eq,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% subset_Compl_self_eq
thf(fact_7957_finite__maxlen,axiom,
    ! [M3: set_list_VEBT_VEBT] :
      ( ( finite3004134309566078307T_VEBT @ M3 )
     => ? [N: nat] :
        ! [X5: list_VEBT_VEBT] :
          ( ( member2936631157270082147T_VEBT @ X5 @ M3 )
         => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X5 ) @ N ) ) ) ).

% finite_maxlen
thf(fact_7958_finite__maxlen,axiom,
    ! [M3: set_list_real] :
      ( ( finite306553202115118035t_real @ M3 )
     => ? [N: nat] :
        ! [X5: list_real] :
          ( ( member_list_real @ X5 @ M3 )
         => ( ord_less_nat @ ( size_size_list_real @ X5 ) @ N ) ) ) ).

% finite_maxlen
thf(fact_7959_finite__maxlen,axiom,
    ! [M3: set_list_o] :
      ( ( finite_finite_list_o @ M3 )
     => ? [N: nat] :
        ! [X5: list_o] :
          ( ( member_list_o @ X5 @ M3 )
         => ( ord_less_nat @ ( size_size_list_o @ X5 ) @ N ) ) ) ).

% finite_maxlen
thf(fact_7960_finite__maxlen,axiom,
    ! [M3: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M3 )
     => ? [N: nat] :
        ! [X5: list_nat] :
          ( ( member_list_nat @ X5 @ M3 )
         => ( ord_less_nat @ ( size_size_list_nat @ X5 ) @ N ) ) ) ).

% finite_maxlen
thf(fact_7961_finite__maxlen,axiom,
    ! [M3: set_list_VEBT_VEBTi] :
      ( ( finite5722435864697464412_VEBTi @ M3 )
     => ? [N: nat] :
        ! [X5: list_VEBT_VEBTi] :
          ( ( member8117914210271334748_VEBTi @ X5 @ M3 )
         => ( ord_less_nat @ ( size_s7982070591426661849_VEBTi @ X5 ) @ N ) ) ) ).

% finite_maxlen
thf(fact_7962_int__cases,axiom,
    ! [Z: int] :
      ( ! [N: nat] :
          ( Z
         != ( semiri1314217659103216013at_int @ N ) )
     => ~ ! [N: nat] :
            ( Z
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ).

% int_cases
thf(fact_7963_int__of__nat__induct,axiom,
    ! [P: int > $o,Z: int] :
      ( ! [N: nat] : ( P @ ( semiri1314217659103216013at_int @ N ) )
     => ( ! [N: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) )
       => ( P @ Z ) ) ) ).

% int_of_nat_induct
thf(fact_7964_minus__int__code_I2_J,axiom,
    ! [L2: int] :
      ( ( minus_minus_int @ zero_zero_int @ L2 )
      = ( uminus_uminus_int @ L2 ) ) ).

% minus_int_code(2)
thf(fact_7965_not__int__zless__negative,axiom,
    ! [N3: nat,M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N3 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% not_int_zless_negative
thf(fact_7966_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).

% finite_atLeastZeroLessThan_int
thf(fact_7967_Compl__insert,axiom,
    ! [X: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ( uminus8041839845116263051T_VEBT @ ( insert_VEBT_VEBT @ X @ A ) )
      = ( minus_5127226145743854075T_VEBT @ ( uminus8041839845116263051T_VEBT @ A ) @ ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% Compl_insert
thf(fact_7968_Compl__insert,axiom,
    ! [X: real,A: set_real] :
      ( ( uminus612125837232591019t_real @ ( insert_real @ X @ A ) )
      = ( minus_minus_set_real @ ( uminus612125837232591019t_real @ A ) @ ( insert_real @ X @ bot_bot_set_real ) ) ) ).

% Compl_insert
thf(fact_7969_Compl__insert,axiom,
    ! [X: $o,A: set_o] :
      ( ( uminus_uminus_set_o @ ( insert_o @ X @ A ) )
      = ( minus_minus_set_o @ ( uminus_uminus_set_o @ A ) @ ( insert_o @ X @ bot_bot_set_o ) ) ) ).

% Compl_insert
thf(fact_7970_Compl__insert,axiom,
    ! [X: int,A: set_int] :
      ( ( uminus1532241313380277803et_int @ ( insert_int @ X @ A ) )
      = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A ) @ ( insert_int @ X @ bot_bot_set_int ) ) ) ).

% Compl_insert
thf(fact_7971_Compl__insert,axiom,
    ! [X: nat,A: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( insert_nat @ X @ A ) )
      = ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% Compl_insert
thf(fact_7972_int__cases4,axiom,
    ! [M: int] :
      ( ! [N: nat] :
          ( M
         != ( semiri1314217659103216013at_int @ N ) )
     => ~ ! [N: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( M
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% int_cases4
thf(fact_7973_int__zle__neg,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N3 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
      = ( ( N3 = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
thf(fact_7974_zmod__zminus1__eq__if,axiom,
    ! [A2: int,B2: int] :
      ( ( ( ( modulo_modulo_int @ A2 @ B2 )
          = zero_zero_int )
       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A2 ) @ B2 )
          = zero_zero_int ) )
      & ( ( ( modulo_modulo_int @ A2 @ B2 )
         != zero_zero_int )
       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A2 ) @ B2 )
          = ( minus_minus_int @ B2 @ ( modulo_modulo_int @ A2 @ B2 ) ) ) ) ) ).

% zmod_zminus1_eq_if
thf(fact_7975_zmod__zminus2__eq__if,axiom,
    ! [A2: int,B2: int] :
      ( ( ( ( modulo_modulo_int @ A2 @ B2 )
          = zero_zero_int )
       => ( ( modulo_modulo_int @ A2 @ ( uminus_uminus_int @ B2 ) )
          = zero_zero_int ) )
      & ( ( ( modulo_modulo_int @ A2 @ B2 )
         != zero_zero_int )
       => ( ( modulo_modulo_int @ A2 @ ( uminus_uminus_int @ B2 ) )
          = ( minus_minus_int @ ( modulo_modulo_int @ A2 @ B2 ) @ B2 ) ) ) ) ).

% zmod_zminus2_eq_if
thf(fact_7976_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N ) )
       => ~ ! [N: nat] :
              ( ( K
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% int_cases3
thf(fact_7977_not__zle__0__negative,axiom,
    ! [N3: nat] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ).

% not_zle_0_negative
thf(fact_7978_negD,axiom,
    ! [X: int] :
      ( ( ord_less_int @ X @ zero_zero_int )
     => ? [N: nat] :
          ( X
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ).

% negD
thf(fact_7979_negative__zless__0,axiom,
    ! [N3: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_7980_verit__less__mono__div__int2,axiom,
    ! [A: int,B: int,N3: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N3 ) )
       => ( ord_less_eq_int @ ( divide_divide_int @ B @ N3 ) @ ( divide_divide_int @ A @ N3 ) ) ) ) ).

% verit_less_mono_div_int2
thf(fact_7981_div__eq__minus1,axiom,
    ! [B2: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B2 )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% div_eq_minus1
thf(fact_7982_ceiling__divide__eq__div,axiom,
    ! [A2: int,B2: int] :
      ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A2 ) @ ( ring_1_of_int_real @ B2 ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A2 ) @ B2 ) ) ) ).

% ceiling_divide_eq_div
thf(fact_7983_ceiling__divide__eq__div,axiom,
    ! [A2: int,B2: int] :
      ( ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A2 ) @ ( ring_1_of_int_rat @ B2 ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A2 ) @ B2 ) ) ) ).

% ceiling_divide_eq_div
thf(fact_7984_neg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ~ ! [N: nat] :
            ( ( K
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% neg_int_cases
thf(fact_7985_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
        = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).

% nat_mult_distrib_neg
thf(fact_7986_minus__mod__int__eq,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L2 )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L2 )
        = ( minus_minus_int @ ( minus_minus_int @ L2 @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L2 ) ) ) ) ).

% minus_mod_int_eq
thf(fact_7987_zmod__minus1,axiom,
    ! [B2: int] :
      ( ( ord_less_int @ zero_zero_int @ B2 )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B2 )
        = ( minus_minus_int @ B2 @ one_one_int ) ) ) ).

% zmod_minus1
thf(fact_7988_zdiv__zminus2__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( ( ( modulo_modulo_int @ A2 @ B2 )
            = zero_zero_int )
         => ( ( divide_divide_int @ A2 @ ( uminus_uminus_int @ B2 ) )
            = ( uminus_uminus_int @ ( divide_divide_int @ A2 @ B2 ) ) ) )
        & ( ( ( modulo_modulo_int @ A2 @ B2 )
           != zero_zero_int )
         => ( ( divide_divide_int @ A2 @ ( uminus_uminus_int @ B2 ) )
            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A2 @ B2 ) ) @ one_one_int ) ) ) ) ) ).

% zdiv_zminus2_eq_if
thf(fact_7989_zdiv__zminus1__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( ( ( ( modulo_modulo_int @ A2 @ B2 )
            = zero_zero_int )
         => ( ( divide_divide_int @ ( uminus_uminus_int @ A2 ) @ B2 )
            = ( uminus_uminus_int @ ( divide_divide_int @ A2 @ B2 ) ) ) )
        & ( ( ( modulo_modulo_int @ A2 @ B2 )
           != zero_zero_int )
         => ( ( divide_divide_int @ ( uminus_uminus_int @ A2 ) @ B2 )
            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A2 @ B2 ) ) @ one_one_int ) ) ) ) ) ).

% zdiv_zminus1_eq_if
thf(fact_7990_zminus1__lemma,axiom,
    ! [A2: int,B2: int,Q3: int,R3: int] :
      ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair_int_int @ Q3 @ R3 ) )
     => ( ( B2 != zero_zero_int )
       => ( eucl_rel_int @ ( uminus_uminus_int @ A2 ) @ B2 @ ( product_Pair_int_int @ ( if_int @ ( R3 = zero_zero_int ) @ ( uminus_uminus_int @ Q3 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q3 ) @ one_one_int ) ) @ ( if_int @ ( R3 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B2 @ R3 ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_7991_minus__1__div__exp__eq__int,axiom,
    ! [N3: nat] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% minus_1_div_exp_eq_int
thf(fact_7992_div__pos__neg__trivial,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
       => ( ( divide_divide_int @ K @ L2 )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% div_pos_neg_trivial
thf(fact_7993_finite__has__minimal2,axiom,
    ! [A: set_real,A2: real] :
      ( ( finite_finite_real @ A )
     => ( ( member_real @ A2 @ A )
       => ? [X4: real] :
            ( ( member_real @ X4 @ A )
            & ( ord_less_eq_real @ X4 @ A2 )
            & ! [Xa2: real] :
                ( ( member_real @ Xa2 @ A )
               => ( ( ord_less_eq_real @ Xa2 @ X4 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_7994_finite__has__minimal2,axiom,
    ! [A: set_Code_integer,A2: code_integer] :
      ( ( finite6017078050557962740nteger @ A )
     => ( ( member_Code_integer @ A2 @ A )
       => ? [X4: code_integer] :
            ( ( member_Code_integer @ X4 @ A )
            & ( ord_le3102999989581377725nteger @ X4 @ A2 )
            & ! [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ A )
               => ( ( ord_le3102999989581377725nteger @ Xa2 @ X4 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_7995_finite__has__minimal2,axiom,
    ! [A: set_set_nat,A2: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( member_set_nat @ A2 @ A )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A )
            & ( ord_less_eq_set_nat @ X4 @ A2 )
            & ! [Xa2: set_nat] :
                ( ( member_set_nat @ Xa2 @ A )
               => ( ( ord_less_eq_set_nat @ Xa2 @ X4 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_7996_finite__has__minimal2,axiom,
    ! [A: set_rat,A2: rat] :
      ( ( finite_finite_rat @ A )
     => ( ( member_rat @ A2 @ A )
       => ? [X4: rat] :
            ( ( member_rat @ X4 @ A )
            & ( ord_less_eq_rat @ X4 @ A2 )
            & ! [Xa2: rat] :
                ( ( member_rat @ Xa2 @ A )
               => ( ( ord_less_eq_rat @ Xa2 @ X4 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_7997_finite__has__minimal2,axiom,
    ! [A: set_num,A2: num] :
      ( ( finite_finite_num @ A )
     => ( ( member_num @ A2 @ A )
       => ? [X4: num] :
            ( ( member_num @ X4 @ A )
            & ( ord_less_eq_num @ X4 @ A2 )
            & ! [Xa2: num] :
                ( ( member_num @ Xa2 @ A )
               => ( ( ord_less_eq_num @ Xa2 @ X4 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_7998_finite__has__minimal2,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A2 @ A )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A )
            & ( ord_less_eq_nat @ X4 @ A2 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A )
               => ( ( ord_less_eq_nat @ Xa2 @ X4 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_7999_finite__has__minimal2,axiom,
    ! [A: set_int,A2: int] :
      ( ( finite_finite_int @ A )
     => ( ( member_int @ A2 @ A )
       => ? [X4: int] :
            ( ( member_int @ X4 @ A )
            & ( ord_less_eq_int @ X4 @ A2 )
            & ! [Xa2: int] :
                ( ( member_int @ Xa2 @ A )
               => ( ( ord_less_eq_int @ Xa2 @ X4 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_8000_finite__has__maximal2,axiom,
    ! [A: set_real,A2: real] :
      ( ( finite_finite_real @ A )
     => ( ( member_real @ A2 @ A )
       => ? [X4: real] :
            ( ( member_real @ X4 @ A )
            & ( ord_less_eq_real @ A2 @ X4 )
            & ! [Xa2: real] :
                ( ( member_real @ Xa2 @ A )
               => ( ( ord_less_eq_real @ X4 @ Xa2 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_8001_finite__has__maximal2,axiom,
    ! [A: set_Code_integer,A2: code_integer] :
      ( ( finite6017078050557962740nteger @ A )
     => ( ( member_Code_integer @ A2 @ A )
       => ? [X4: code_integer] :
            ( ( member_Code_integer @ X4 @ A )
            & ( ord_le3102999989581377725nteger @ A2 @ X4 )
            & ! [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ A )
               => ( ( ord_le3102999989581377725nteger @ X4 @ Xa2 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_8002_finite__has__maximal2,axiom,
    ! [A: set_set_nat,A2: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( member_set_nat @ A2 @ A )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A )
            & ( ord_less_eq_set_nat @ A2 @ X4 )
            & ! [Xa2: set_nat] :
                ( ( member_set_nat @ Xa2 @ A )
               => ( ( ord_less_eq_set_nat @ X4 @ Xa2 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_8003_finite__has__maximal2,axiom,
    ! [A: set_rat,A2: rat] :
      ( ( finite_finite_rat @ A )
     => ( ( member_rat @ A2 @ A )
       => ? [X4: rat] :
            ( ( member_rat @ X4 @ A )
            & ( ord_less_eq_rat @ A2 @ X4 )
            & ! [Xa2: rat] :
                ( ( member_rat @ Xa2 @ A )
               => ( ( ord_less_eq_rat @ X4 @ Xa2 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_8004_finite__has__maximal2,axiom,
    ! [A: set_num,A2: num] :
      ( ( finite_finite_num @ A )
     => ( ( member_num @ A2 @ A )
       => ? [X4: num] :
            ( ( member_num @ X4 @ A )
            & ( ord_less_eq_num @ A2 @ X4 )
            & ! [Xa2: num] :
                ( ( member_num @ Xa2 @ A )
               => ( ( ord_less_eq_num @ X4 @ Xa2 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_8005_finite__has__maximal2,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A2 @ A )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A )
            & ( ord_less_eq_nat @ A2 @ X4 )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A )
               => ( ( ord_less_eq_nat @ X4 @ Xa2 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_8006_finite__has__maximal2,axiom,
    ! [A: set_int,A2: int] :
      ( ( finite_finite_int @ A )
     => ( ( member_int @ A2 @ A )
       => ? [X4: int] :
            ( ( member_int @ X4 @ A )
            & ( ord_less_eq_int @ A2 @ X4 )
            & ! [Xa2: int] :
                ( ( member_int @ Xa2 @ A )
               => ( ( ord_less_eq_int @ X4 @ Xa2 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_8007_finite_OemptyI,axiom,
    finite6017078050557962740nteger @ bot_bo3990330152332043303nteger ).

% finite.emptyI
thf(fact_8008_finite_OemptyI,axiom,
    finite3207457112153483333omplex @ bot_bot_set_complex ).

% finite.emptyI
thf(fact_8009_finite_OemptyI,axiom,
    finite_finite_real @ bot_bot_set_real ).

% finite.emptyI
thf(fact_8010_finite_OemptyI,axiom,
    finite_finite_o @ bot_bot_set_o ).

% finite.emptyI
thf(fact_8011_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_8012_finite_OemptyI,axiom,
    finite_finite_int @ bot_bot_set_int ).

% finite.emptyI
thf(fact_8013_infinite__imp__nonempty,axiom,
    ! [S: set_Code_integer] :
      ( ~ ( finite6017078050557962740nteger @ S )
     => ( S != bot_bo3990330152332043303nteger ) ) ).

% infinite_imp_nonempty
thf(fact_8014_infinite__imp__nonempty,axiom,
    ! [S: set_complex] :
      ( ~ ( finite3207457112153483333omplex @ S )
     => ( S != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
thf(fact_8015_infinite__imp__nonempty,axiom,
    ! [S: set_real] :
      ( ~ ( finite_finite_real @ S )
     => ( S != bot_bot_set_real ) ) ).

% infinite_imp_nonempty
thf(fact_8016_infinite__imp__nonempty,axiom,
    ! [S: set_o] :
      ( ~ ( finite_finite_o @ S )
     => ( S != bot_bot_set_o ) ) ).

% infinite_imp_nonempty
thf(fact_8017_infinite__imp__nonempty,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( S != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_8018_infinite__imp__nonempty,axiom,
    ! [S: set_int] :
      ( ~ ( finite_finite_int @ S )
     => ( S != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
thf(fact_8019_rev__finite__subset,axiom,
    ! [B: set_int,A: set_int] :
      ( ( finite_finite_int @ B )
     => ( ( ord_less_eq_set_int @ A @ B )
       => ( finite_finite_int @ A ) ) ) ).

% rev_finite_subset
thf(fact_8020_rev__finite__subset,axiom,
    ! [B: set_Code_integer,A: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ B )
     => ( ( ord_le7084787975880047091nteger @ A @ B )
       => ( finite6017078050557962740nteger @ A ) ) ) ).

% rev_finite_subset
thf(fact_8021_rev__finite__subset,axiom,
    ! [B: set_complex,A: set_complex] :
      ( ( finite3207457112153483333omplex @ B )
     => ( ( ord_le211207098394363844omplex @ A @ B )
       => ( finite3207457112153483333omplex @ A ) ) ) ).

% rev_finite_subset
thf(fact_8022_rev__finite__subset,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( finite_finite_nat @ A ) ) ) ).

% rev_finite_subset
thf(fact_8023_infinite__super,axiom,
    ! [S: set_int,T4: set_int] :
      ( ( ord_less_eq_set_int @ S @ T4 )
     => ( ~ ( finite_finite_int @ S )
       => ~ ( finite_finite_int @ T4 ) ) ) ).

% infinite_super
thf(fact_8024_infinite__super,axiom,
    ! [S: set_Code_integer,T4: set_Code_integer] :
      ( ( ord_le7084787975880047091nteger @ S @ T4 )
     => ( ~ ( finite6017078050557962740nteger @ S )
       => ~ ( finite6017078050557962740nteger @ T4 ) ) ) ).

% infinite_super
thf(fact_8025_infinite__super,axiom,
    ! [S: set_complex,T4: set_complex] :
      ( ( ord_le211207098394363844omplex @ S @ T4 )
     => ( ~ ( finite3207457112153483333omplex @ S )
       => ~ ( finite3207457112153483333omplex @ T4 ) ) ) ).

% infinite_super
thf(fact_8026_infinite__super,axiom,
    ! [S: set_nat,T4: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ T4 )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ T4 ) ) ) ).

% infinite_super
thf(fact_8027_finite__subset,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( finite_finite_int @ B )
       => ( finite_finite_int @ A ) ) ) ).

% finite_subset
thf(fact_8028_finite__subset,axiom,
    ! [A: set_Code_integer,B: set_Code_integer] :
      ( ( ord_le7084787975880047091nteger @ A @ B )
     => ( ( finite6017078050557962740nteger @ B )
       => ( finite6017078050557962740nteger @ A ) ) ) ).

% finite_subset
thf(fact_8029_finite__subset,axiom,
    ! [A: set_complex,B: set_complex] :
      ( ( ord_le211207098394363844omplex @ A @ B )
     => ( ( finite3207457112153483333omplex @ B )
       => ( finite3207457112153483333omplex @ A ) ) ) ).

% finite_subset
thf(fact_8030_finite__subset,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( finite_finite_nat @ B )
       => ( finite_finite_nat @ A ) ) ) ).

% finite_subset
thf(fact_8031_finite_OinsertI,axiom,
    ! [A: set_VEBT_VEBT,A2: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A )
     => ( finite5795047828879050333T_VEBT @ ( insert_VEBT_VEBT @ A2 @ A ) ) ) ).

% finite.insertI
thf(fact_8032_finite_OinsertI,axiom,
    ! [A: set_real,A2: real] :
      ( ( finite_finite_real @ A )
     => ( finite_finite_real @ ( insert_real @ A2 @ A ) ) ) ).

% finite.insertI
thf(fact_8033_finite_OinsertI,axiom,
    ! [A: set_o,A2: $o] :
      ( ( finite_finite_o @ A )
     => ( finite_finite_o @ ( insert_o @ A2 @ A ) ) ) ).

% finite.insertI
thf(fact_8034_finite_OinsertI,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( finite_finite_nat @ ( insert_nat @ A2 @ A ) ) ) ).

% finite.insertI
thf(fact_8035_finite_OinsertI,axiom,
    ! [A: set_int,A2: int] :
      ( ( finite_finite_int @ A )
     => ( finite_finite_int @ ( insert_int @ A2 @ A ) ) ) ).

% finite.insertI
thf(fact_8036_finite_OinsertI,axiom,
    ! [A: set_Code_integer,A2: code_integer] :
      ( ( finite6017078050557962740nteger @ A )
     => ( finite6017078050557962740nteger @ ( insert_Code_integer @ A2 @ A ) ) ) ).

% finite.insertI
thf(fact_8037_finite_OinsertI,axiom,
    ! [A: set_complex,A2: complex] :
      ( ( finite3207457112153483333omplex @ A )
     => ( finite3207457112153483333omplex @ ( insert_complex @ A2 @ A ) ) ) ).

% finite.insertI
thf(fact_8038_Diff__infinite__finite,axiom,
    ! [T4: set_int,S: set_int] :
      ( ( finite_finite_int @ T4 )
     => ( ~ ( finite_finite_int @ S )
       => ~ ( finite_finite_int @ ( minus_minus_set_int @ S @ T4 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_8039_Diff__infinite__finite,axiom,
    ! [T4: set_Code_integer,S: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ T4 )
     => ( ~ ( finite6017078050557962740nteger @ S )
       => ~ ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ S @ T4 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_8040_Diff__infinite__finite,axiom,
    ! [T4: set_complex,S: set_complex] :
      ( ( finite3207457112153483333omplex @ T4 )
     => ( ~ ( finite3207457112153483333omplex @ S )
       => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S @ T4 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_8041_Diff__infinite__finite,axiom,
    ! [T4: set_nat,S: set_nat] :
      ( ( finite_finite_nat @ T4 )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T4 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_8042_finite__psubset__induct,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A )
     => ( ! [A8: set_nat] :
            ( ( finite_finite_nat @ A8 )
           => ( ! [B9: set_nat] :
                  ( ( ord_less_set_nat @ B9 @ A8 )
                 => ( P @ B9 ) )
             => ( P @ A8 ) ) )
       => ( P @ A ) ) ) ).

% finite_psubset_induct
thf(fact_8043_finite__psubset__induct,axiom,
    ! [A: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A )
     => ( ! [A8: set_int] :
            ( ( finite_finite_int @ A8 )
           => ( ! [B9: set_int] :
                  ( ( ord_less_set_int @ B9 @ A8 )
                 => ( P @ B9 ) )
             => ( P @ A8 ) ) )
       => ( P @ A ) ) ) ).

% finite_psubset_induct
thf(fact_8044_finite__psubset__induct,axiom,
    ! [A: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A )
     => ( ! [A8: set_Code_integer] :
            ( ( finite6017078050557962740nteger @ A8 )
           => ( ! [B9: set_Code_integer] :
                  ( ( ord_le1307284697595431911nteger @ B9 @ A8 )
                 => ( P @ B9 ) )
             => ( P @ A8 ) ) )
       => ( P @ A ) ) ) ).

% finite_psubset_induct
thf(fact_8045_finite__psubset__induct,axiom,
    ! [A: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ A )
     => ( ! [A8: set_complex] :
            ( ( finite3207457112153483333omplex @ A8 )
           => ( ! [B9: set_complex] :
                  ( ( ord_less_set_complex @ B9 @ A8 )
                 => ( P @ B9 ) )
             => ( P @ A8 ) ) )
       => ( P @ A ) ) ) ).

% finite_psubset_induct
thf(fact_8046_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_8047_m1mod2k,axiom,
    ! [N3: nat] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ).

% m1mod2k
thf(fact_8048_sb__dec__lem_H,axiom,
    ! [K: nat,A2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) @ A2 )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ ( plus_plus_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ A2 ) ) ) ).

% sb_dec_lem'
thf(fact_8049_m1mod22k,axiom,
    ! [N3: nat] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) @ one_one_int ) ) ).

% m1mod22k
thf(fact_8050_sb__inc__lem_H,axiom,
    ! [A2: int,K: nat] :
      ( ( ord_less_int @ A2 @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ ( plus_plus_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ K ) ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ K ) ) ) ) ) ).

% sb_inc_lem'
thf(fact_8051_sb__dec__lem,axiom,
    ! [K: nat,A2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ A2 ) )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ ( plus_plus_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ A2 ) ) ) ).

% sb_dec_lem
thf(fact_8052_finite__has__minimal,axiom,
    ! [A: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A )
     => ( ( A != bot_bo3990330152332043303nteger )
       => ? [X4: code_integer] :
            ( ( member_Code_integer @ X4 @ A )
            & ! [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ A )
               => ( ( ord_le3102999989581377725nteger @ Xa2 @ X4 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_8053_finite__has__minimal,axiom,
    ! [A: set_real] :
      ( ( finite_finite_real @ A )
     => ( ( A != bot_bot_set_real )
       => ? [X4: real] :
            ( ( member_real @ X4 @ A )
            & ! [Xa2: real] :
                ( ( member_real @ Xa2 @ A )
               => ( ( ord_less_eq_real @ Xa2 @ X4 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_8054_finite__has__minimal,axiom,
    ! [A: set_o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ? [X4: $o] :
            ( ( member_o @ X4 @ A )
            & ! [Xa2: $o] :
                ( ( member_o @ Xa2 @ A )
               => ( ( ord_less_eq_o @ Xa2 @ X4 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_8055_finite__has__minimal,axiom,
    ! [A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A )
            & ! [Xa2: set_nat] :
                ( ( member_set_nat @ Xa2 @ A )
               => ( ( ord_less_eq_set_nat @ Xa2 @ X4 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_8056_finite__has__minimal,axiom,
    ! [A: set_rat] :
      ( ( finite_finite_rat @ A )
     => ( ( A != bot_bot_set_rat )
       => ? [X4: rat] :
            ( ( member_rat @ X4 @ A )
            & ! [Xa2: rat] :
                ( ( member_rat @ Xa2 @ A )
               => ( ( ord_less_eq_rat @ Xa2 @ X4 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_8057_finite__has__minimal,axiom,
    ! [A: set_num] :
      ( ( finite_finite_num @ A )
     => ( ( A != bot_bot_set_num )
       => ? [X4: num] :
            ( ( member_num @ X4 @ A )
            & ! [Xa2: num] :
                ( ( member_num @ Xa2 @ A )
               => ( ( ord_less_eq_num @ Xa2 @ X4 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_8058_finite__has__minimal,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A )
               => ( ( ord_less_eq_nat @ Xa2 @ X4 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_8059_finite__has__minimal,axiom,
    ! [A: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ? [X4: int] :
            ( ( member_int @ X4 @ A )
            & ! [Xa2: int] :
                ( ( member_int @ Xa2 @ A )
               => ( ( ord_less_eq_int @ Xa2 @ X4 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_8060_finite__has__maximal,axiom,
    ! [A: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A )
     => ( ( A != bot_bo3990330152332043303nteger )
       => ? [X4: code_integer] :
            ( ( member_Code_integer @ X4 @ A )
            & ! [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ A )
               => ( ( ord_le3102999989581377725nteger @ X4 @ Xa2 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_8061_finite__has__maximal,axiom,
    ! [A: set_real] :
      ( ( finite_finite_real @ A )
     => ( ( A != bot_bot_set_real )
       => ? [X4: real] :
            ( ( member_real @ X4 @ A )
            & ! [Xa2: real] :
                ( ( member_real @ Xa2 @ A )
               => ( ( ord_less_eq_real @ X4 @ Xa2 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_8062_finite__has__maximal,axiom,
    ! [A: set_o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ? [X4: $o] :
            ( ( member_o @ X4 @ A )
            & ! [Xa2: $o] :
                ( ( member_o @ Xa2 @ A )
               => ( ( ord_less_eq_o @ X4 @ Xa2 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_8063_finite__has__maximal,axiom,
    ! [A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A )
            & ! [Xa2: set_nat] :
                ( ( member_set_nat @ Xa2 @ A )
               => ( ( ord_less_eq_set_nat @ X4 @ Xa2 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_8064_finite__has__maximal,axiom,
    ! [A: set_rat] :
      ( ( finite_finite_rat @ A )
     => ( ( A != bot_bot_set_rat )
       => ? [X4: rat] :
            ( ( member_rat @ X4 @ A )
            & ! [Xa2: rat] :
                ( ( member_rat @ Xa2 @ A )
               => ( ( ord_less_eq_rat @ X4 @ Xa2 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_8065_finite__has__maximal,axiom,
    ! [A: set_num] :
      ( ( finite_finite_num @ A )
     => ( ( A != bot_bot_set_num )
       => ? [X4: num] :
            ( ( member_num @ X4 @ A )
            & ! [Xa2: num] :
                ( ( member_num @ Xa2 @ A )
               => ( ( ord_less_eq_num @ X4 @ Xa2 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_8066_finite__has__maximal,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A )
            & ! [Xa2: nat] :
                ( ( member_nat @ Xa2 @ A )
               => ( ( ord_less_eq_nat @ X4 @ Xa2 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_8067_finite__has__maximal,axiom,
    ! [A: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ? [X4: int] :
            ( ( member_int @ X4 @ A )
            & ! [Xa2: int] :
                ( ( member_int @ Xa2 @ A )
               => ( ( ord_less_eq_int @ X4 @ Xa2 )
                 => ( X4 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_8068_finite_Ocases,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( ( A2 != bot_bo8194388402131092736T_VEBT )
       => ~ ! [A8: set_VEBT_VEBT] :
              ( ? [A3: vEBT_VEBT] :
                  ( A2
                  = ( insert_VEBT_VEBT @ A3 @ A8 ) )
             => ~ ( finite5795047828879050333T_VEBT @ A8 ) ) ) ) ).

% finite.cases
thf(fact_8069_finite_Ocases,axiom,
    ! [A2: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A2 )
     => ( ( A2 != bot_bo3990330152332043303nteger )
       => ~ ! [A8: set_Code_integer] :
              ( ? [A3: code_integer] :
                  ( A2
                  = ( insert_Code_integer @ A3 @ A8 ) )
             => ~ ( finite6017078050557962740nteger @ A8 ) ) ) ) ).

% finite.cases
thf(fact_8070_finite_Ocases,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ~ ! [A8: set_complex] :
              ( ? [A3: complex] :
                  ( A2
                  = ( insert_complex @ A3 @ A8 ) )
             => ~ ( finite3207457112153483333omplex @ A8 ) ) ) ) ).

% finite.cases
thf(fact_8071_finite_Ocases,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ~ ! [A8: set_real] :
              ( ? [A3: real] :
                  ( A2
                  = ( insert_real @ A3 @ A8 ) )
             => ~ ( finite_finite_real @ A8 ) ) ) ) ).

% finite.cases
thf(fact_8072_finite_Ocases,axiom,
    ! [A2: set_o] :
      ( ( finite_finite_o @ A2 )
     => ( ( A2 != bot_bot_set_o )
       => ~ ! [A8: set_o] :
              ( ? [A3: $o] :
                  ( A2
                  = ( insert_o @ A3 @ A8 ) )
             => ~ ( finite_finite_o @ A8 ) ) ) ) ).

% finite.cases
thf(fact_8073_finite_Ocases,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ~ ! [A8: set_nat] :
              ( ? [A3: nat] :
                  ( A2
                  = ( insert_nat @ A3 @ A8 ) )
             => ~ ( finite_finite_nat @ A8 ) ) ) ) ).

% finite.cases
thf(fact_8074_finite_Ocases,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ~ ! [A8: set_int] :
              ( ? [A3: int] :
                  ( A2
                  = ( insert_int @ A3 @ A8 ) )
             => ~ ( finite_finite_int @ A8 ) ) ) ) ).

% finite.cases
thf(fact_8075_finite_Osimps,axiom,
    ( finite5795047828879050333T_VEBT
    = ( ^ [A7: set_VEBT_VEBT] :
          ( ( A7 = bot_bo8194388402131092736T_VEBT )
          | ? [A6: set_VEBT_VEBT,B7: vEBT_VEBT] :
              ( ( A7
                = ( insert_VEBT_VEBT @ B7 @ A6 ) )
              & ( finite5795047828879050333T_VEBT @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_8076_finite_Osimps,axiom,
    ( finite6017078050557962740nteger
    = ( ^ [A7: set_Code_integer] :
          ( ( A7 = bot_bo3990330152332043303nteger )
          | ? [A6: set_Code_integer,B7: code_integer] :
              ( ( A7
                = ( insert_Code_integer @ B7 @ A6 ) )
              & ( finite6017078050557962740nteger @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_8077_finite_Osimps,axiom,
    ( finite3207457112153483333omplex
    = ( ^ [A7: set_complex] :
          ( ( A7 = bot_bot_set_complex )
          | ? [A6: set_complex,B7: complex] :
              ( ( A7
                = ( insert_complex @ B7 @ A6 ) )
              & ( finite3207457112153483333omplex @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_8078_finite_Osimps,axiom,
    ( finite_finite_real
    = ( ^ [A7: set_real] :
          ( ( A7 = bot_bot_set_real )
          | ? [A6: set_real,B7: real] :
              ( ( A7
                = ( insert_real @ B7 @ A6 ) )
              & ( finite_finite_real @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_8079_finite_Osimps,axiom,
    ( finite_finite_o
    = ( ^ [A7: set_o] :
          ( ( A7 = bot_bot_set_o )
          | ? [A6: set_o,B7: $o] :
              ( ( A7
                = ( insert_o @ B7 @ A6 ) )
              & ( finite_finite_o @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_8080_finite_Osimps,axiom,
    ( finite_finite_nat
    = ( ^ [A7: set_nat] :
          ( ( A7 = bot_bot_set_nat )
          | ? [A6: set_nat,B7: nat] :
              ( ( A7
                = ( insert_nat @ B7 @ A6 ) )
              & ( finite_finite_nat @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_8081_finite_Osimps,axiom,
    ( finite_finite_int
    = ( ^ [A7: set_int] :
          ( ( A7 = bot_bot_set_int )
          | ? [A6: set_int,B7: int] :
              ( ( A7
                = ( insert_int @ B7 @ A6 ) )
              & ( finite_finite_int @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_8082_finite__induct,axiom,
    ! [F: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ F )
     => ( ( P @ bot_bo8194388402131092736T_VEBT )
       => ( ! [X4: vEBT_VEBT,F6: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ F6 )
             => ( ~ ( member_VEBT_VEBT @ X4 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_VEBT_VEBT @ X4 @ F6 ) ) ) ) )
         => ( P @ F ) ) ) ) ).

% finite_induct
thf(fact_8083_finite__induct,axiom,
    ! [F: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ F )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [X4: code_integer,F6: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ F6 )
             => ( ~ ( member_Code_integer @ X4 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_Code_integer @ X4 @ F6 ) ) ) ) )
         => ( P @ F ) ) ) ) ).

% finite_induct
thf(fact_8084_finite__induct,axiom,
    ! [F: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X4: complex,F6: set_complex] :
              ( ( finite3207457112153483333omplex @ F6 )
             => ( ~ ( member_complex @ X4 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_complex @ X4 @ F6 ) ) ) ) )
         => ( P @ F ) ) ) ) ).

% finite_induct
thf(fact_8085_finite__induct,axiom,
    ! [F: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X4: real,F6: set_real] :
              ( ( finite_finite_real @ F6 )
             => ( ~ ( member_real @ X4 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_real @ X4 @ F6 ) ) ) ) )
         => ( P @ F ) ) ) ) ).

% finite_induct
thf(fact_8086_finite__induct,axiom,
    ! [F: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X4: $o,F6: set_o] :
              ( ( finite_finite_o @ F6 )
             => ( ~ ( member_o @ X4 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_o @ X4 @ F6 ) ) ) ) )
         => ( P @ F ) ) ) ) ).

% finite_induct
thf(fact_8087_finite__induct,axiom,
    ! [F: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X4: nat,F6: set_nat] :
              ( ( finite_finite_nat @ F6 )
             => ( ~ ( member_nat @ X4 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_nat @ X4 @ F6 ) ) ) ) )
         => ( P @ F ) ) ) ) ).

% finite_induct
thf(fact_8088_finite__induct,axiom,
    ! [F: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X4: int,F6: set_int] :
              ( ( finite_finite_int @ F6 )
             => ( ~ ( member_int @ X4 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_int @ X4 @ F6 ) ) ) ) )
         => ( P @ F ) ) ) ) ).

% finite_induct
thf(fact_8089_finite__ne__induct,axiom,
    ! [F: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ F )
     => ( ( F != bot_bo8194388402131092736T_VEBT )
       => ( ! [X4: vEBT_VEBT] : ( P @ ( insert_VEBT_VEBT @ X4 @ bot_bo8194388402131092736T_VEBT ) )
         => ( ! [X4: vEBT_VEBT,F6: set_VEBT_VEBT] :
                ( ( finite5795047828879050333T_VEBT @ F6 )
               => ( ( F6 != bot_bo8194388402131092736T_VEBT )
                 => ( ~ ( member_VEBT_VEBT @ X4 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_VEBT_VEBT @ X4 @ F6 ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_ne_induct
thf(fact_8090_finite__ne__induct,axiom,
    ! [F: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ F )
     => ( ( F != bot_bo3990330152332043303nteger )
       => ( ! [X4: code_integer] : ( P @ ( insert_Code_integer @ X4 @ bot_bo3990330152332043303nteger ) )
         => ( ! [X4: code_integer,F6: set_Code_integer] :
                ( ( finite6017078050557962740nteger @ F6 )
               => ( ( F6 != bot_bo3990330152332043303nteger )
                 => ( ~ ( member_Code_integer @ X4 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_Code_integer @ X4 @ F6 ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_ne_induct
thf(fact_8091_finite__ne__induct,axiom,
    ! [F: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F )
     => ( ( F != bot_bot_set_complex )
       => ( ! [X4: complex] : ( P @ ( insert_complex @ X4 @ bot_bot_set_complex ) )
         => ( ! [X4: complex,F6: set_complex] :
                ( ( finite3207457112153483333omplex @ F6 )
               => ( ( F6 != bot_bot_set_complex )
                 => ( ~ ( member_complex @ X4 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_complex @ X4 @ F6 ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_ne_induct
thf(fact_8092_finite__ne__induct,axiom,
    ! [F: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F )
     => ( ( F != bot_bot_set_real )
       => ( ! [X4: real] : ( P @ ( insert_real @ X4 @ bot_bot_set_real ) )
         => ( ! [X4: real,F6: set_real] :
                ( ( finite_finite_real @ F6 )
               => ( ( F6 != bot_bot_set_real )
                 => ( ~ ( member_real @ X4 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_real @ X4 @ F6 ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_ne_induct
thf(fact_8093_finite__ne__induct,axiom,
    ! [F: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F )
     => ( ( F != bot_bot_set_o )
       => ( ! [X4: $o] : ( P @ ( insert_o @ X4 @ bot_bot_set_o ) )
         => ( ! [X4: $o,F6: set_o] :
                ( ( finite_finite_o @ F6 )
               => ( ( F6 != bot_bot_set_o )
                 => ( ~ ( member_o @ X4 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_o @ X4 @ F6 ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_ne_induct
thf(fact_8094_finite__ne__induct,axiom,
    ! [F: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F )
     => ( ( F != bot_bot_set_nat )
       => ( ! [X4: nat] : ( P @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
         => ( ! [X4: nat,F6: set_nat] :
                ( ( finite_finite_nat @ F6 )
               => ( ( F6 != bot_bot_set_nat )
                 => ( ~ ( member_nat @ X4 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_nat @ X4 @ F6 ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_ne_induct
thf(fact_8095_finite__ne__induct,axiom,
    ! [F: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F )
     => ( ( F != bot_bot_set_int )
       => ( ! [X4: int] : ( P @ ( insert_int @ X4 @ bot_bot_set_int ) )
         => ( ! [X4: int,F6: set_int] :
                ( ( finite_finite_int @ F6 )
               => ( ( F6 != bot_bot_set_int )
                 => ( ~ ( member_int @ X4 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_int @ X4 @ F6 ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_ne_induct
thf(fact_8096_infinite__finite__induct,axiom,
    ! [P: set_VEBT_VEBT > $o,A: set_VEBT_VEBT] :
      ( ! [A8: set_VEBT_VEBT] :
          ( ~ ( finite5795047828879050333T_VEBT @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bo8194388402131092736T_VEBT )
       => ( ! [X4: vEBT_VEBT,F6: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ F6 )
             => ( ~ ( member_VEBT_VEBT @ X4 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_VEBT_VEBT @ X4 @ F6 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_8097_infinite__finite__induct,axiom,
    ! [P: set_Code_integer > $o,A: set_Code_integer] :
      ( ! [A8: set_Code_integer] :
          ( ~ ( finite6017078050557962740nteger @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [X4: code_integer,F6: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ F6 )
             => ( ~ ( member_Code_integer @ X4 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_Code_integer @ X4 @ F6 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_8098_infinite__finite__induct,axiom,
    ! [P: set_complex > $o,A: set_complex] :
      ( ! [A8: set_complex] :
          ( ~ ( finite3207457112153483333omplex @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X4: complex,F6: set_complex] :
              ( ( finite3207457112153483333omplex @ F6 )
             => ( ~ ( member_complex @ X4 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_complex @ X4 @ F6 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_8099_infinite__finite__induct,axiom,
    ! [P: set_real > $o,A: set_real] :
      ( ! [A8: set_real] :
          ( ~ ( finite_finite_real @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X4: real,F6: set_real] :
              ( ( finite_finite_real @ F6 )
             => ( ~ ( member_real @ X4 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_real @ X4 @ F6 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_8100_infinite__finite__induct,axiom,
    ! [P: set_o > $o,A: set_o] :
      ( ! [A8: set_o] :
          ( ~ ( finite_finite_o @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X4: $o,F6: set_o] :
              ( ( finite_finite_o @ F6 )
             => ( ~ ( member_o @ X4 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_o @ X4 @ F6 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_8101_infinite__finite__induct,axiom,
    ! [P: set_nat > $o,A: set_nat] :
      ( ! [A8: set_nat] :
          ( ~ ( finite_finite_nat @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X4: nat,F6: set_nat] :
              ( ( finite_finite_nat @ F6 )
             => ( ~ ( member_nat @ X4 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_nat @ X4 @ F6 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_8102_infinite__finite__induct,axiom,
    ! [P: set_int > $o,A: set_int] :
      ( ! [A8: set_int] :
          ( ~ ( finite_finite_int @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X4: int,F6: set_int] :
              ( ( finite_finite_int @ F6 )
             => ( ~ ( member_int @ X4 @ F6 )
               => ( ( P @ F6 )
                 => ( P @ ( insert_int @ X4 @ F6 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_8103_finite__subset__induct,axiom,
    ! [F: set_VEBT_VEBT,A: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ F )
     => ( ( ord_le4337996190870823476T_VEBT @ F @ A )
       => ( ( P @ bot_bo8194388402131092736T_VEBT )
         => ( ! [A3: vEBT_VEBT,F6: set_VEBT_VEBT] :
                ( ( finite5795047828879050333T_VEBT @ F6 )
               => ( ( member_VEBT_VEBT @ A3 @ A )
                 => ( ~ ( member_VEBT_VEBT @ A3 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_VEBT_VEBT @ A3 @ F6 ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_subset_induct
thf(fact_8104_finite__subset__induct,axiom,
    ! [F: set_Code_integer,A: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ F )
     => ( ( ord_le7084787975880047091nteger @ F @ A )
       => ( ( P @ bot_bo3990330152332043303nteger )
         => ( ! [A3: code_integer,F6: set_Code_integer] :
                ( ( finite6017078050557962740nteger @ F6 )
               => ( ( member_Code_integer @ A3 @ A )
                 => ( ~ ( member_Code_integer @ A3 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_Code_integer @ A3 @ F6 ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_subset_induct
thf(fact_8105_finite__subset__induct,axiom,
    ! [F: set_complex,A: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F )
     => ( ( ord_le211207098394363844omplex @ F @ A )
       => ( ( P @ bot_bot_set_complex )
         => ( ! [A3: complex,F6: set_complex] :
                ( ( finite3207457112153483333omplex @ F6 )
               => ( ( member_complex @ A3 @ A )
                 => ( ~ ( member_complex @ A3 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_complex @ A3 @ F6 ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_subset_induct
thf(fact_8106_finite__subset__induct,axiom,
    ! [F: set_real,A: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F )
     => ( ( ord_less_eq_set_real @ F @ A )
       => ( ( P @ bot_bot_set_real )
         => ( ! [A3: real,F6: set_real] :
                ( ( finite_finite_real @ F6 )
               => ( ( member_real @ A3 @ A )
                 => ( ~ ( member_real @ A3 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_real @ A3 @ F6 ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_subset_induct
thf(fact_8107_finite__subset__induct,axiom,
    ! [F: set_o,A: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F )
     => ( ( ord_less_eq_set_o @ F @ A )
       => ( ( P @ bot_bot_set_o )
         => ( ! [A3: $o,F6: set_o] :
                ( ( finite_finite_o @ F6 )
               => ( ( member_o @ A3 @ A )
                 => ( ~ ( member_o @ A3 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_o @ A3 @ F6 ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_subset_induct
thf(fact_8108_finite__subset__induct,axiom,
    ! [F: set_int,A: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F )
     => ( ( ord_less_eq_set_int @ F @ A )
       => ( ( P @ bot_bot_set_int )
         => ( ! [A3: int,F6: set_int] :
                ( ( finite_finite_int @ F6 )
               => ( ( member_int @ A3 @ A )
                 => ( ~ ( member_int @ A3 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_int @ A3 @ F6 ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_subset_induct
thf(fact_8109_finite__subset__induct,axiom,
    ! [F: set_nat,A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F )
     => ( ( ord_less_eq_set_nat @ F @ A )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A3: nat,F6: set_nat] :
                ( ( finite_finite_nat @ F6 )
               => ( ( member_nat @ A3 @ A )
                 => ( ~ ( member_nat @ A3 @ F6 )
                   => ( ( P @ F6 )
                     => ( P @ ( insert_nat @ A3 @ F6 ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_subset_induct
thf(fact_8110_finite__subset__induct_H,axiom,
    ! [F: set_VEBT_VEBT,A: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ F )
     => ( ( ord_le4337996190870823476T_VEBT @ F @ A )
       => ( ( P @ bot_bo8194388402131092736T_VEBT )
         => ( ! [A3: vEBT_VEBT,F6: set_VEBT_VEBT] :
                ( ( finite5795047828879050333T_VEBT @ F6 )
               => ( ( member_VEBT_VEBT @ A3 @ A )
                 => ( ( ord_le4337996190870823476T_VEBT @ F6 @ A )
                   => ( ~ ( member_VEBT_VEBT @ A3 @ F6 )
                     => ( ( P @ F6 )
                       => ( P @ ( insert_VEBT_VEBT @ A3 @ F6 ) ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_8111_finite__subset__induct_H,axiom,
    ! [F: set_Code_integer,A: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ F )
     => ( ( ord_le7084787975880047091nteger @ F @ A )
       => ( ( P @ bot_bo3990330152332043303nteger )
         => ( ! [A3: code_integer,F6: set_Code_integer] :
                ( ( finite6017078050557962740nteger @ F6 )
               => ( ( member_Code_integer @ A3 @ A )
                 => ( ( ord_le7084787975880047091nteger @ F6 @ A )
                   => ( ~ ( member_Code_integer @ A3 @ F6 )
                     => ( ( P @ F6 )
                       => ( P @ ( insert_Code_integer @ A3 @ F6 ) ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_8112_finite__subset__induct_H,axiom,
    ! [F: set_complex,A: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F )
     => ( ( ord_le211207098394363844omplex @ F @ A )
       => ( ( P @ bot_bot_set_complex )
         => ( ! [A3: complex,F6: set_complex] :
                ( ( finite3207457112153483333omplex @ F6 )
               => ( ( member_complex @ A3 @ A )
                 => ( ( ord_le211207098394363844omplex @ F6 @ A )
                   => ( ~ ( member_complex @ A3 @ F6 )
                     => ( ( P @ F6 )
                       => ( P @ ( insert_complex @ A3 @ F6 ) ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_8113_finite__subset__induct_H,axiom,
    ! [F: set_real,A: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F )
     => ( ( ord_less_eq_set_real @ F @ A )
       => ( ( P @ bot_bot_set_real )
         => ( ! [A3: real,F6: set_real] :
                ( ( finite_finite_real @ F6 )
               => ( ( member_real @ A3 @ A )
                 => ( ( ord_less_eq_set_real @ F6 @ A )
                   => ( ~ ( member_real @ A3 @ F6 )
                     => ( ( P @ F6 )
                       => ( P @ ( insert_real @ A3 @ F6 ) ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_8114_finite__subset__induct_H,axiom,
    ! [F: set_o,A: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F )
     => ( ( ord_less_eq_set_o @ F @ A )
       => ( ( P @ bot_bot_set_o )
         => ( ! [A3: $o,F6: set_o] :
                ( ( finite_finite_o @ F6 )
               => ( ( member_o @ A3 @ A )
                 => ( ( ord_less_eq_set_o @ F6 @ A )
                   => ( ~ ( member_o @ A3 @ F6 )
                     => ( ( P @ F6 )
                       => ( P @ ( insert_o @ A3 @ F6 ) ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_8115_finite__subset__induct_H,axiom,
    ! [F: set_int,A: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F )
     => ( ( ord_less_eq_set_int @ F @ A )
       => ( ( P @ bot_bot_set_int )
         => ( ! [A3: int,F6: set_int] :
                ( ( finite_finite_int @ F6 )
               => ( ( member_int @ A3 @ A )
                 => ( ( ord_less_eq_set_int @ F6 @ A )
                   => ( ~ ( member_int @ A3 @ F6 )
                     => ( ( P @ F6 )
                       => ( P @ ( insert_int @ A3 @ F6 ) ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_8116_finite__subset__induct_H,axiom,
    ! [F: set_nat,A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F )
     => ( ( ord_less_eq_set_nat @ F @ A )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A3: nat,F6: set_nat] :
                ( ( finite_finite_nat @ F6 )
               => ( ( member_nat @ A3 @ A )
                 => ( ( ord_less_eq_set_nat @ F6 @ A )
                   => ( ~ ( member_nat @ A3 @ F6 )
                     => ( ( P @ F6 )
                       => ( P @ ( insert_nat @ A3 @ F6 ) ) ) ) ) ) )
           => ( P @ F ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_8117_infinite__remove,axiom,
    ! [S: set_VEBT_VEBT,A2: vEBT_VEBT] :
      ( ~ ( finite5795047828879050333T_VEBT @ S )
     => ~ ( finite5795047828879050333T_VEBT @ ( minus_5127226145743854075T_VEBT @ S @ ( insert_VEBT_VEBT @ A2 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% infinite_remove
thf(fact_8118_infinite__remove,axiom,
    ! [S: set_Code_integer,A2: code_integer] :
      ( ~ ( finite6017078050557962740nteger @ S )
     => ~ ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ S @ ( insert_Code_integer @ A2 @ bot_bo3990330152332043303nteger ) ) ) ) ).

% infinite_remove
thf(fact_8119_infinite__remove,axiom,
    ! [S: set_complex,A2: complex] :
      ( ~ ( finite3207457112153483333omplex @ S )
     => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ).

% infinite_remove
thf(fact_8120_infinite__remove,axiom,
    ! [S: set_real,A2: real] :
      ( ~ ( finite_finite_real @ S )
     => ~ ( finite_finite_real @ ( minus_minus_set_real @ S @ ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ).

% infinite_remove
thf(fact_8121_infinite__remove,axiom,
    ! [S: set_o,A2: $o] :
      ( ~ ( finite_finite_o @ S )
     => ~ ( finite_finite_o @ ( minus_minus_set_o @ S @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).

% infinite_remove
thf(fact_8122_infinite__remove,axiom,
    ! [S: set_int,A2: int] :
      ( ~ ( finite_finite_int @ S )
     => ~ ( finite_finite_int @ ( minus_minus_set_int @ S @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ).

% infinite_remove
thf(fact_8123_infinite__remove,axiom,
    ! [S: set_nat,A2: nat] :
      ( ~ ( finite_finite_nat @ S )
     => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).

% infinite_remove
thf(fact_8124_infinite__coinduct,axiom,
    ! [X2: set_VEBT_VEBT > $o,A: set_VEBT_VEBT] :
      ( ( X2 @ A )
     => ( ! [A8: set_VEBT_VEBT] :
            ( ( X2 @ A8 )
           => ? [X5: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X5 @ A8 )
                & ( ( X2 @ ( minus_5127226145743854075T_VEBT @ A8 @ ( insert_VEBT_VEBT @ X5 @ bot_bo8194388402131092736T_VEBT ) ) )
                  | ~ ( finite5795047828879050333T_VEBT @ ( minus_5127226145743854075T_VEBT @ A8 @ ( insert_VEBT_VEBT @ X5 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
       => ~ ( finite5795047828879050333T_VEBT @ A ) ) ) ).

% infinite_coinduct
thf(fact_8125_infinite__coinduct,axiom,
    ! [X2: set_Code_integer > $o,A: set_Code_integer] :
      ( ( X2 @ A )
     => ( ! [A8: set_Code_integer] :
            ( ( X2 @ A8 )
           => ? [X5: code_integer] :
                ( ( member_Code_integer @ X5 @ A8 )
                & ( ( X2 @ ( minus_2355218937544613996nteger @ A8 @ ( insert_Code_integer @ X5 @ bot_bo3990330152332043303nteger ) ) )
                  | ~ ( finite6017078050557962740nteger @ ( minus_2355218937544613996nteger @ A8 @ ( insert_Code_integer @ X5 @ bot_bo3990330152332043303nteger ) ) ) ) ) )
       => ~ ( finite6017078050557962740nteger @ A ) ) ) ).

% infinite_coinduct
thf(fact_8126_infinite__coinduct,axiom,
    ! [X2: set_complex > $o,A: set_complex] :
      ( ( X2 @ A )
     => ( ! [A8: set_complex] :
            ( ( X2 @ A8 )
           => ? [X5: complex] :
                ( ( member_complex @ X5 @ A8 )
                & ( ( X2 @ ( minus_811609699411566653omplex @ A8 @ ( insert_complex @ X5 @ bot_bot_set_complex ) ) )
                  | ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A8 @ ( insert_complex @ X5 @ bot_bot_set_complex ) ) ) ) ) )
       => ~ ( finite3207457112153483333omplex @ A ) ) ) ).

% infinite_coinduct
thf(fact_8127_infinite__coinduct,axiom,
    ! [X2: set_real > $o,A: set_real] :
      ( ( X2 @ A )
     => ( ! [A8: set_real] :
            ( ( X2 @ A8 )
           => ? [X5: real] :
                ( ( member_real @ X5 @ A8 )
                & ( ( X2 @ ( minus_minus_set_real @ A8 @ ( insert_real @ X5 @ bot_bot_set_real ) ) )
                  | ~ ( finite_finite_real @ ( minus_minus_set_real @ A8 @ ( insert_real @ X5 @ bot_bot_set_real ) ) ) ) ) )
       => ~ ( finite_finite_real @ A ) ) ) ).

% infinite_coinduct
thf(fact_8128_infinite__coinduct,axiom,
    ! [X2: set_o > $o,A: set_o] :
      ( ( X2 @ A )
     => ( ! [A8: set_o] :
            ( ( X2 @ A8 )
           => ? [X5: $o] :
                ( ( member_o @ X5 @ A8 )
                & ( ( X2 @ ( minus_minus_set_o @ A8 @ ( insert_o @ X5 @ bot_bot_set_o ) ) )
                  | ~ ( finite_finite_o @ ( minus_minus_set_o @ A8 @ ( insert_o @ X5 @ bot_bot_set_o ) ) ) ) ) )
       => ~ ( finite_finite_o @ A ) ) ) ).

% infinite_coinduct
thf(fact_8129_infinite__coinduct,axiom,
    ! [X2: set_int > $o,A: set_int] :
      ( ( X2 @ A )
     => ( ! [A8: set_int] :
            ( ( X2 @ A8 )
           => ? [X5: int] :
                ( ( member_int @ X5 @ A8 )
                & ( ( X2 @ ( minus_minus_set_int @ A8 @ ( insert_int @ X5 @ bot_bot_set_int ) ) )
                  | ~ ( finite_finite_int @ ( minus_minus_set_int @ A8 @ ( insert_int @ X5 @ bot_bot_set_int ) ) ) ) ) )
       => ~ ( finite_finite_int @ A ) ) ) ).

% infinite_coinduct
thf(fact_8130_infinite__coinduct,axiom,
    ! [X2: set_nat > $o,A: set_nat] :
      ( ( X2 @ A )
     => ( ! [A8: set_nat] :
            ( ( X2 @ A8 )
           => ? [X5: nat] :
                ( ( member_nat @ X5 @ A8 )
                & ( ( X2 @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
                  | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) )
       => ~ ( finite_finite_nat @ A ) ) ) ).

% infinite_coinduct
thf(fact_8131_finite__empty__induct,axiom,
    ! [A: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ A )
     => ( ( P @ A )
       => ( ! [A3: vEBT_VEBT,A8: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ A8 )
             => ( ( member_VEBT_VEBT @ A3 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_5127226145743854075T_VEBT @ A8 @ ( insert_VEBT_VEBT @ A3 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
         => ( P @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% finite_empty_induct
thf(fact_8132_finite__empty__induct,axiom,
    ! [A: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A )
     => ( ( P @ A )
       => ( ! [A3: code_integer,A8: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A8 )
             => ( ( member_Code_integer @ A3 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_2355218937544613996nteger @ A8 @ ( insert_Code_integer @ A3 @ bot_bo3990330152332043303nteger ) ) ) ) ) )
         => ( P @ bot_bo3990330152332043303nteger ) ) ) ) ).

% finite_empty_induct
thf(fact_8133_finite__empty__induct,axiom,
    ! [A: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ A )
     => ( ( P @ A )
       => ( ! [A3: complex,A8: set_complex] :
              ( ( finite3207457112153483333omplex @ A8 )
             => ( ( member_complex @ A3 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_811609699411566653omplex @ A8 @ ( insert_complex @ A3 @ bot_bot_set_complex ) ) ) ) ) )
         => ( P @ bot_bot_set_complex ) ) ) ) ).

% finite_empty_induct
thf(fact_8134_finite__empty__induct,axiom,
    ! [A: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A )
     => ( ( P @ A )
       => ( ! [A3: real,A8: set_real] :
              ( ( finite_finite_real @ A8 )
             => ( ( member_real @ A3 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_minus_set_real @ A8 @ ( insert_real @ A3 @ bot_bot_set_real ) ) ) ) ) )
         => ( P @ bot_bot_set_real ) ) ) ) ).

% finite_empty_induct
thf(fact_8135_finite__empty__induct,axiom,
    ! [A: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ A )
     => ( ( P @ A )
       => ( ! [A3: $o,A8: set_o] :
              ( ( finite_finite_o @ A8 )
             => ( ( member_o @ A3 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_minus_set_o @ A8 @ ( insert_o @ A3 @ bot_bot_set_o ) ) ) ) ) )
         => ( P @ bot_bot_set_o ) ) ) ) ).

% finite_empty_induct
thf(fact_8136_finite__empty__induct,axiom,
    ! [A: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A )
     => ( ( P @ A )
       => ( ! [A3: int,A8: set_int] :
              ( ( finite_finite_int @ A8 )
             => ( ( member_int @ A3 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_minus_set_int @ A8 @ ( insert_int @ A3 @ bot_bot_set_int ) ) ) ) ) )
         => ( P @ bot_bot_set_int ) ) ) ) ).

% finite_empty_induct
thf(fact_8137_finite__empty__induct,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A )
     => ( ( P @ A )
       => ( ! [A3: nat,A8: set_nat] :
              ( ( finite_finite_nat @ A8 )
             => ( ( member_nat @ A3 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ) )
         => ( P @ bot_bot_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_8138_remove__induct,axiom,
    ! [P: set_VEBT_VEBT > $o,B: set_VEBT_VEBT] :
      ( ( P @ bot_bo8194388402131092736T_VEBT )
     => ( ( ~ ( finite5795047828879050333T_VEBT @ B )
         => ( P @ B ) )
       => ( ! [A8: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ A8 )
             => ( ( A8 != bot_bo8194388402131092736T_VEBT )
               => ( ( ord_le4337996190870823476T_VEBT @ A8 @ B )
                 => ( ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ A8 )
                       => ( P @ ( minus_5127226145743854075T_VEBT @ A8 @ ( insert_VEBT_VEBT @ X5 @ bot_bo8194388402131092736T_VEBT ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_8139_remove__induct,axiom,
    ! [P: set_Code_integer > $o,B: set_Code_integer] :
      ( ( P @ bot_bo3990330152332043303nteger )
     => ( ( ~ ( finite6017078050557962740nteger @ B )
         => ( P @ B ) )
       => ( ! [A8: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A8 )
             => ( ( A8 != bot_bo3990330152332043303nteger )
               => ( ( ord_le7084787975880047091nteger @ A8 @ B )
                 => ( ! [X5: code_integer] :
                        ( ( member_Code_integer @ X5 @ A8 )
                       => ( P @ ( minus_2355218937544613996nteger @ A8 @ ( insert_Code_integer @ X5 @ bot_bo3990330152332043303nteger ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_8140_remove__induct,axiom,
    ! [P: set_complex > $o,B: set_complex] :
      ( ( P @ bot_bot_set_complex )
     => ( ( ~ ( finite3207457112153483333omplex @ B )
         => ( P @ B ) )
       => ( ! [A8: set_complex] :
              ( ( finite3207457112153483333omplex @ A8 )
             => ( ( A8 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A8 @ B )
                 => ( ! [X5: complex] :
                        ( ( member_complex @ X5 @ A8 )
                       => ( P @ ( minus_811609699411566653omplex @ A8 @ ( insert_complex @ X5 @ bot_bot_set_complex ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_8141_remove__induct,axiom,
    ! [P: set_real > $o,B: set_real] :
      ( ( P @ bot_bot_set_real )
     => ( ( ~ ( finite_finite_real @ B )
         => ( P @ B ) )
       => ( ! [A8: set_real] :
              ( ( finite_finite_real @ A8 )
             => ( ( A8 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A8 @ B )
                 => ( ! [X5: real] :
                        ( ( member_real @ X5 @ A8 )
                       => ( P @ ( minus_minus_set_real @ A8 @ ( insert_real @ X5 @ bot_bot_set_real ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_8142_remove__induct,axiom,
    ! [P: set_o > $o,B: set_o] :
      ( ( P @ bot_bot_set_o )
     => ( ( ~ ( finite_finite_o @ B )
         => ( P @ B ) )
       => ( ! [A8: set_o] :
              ( ( finite_finite_o @ A8 )
             => ( ( A8 != bot_bot_set_o )
               => ( ( ord_less_eq_set_o @ A8 @ B )
                 => ( ! [X5: $o] :
                        ( ( member_o @ X5 @ A8 )
                       => ( P @ ( minus_minus_set_o @ A8 @ ( insert_o @ X5 @ bot_bot_set_o ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_8143_remove__induct,axiom,
    ! [P: set_int > $o,B: set_int] :
      ( ( P @ bot_bot_set_int )
     => ( ( ~ ( finite_finite_int @ B )
         => ( P @ B ) )
       => ( ! [A8: set_int] :
              ( ( finite_finite_int @ A8 )
             => ( ( A8 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A8 @ B )
                 => ( ! [X5: int] :
                        ( ( member_int @ X5 @ A8 )
                       => ( P @ ( minus_minus_set_int @ A8 @ ( insert_int @ X5 @ bot_bot_set_int ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_8144_remove__induct,axiom,
    ! [P: set_nat > $o,B: set_nat] :
      ( ( P @ bot_bot_set_nat )
     => ( ( ~ ( finite_finite_nat @ B )
         => ( P @ B ) )
       => ( ! [A8: set_nat] :
              ( ( finite_finite_nat @ A8 )
             => ( ( A8 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A8 @ B )
                 => ( ! [X5: nat] :
                        ( ( member_nat @ X5 @ A8 )
                       => ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% remove_induct
thf(fact_8145_finite__remove__induct,axiom,
    ! [B: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ B )
     => ( ( P @ bot_bo8194388402131092736T_VEBT )
       => ( ! [A8: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ A8 )
             => ( ( A8 != bot_bo8194388402131092736T_VEBT )
               => ( ( ord_le4337996190870823476T_VEBT @ A8 @ B )
                 => ( ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ A8 )
                       => ( P @ ( minus_5127226145743854075T_VEBT @ A8 @ ( insert_VEBT_VEBT @ X5 @ bot_bo8194388402131092736T_VEBT ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_8146_finite__remove__induct,axiom,
    ! [B: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ B )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [A8: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A8 )
             => ( ( A8 != bot_bo3990330152332043303nteger )
               => ( ( ord_le7084787975880047091nteger @ A8 @ B )
                 => ( ! [X5: code_integer] :
                        ( ( member_Code_integer @ X5 @ A8 )
                       => ( P @ ( minus_2355218937544613996nteger @ A8 @ ( insert_Code_integer @ X5 @ bot_bo3990330152332043303nteger ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_8147_finite__remove__induct,axiom,
    ! [B: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ B )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [A8: set_complex] :
              ( ( finite3207457112153483333omplex @ A8 )
             => ( ( A8 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A8 @ B )
                 => ( ! [X5: complex] :
                        ( ( member_complex @ X5 @ A8 )
                       => ( P @ ( minus_811609699411566653omplex @ A8 @ ( insert_complex @ X5 @ bot_bot_set_complex ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_8148_finite__remove__induct,axiom,
    ! [B: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ B )
     => ( ( P @ bot_bot_set_real )
       => ( ! [A8: set_real] :
              ( ( finite_finite_real @ A8 )
             => ( ( A8 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A8 @ B )
                 => ( ! [X5: real] :
                        ( ( member_real @ X5 @ A8 )
                       => ( P @ ( minus_minus_set_real @ A8 @ ( insert_real @ X5 @ bot_bot_set_real ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_8149_finite__remove__induct,axiom,
    ! [B: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ B )
     => ( ( P @ bot_bot_set_o )
       => ( ! [A8: set_o] :
              ( ( finite_finite_o @ A8 )
             => ( ( A8 != bot_bot_set_o )
               => ( ( ord_less_eq_set_o @ A8 @ B )
                 => ( ! [X5: $o] :
                        ( ( member_o @ X5 @ A8 )
                       => ( P @ ( minus_minus_set_o @ A8 @ ( insert_o @ X5 @ bot_bot_set_o ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_8150_finite__remove__induct,axiom,
    ! [B: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ B )
     => ( ( P @ bot_bot_set_int )
       => ( ! [A8: set_int] :
              ( ( finite_finite_int @ A8 )
             => ( ( A8 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A8 @ B )
                 => ( ! [X5: int] :
                        ( ( member_int @ X5 @ A8 )
                       => ( P @ ( minus_minus_set_int @ A8 @ ( insert_int @ X5 @ bot_bot_set_int ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_8151_finite__remove__induct,axiom,
    ! [B: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ B )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [A8: set_nat] :
              ( ( finite_finite_nat @ A8 )
             => ( ( A8 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A8 @ B )
                 => ( ! [X5: nat] :
                        ( ( member_nat @ X5 @ A8 )
                       => ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B ) ) ) ) ).

% finite_remove_induct
thf(fact_8152_finite__induct__select,axiom,
    ! [S: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ S )
     => ( ( P @ bot_bo8194388402131092736T_VEBT )
       => ( ! [T5: set_VEBT_VEBT] :
              ( ( ord_le3480810397992357184T_VEBT @ T5 @ S )
             => ( ( P @ T5 )
               => ? [X5: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X5 @ ( minus_5127226145743854075T_VEBT @ S @ T5 ) )
                    & ( P @ ( insert_VEBT_VEBT @ X5 @ T5 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_8153_finite__induct__select,axiom,
    ! [S: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [T5: set_Code_integer] :
              ( ( ord_le1307284697595431911nteger @ T5 @ S )
             => ( ( P @ T5 )
               => ? [X5: code_integer] :
                    ( ( member_Code_integer @ X5 @ ( minus_2355218937544613996nteger @ S @ T5 ) )
                    & ( P @ ( insert_Code_integer @ X5 @ T5 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_8154_finite__induct__select,axiom,
    ! [S: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [T5: set_complex] :
              ( ( ord_less_set_complex @ T5 @ S )
             => ( ( P @ T5 )
               => ? [X5: complex] :
                    ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ S @ T5 ) )
                    & ( P @ ( insert_complex @ X5 @ T5 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_8155_finite__induct__select,axiom,
    ! [S: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ S )
     => ( ( P @ bot_bot_set_real )
       => ( ! [T5: set_real] :
              ( ( ord_less_set_real @ T5 @ S )
             => ( ( P @ T5 )
               => ? [X5: real] :
                    ( ( member_real @ X5 @ ( minus_minus_set_real @ S @ T5 ) )
                    & ( P @ ( insert_real @ X5 @ T5 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_8156_finite__induct__select,axiom,
    ! [S: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ S )
     => ( ( P @ bot_bot_set_o )
       => ( ! [T5: set_o] :
              ( ( ord_less_set_o @ T5 @ S )
             => ( ( P @ T5 )
               => ? [X5: $o] :
                    ( ( member_o @ X5 @ ( minus_minus_set_o @ S @ T5 ) )
                    & ( P @ ( insert_o @ X5 @ T5 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_8157_finite__induct__select,axiom,
    ! [S: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ S )
     => ( ( P @ bot_bot_set_int )
       => ( ! [T5: set_int] :
              ( ( ord_less_set_int @ T5 @ S )
             => ( ( P @ T5 )
               => ? [X5: int] :
                    ( ( member_int @ X5 @ ( minus_minus_set_int @ S @ T5 ) )
                    & ( P @ ( insert_int @ X5 @ T5 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_8158_finite__induct__select,axiom,
    ! [S: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ S )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [T5: set_nat] :
              ( ( ord_less_set_nat @ T5 @ S )
             => ( ( P @ T5 )
               => ? [X5: nat] :
                    ( ( member_nat @ X5 @ ( minus_minus_set_nat @ S @ T5 ) )
                    & ( P @ ( insert_nat @ X5 @ T5 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_8159_minus__one__div__numeral,axiom,
    ! [N3: num] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N3 ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N3 ) ) ) ) ).

% minus_one_div_numeral
thf(fact_8160_one__div__minus__numeral,axiom,
    ! [N3: num] :
      ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N3 ) ) ) ) ).

% one_div_minus_numeral
thf(fact_8161_finite__nth__roots,axiom,
    ! [N3: nat,C2: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z5: complex] :
              ( ( power_power_complex @ Z5 @ N3 )
              = C2 ) ) ) ) ).

% finite_nth_roots
thf(fact_8162_compl__le__compl__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) )
      = ( ord_less_eq_set_nat @ Y @ X ) ) ).

% compl_le_compl_iff
thf(fact_8163_Compl__iff,axiom,
    ! [C2: nat,A: set_nat] :
      ( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A ) )
      = ( ~ ( member_nat @ C2 @ A ) ) ) ).

% Compl_iff
thf(fact_8164_Compl__iff,axiom,
    ! [C2: real,A: set_real] :
      ( ( member_real @ C2 @ ( uminus612125837232591019t_real @ A ) )
      = ( ~ ( member_real @ C2 @ A ) ) ) ).

% Compl_iff
thf(fact_8165_Compl__iff,axiom,
    ! [C2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ C2 @ ( uminus8041839845116263051T_VEBT @ A ) )
      = ( ~ ( member_VEBT_VEBT @ C2 @ A ) ) ) ).

% Compl_iff
thf(fact_8166_Compl__iff,axiom,
    ! [C2: int,A: set_int] :
      ( ( member_int @ C2 @ ( uminus1532241313380277803et_int @ A ) )
      = ( ~ ( member_int @ C2 @ A ) ) ) ).

% Compl_iff
thf(fact_8167_Compl__iff,axiom,
    ! [C2: complex,A: set_complex] :
      ( ( member_complex @ C2 @ ( uminus8566677241136511917omplex @ A ) )
      = ( ~ ( member_complex @ C2 @ A ) ) ) ).

% Compl_iff
thf(fact_8168_ComplI,axiom,
    ! [C2: nat,A: set_nat] :
      ( ~ ( member_nat @ C2 @ A )
     => ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A ) ) ) ).

% ComplI
thf(fact_8169_ComplI,axiom,
    ! [C2: real,A: set_real] :
      ( ~ ( member_real @ C2 @ A )
     => ( member_real @ C2 @ ( uminus612125837232591019t_real @ A ) ) ) ).

% ComplI
thf(fact_8170_ComplI,axiom,
    ! [C2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ C2 @ A )
     => ( member_VEBT_VEBT @ C2 @ ( uminus8041839845116263051T_VEBT @ A ) ) ) ).

% ComplI
thf(fact_8171_ComplI,axiom,
    ! [C2: int,A: set_int] :
      ( ~ ( member_int @ C2 @ A )
     => ( member_int @ C2 @ ( uminus1532241313380277803et_int @ A ) ) ) ).

% ComplI
thf(fact_8172_ComplI,axiom,
    ! [C2: complex,A: set_complex] :
      ( ~ ( member_complex @ C2 @ A )
     => ( member_complex @ C2 @ ( uminus8566677241136511917omplex @ A ) ) ) ).

% ComplI
thf(fact_8173_minus__numeral__div__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N3 ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N3 ) ) ) ) ).

% minus_numeral_div_numeral
thf(fact_8174_numeral__div__minus__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N3 ) ) ) ) ).

% numeral_div_minus_numeral
thf(fact_8175_ComplD,axiom,
    ! [C2: nat,A: set_nat] :
      ( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A ) )
     => ~ ( member_nat @ C2 @ A ) ) ).

% ComplD
thf(fact_8176_ComplD,axiom,
    ! [C2: real,A: set_real] :
      ( ( member_real @ C2 @ ( uminus612125837232591019t_real @ A ) )
     => ~ ( member_real @ C2 @ A ) ) ).

% ComplD
thf(fact_8177_ComplD,axiom,
    ! [C2: vEBT_VEBT,A: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ C2 @ ( uminus8041839845116263051T_VEBT @ A ) )
     => ~ ( member_VEBT_VEBT @ C2 @ A ) ) ).

% ComplD
thf(fact_8178_ComplD,axiom,
    ! [C2: int,A: set_int] :
      ( ( member_int @ C2 @ ( uminus1532241313380277803et_int @ A ) )
     => ~ ( member_int @ C2 @ A ) ) ).

% ComplD
thf(fact_8179_ComplD,axiom,
    ! [C2: complex,A: set_complex] :
      ( ( member_complex @ C2 @ ( uminus8566677241136511917omplex @ A ) )
     => ~ ( member_complex @ C2 @ A ) ) ).

% ComplD
thf(fact_8180_Compl__eq,axiom,
    ( uminus612125837232591019t_real
    = ( ^ [A6: set_real] :
          ( collect_real
          @ ^ [X3: real] :
              ~ ( member_real @ X3 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_8181_Compl__eq,axiom,
    ( uminus8041839845116263051T_VEBT
    = ( ^ [A6: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ^ [X3: vEBT_VEBT] :
              ~ ( member_VEBT_VEBT @ X3 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_8182_Compl__eq,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A6: set_nat] :
          ( collect_nat
          @ ^ [X3: nat] :
              ~ ( member_nat @ X3 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_8183_Compl__eq,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A6: set_int] :
          ( collect_int
          @ ^ [X3: int] :
              ~ ( member_int @ X3 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_8184_Compl__eq,axiom,
    ( uminus8566677241136511917omplex
    = ( ^ [A6: set_complex] :
          ( collect_complex
          @ ^ [X3: complex] :
              ~ ( member_complex @ X3 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_8185_Compl__eq,axiom,
    ( uminus6221592323253981072nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ^ [X3: product_prod_int_int] :
              ~ ( member5262025264175285858nt_int @ X3 @ A6 ) ) ) ) ).

% Compl_eq
thf(fact_8186_Collect__neg__eq,axiom,
    ! [P: nat > $o] :
      ( ( collect_nat
        @ ^ [X3: nat] :
            ~ ( P @ X3 ) )
      = ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_8187_Collect__neg__eq,axiom,
    ! [P: int > $o] :
      ( ( collect_int
        @ ^ [X3: int] :
            ~ ( P @ X3 ) )
      = ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ).

% Collect_neg_eq
thf(fact_8188_Collect__neg__eq,axiom,
    ! [P: complex > $o] :
      ( ( collect_complex
        @ ^ [X3: complex] :
            ~ ( P @ X3 ) )
      = ( uminus8566677241136511917omplex @ ( collect_complex @ P ) ) ) ).

% Collect_neg_eq
thf(fact_8189_Collect__neg__eq,axiom,
    ! [P: product_prod_int_int > $o] :
      ( ( collec213857154873943460nt_int
        @ ^ [X3: product_prod_int_int] :
            ~ ( P @ X3 ) )
      = ( uminus6221592323253981072nt_int @ ( collec213857154873943460nt_int @ P ) ) ) ).

% Collect_neg_eq
thf(fact_8190_uminus__set__def,axiom,
    ( uminus612125837232591019t_real
    = ( ^ [A6: set_real] :
          ( collect_real
          @ ( uminus_uminus_real_o
            @ ^ [X3: real] : ( member_real @ X3 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_8191_uminus__set__def,axiom,
    ( uminus8041839845116263051T_VEBT
    = ( ^ [A6: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ( uminus2746543603091002386VEBT_o
            @ ^ [X3: vEBT_VEBT] : ( member_VEBT_VEBT @ X3 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_8192_uminus__set__def,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A6: set_nat] :
          ( collect_nat
          @ ( uminus_uminus_nat_o
            @ ^ [X3: nat] : ( member_nat @ X3 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_8193_uminus__set__def,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A6: set_int] :
          ( collect_int
          @ ( uminus_uminus_int_o
            @ ^ [X3: int] : ( member_int @ X3 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_8194_uminus__set__def,axiom,
    ( uminus8566677241136511917omplex
    = ( ^ [A6: set_complex] :
          ( collect_complex
          @ ( uminus1680532995456772888plex_o
            @ ^ [X3: complex] : ( member_complex @ X3 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_8195_uminus__set__def,axiom,
    ( uminus6221592323253981072nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ( uminus7117520113953359693_int_o
            @ ^ [X3: product_prod_int_int] : ( member5262025264175285858nt_int @ X3 @ A6 ) ) ) ) ) ).

% uminus_set_def
thf(fact_8196_compl__mono,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ ( uminus5710092332889474511et_nat @ X ) ) ) ).

% compl_mono
thf(fact_8197_compl__le__swap1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) )
     => ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).

% compl_le_swap1
thf(fact_8198_compl__le__swap2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ X )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) ) ).

% compl_le_swap2
thf(fact_8199_complex__mod__minus__le__complex__mod,axiom,
    ! [X: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).

% complex_mod_minus_le_complex_mod
thf(fact_8200_complex__mod__triangle__ineq2,axiom,
    ! [B2: complex,A2: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B2 @ A2 ) ) @ ( real_V1022390504157884413omplex @ B2 ) ) @ ( real_V1022390504157884413omplex @ A2 ) ) ).

% complex_mod_triangle_ineq2
thf(fact_8201_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_8202_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_8203_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_8204_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_8205_set__encode__insert,axiom,
    ! [A: set_nat,N3: nat] :
      ( ( finite_finite_nat @ A )
     => ( ~ ( member_nat @ N3 @ A )
       => ( ( nat_set_encode @ ( insert_nat @ N3 @ A ) )
          = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( nat_set_encode @ A ) ) ) ) ) ).

% set_encode_insert
thf(fact_8206_diff__preserves__multiset,axiom,
    ! [M3: product_prod_int_int > nat,N7: product_prod_int_int > nat] :
      ( ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X3: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( M3 @ X3 ) ) ) )
     => ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X3: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M3 @ X3 ) @ ( N7 @ X3 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_8207_diff__preserves__multiset,axiom,
    ! [M3: nat > nat,N7: nat > nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] : ( ord_less_nat @ zero_zero_nat @ ( M3 @ X3 ) ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M3 @ X3 ) @ ( N7 @ X3 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_8208_diff__preserves__multiset,axiom,
    ! [M3: int > nat,N7: int > nat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X3: int] : ( ord_less_nat @ zero_zero_nat @ ( M3 @ X3 ) ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X3: int] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M3 @ X3 ) @ ( N7 @ X3 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_8209_diff__preserves__multiset,axiom,
    ! [M3: code_integer > nat,N7: code_integer > nat] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X3: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( M3 @ X3 ) ) ) )
     => ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X3: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M3 @ X3 ) @ ( N7 @ X3 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_8210_diff__preserves__multiset,axiom,
    ! [M3: complex > nat,N7: complex > nat] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X3: complex] : ( ord_less_nat @ zero_zero_nat @ ( M3 @ X3 ) ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X3: complex] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M3 @ X3 ) @ ( N7 @ X3 ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_8211_add__mset__in__multiset,axiom,
    ! [M3: product_prod_int_int > nat,A2: product_prod_int_int] :
      ( ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X3: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( M3 @ X3 ) ) ) )
     => ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X3: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X3 = A2 ) @ ( suc @ ( M3 @ X3 ) ) @ ( M3 @ X3 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_8212_add__mset__in__multiset,axiom,
    ! [M3: nat > nat,A2: nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] : ( ord_less_nat @ zero_zero_nat @ ( M3 @ X3 ) ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X3 = A2 ) @ ( suc @ ( M3 @ X3 ) ) @ ( M3 @ X3 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_8213_add__mset__in__multiset,axiom,
    ! [M3: int > nat,A2: int] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X3: int] : ( ord_less_nat @ zero_zero_nat @ ( M3 @ X3 ) ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X3: int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X3 = A2 ) @ ( suc @ ( M3 @ X3 ) ) @ ( M3 @ X3 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_8214_add__mset__in__multiset,axiom,
    ! [M3: code_integer > nat,A2: code_integer] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X3: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( M3 @ X3 ) ) ) )
     => ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X3: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X3 = A2 ) @ ( suc @ ( M3 @ X3 ) ) @ ( M3 @ X3 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_8215_add__mset__in__multiset,axiom,
    ! [M3: complex > nat,A2: complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X3: complex] : ( ord_less_nat @ zero_zero_nat @ ( M3 @ X3 ) ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X3: complex] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X3 = A2 ) @ ( suc @ ( M3 @ X3 ) ) @ ( M3 @ X3 ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_8216_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
            @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N2 ) ) ) ) ) ) ) ).

% ln_series
thf(fact_8217_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_8218_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_8219_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_8220_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_8221_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_8222_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ one_one_real )
    = one_one_real ) ).

% dbl_dec_simps(3)
thf(fact_8223_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
    = one_one_rat ) ).

% dbl_dec_simps(3)
thf(fact_8224_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ one_one_int )
    = one_one_int ) ).

% dbl_dec_simps(3)
thf(fact_8225_set__encode__empty,axiom,
    ( ( nat_set_encode @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% set_encode_empty
thf(fact_8226_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_dec_simps(2)
thf(fact_8227_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_dec_simps(2)
thf(fact_8228_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_dec_simps(2)
thf(fact_8229_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_dec_simps(2)
thf(fact_8230_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_dec_simps(2)
thf(fact_8231_powser__zero,axiom,
    ! [F2: nat > complex] :
      ( ( suminf_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( F2 @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) )
      = ( F2 @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_8232_powser__zero,axiom,
    ! [F2: nat > real] :
      ( ( suminf_real
        @ ^ [N2: nat] : ( times_times_real @ ( F2 @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) )
      = ( F2 @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_8233_set__encode__inf,axiom,
    ! [A: set_nat] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( nat_set_encode @ A )
        = zero_zero_nat ) ) ).

% set_encode_inf
thf(fact_8234_dbl__dec__def,axiom,
    ( neg_nu6075765906172075777c_real
    = ( ^ [X3: real] : ( minus_minus_real @ ( plus_plus_real @ X3 @ X3 ) @ one_one_real ) ) ) ).

% dbl_dec_def
thf(fact_8235_dbl__dec__def,axiom,
    ( neg_nu3179335615603231917ec_rat
    = ( ^ [X3: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X3 @ X3 ) @ one_one_rat ) ) ) ).

% dbl_dec_def
thf(fact_8236_dbl__dec__def,axiom,
    ( neg_nu3811975205180677377ec_int
    = ( ^ [X3: int] : ( minus_minus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).

% dbl_dec_def
thf(fact_8237_even__set__encode__iff,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A ) )
        = ( ~ ( member_nat @ zero_zero_nat @ A ) ) ) ) ).

% even_set_encode_iff
thf(fact_8238_filter__preserves__multiset,axiom,
    ! [M3: product_prod_int_int > nat,P: product_prod_int_int > $o] :
      ( ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X3: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( M3 @ X3 ) ) ) )
     => ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X3: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X3 ) @ ( M3 @ X3 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_8239_filter__preserves__multiset,axiom,
    ! [M3: nat > nat,P: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] : ( ord_less_nat @ zero_zero_nat @ ( M3 @ X3 ) ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X3 ) @ ( M3 @ X3 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_8240_filter__preserves__multiset,axiom,
    ! [M3: int > nat,P: int > $o] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X3: int] : ( ord_less_nat @ zero_zero_nat @ ( M3 @ X3 ) ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X3: int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X3 ) @ ( M3 @ X3 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_8241_filter__preserves__multiset,axiom,
    ! [M3: code_integer > nat,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X3: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( M3 @ X3 ) ) ) )
     => ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X3: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X3 ) @ ( M3 @ X3 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_8242_filter__preserves__multiset,axiom,
    ! [M3: complex > nat,P: complex > $o] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X3: complex] : ( ord_less_nat @ zero_zero_nat @ ( M3 @ X3 ) ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X3: complex] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X3 ) @ ( M3 @ X3 ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_8243_suminf__geometric,axiom,
    ! [C2: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C2 ) @ one_one_real )
     => ( ( suminf_real @ ( power_power_real @ C2 ) )
        = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C2 ) ) ) ) ).

% suminf_geometric
thf(fact_8244_suminf__geometric,axiom,
    ! [C2: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C2 ) @ one_one_real )
     => ( ( suminf_complex @ ( power_power_complex @ C2 ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C2 ) ) ) ) ).

% suminf_geometric
thf(fact_8245_suminf__zero,axiom,
    ( ( suminf_complex
      @ ^ [N2: nat] : zero_zero_complex )
    = zero_zero_complex ) ).

% suminf_zero
thf(fact_8246_suminf__zero,axiom,
    ( ( suminf_real
      @ ^ [N2: nat] : zero_zero_real )
    = zero_zero_real ) ).

% suminf_zero
thf(fact_8247_suminf__zero,axiom,
    ( ( suminf_nat
      @ ^ [N2: nat] : zero_zero_nat )
    = zero_zero_nat ) ).

% suminf_zero
thf(fact_8248_suminf__zero,axiom,
    ( ( suminf_int
      @ ^ [N2: nat] : zero_zero_int )
    = zero_zero_int ) ).

% suminf_zero
thf(fact_8249_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_8250_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_8251_finite__linorder__max__induct,axiom,
    ! [A: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [B3: code_integer,A8: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A8 )
             => ( ! [X5: code_integer] :
                    ( ( member_Code_integer @ X5 @ A8 )
                   => ( ord_le6747313008572928689nteger @ X5 @ B3 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_Code_integer @ B3 @ A8 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_8252_finite__linorder__max__induct,axiom,
    ! [A: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ A )
     => ( ( P @ bot_bot_set_o )
       => ( ! [B3: $o,A8: set_o] :
              ( ( finite_finite_o @ A8 )
             => ( ! [X5: $o] :
                    ( ( member_o @ X5 @ A8 )
                   => ( ord_less_o @ X5 @ B3 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_o @ B3 @ A8 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_8253_finite__linorder__max__induct,axiom,
    ! [A: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B3: real,A8: set_real] :
              ( ( finite_finite_real @ A8 )
             => ( ! [X5: real] :
                    ( ( member_real @ X5 @ A8 )
                   => ( ord_less_real @ X5 @ B3 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_real @ B3 @ A8 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_8254_finite__linorder__max__induct,axiom,
    ! [A: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B3: rat,A8: set_rat] :
              ( ( finite_finite_rat @ A8 )
             => ( ! [X5: rat] :
                    ( ( member_rat @ X5 @ A8 )
                   => ( ord_less_rat @ X5 @ B3 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_rat @ B3 @ A8 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_8255_finite__linorder__max__induct,axiom,
    ! [A: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B3: num,A8: set_num] :
              ( ( finite_finite_num @ A8 )
             => ( ! [X5: num] :
                    ( ( member_num @ X5 @ A8 )
                   => ( ord_less_num @ X5 @ B3 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_num @ B3 @ A8 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_8256_finite__linorder__max__induct,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B3: nat,A8: set_nat] :
              ( ( finite_finite_nat @ A8 )
             => ( ! [X5: nat] :
                    ( ( member_nat @ X5 @ A8 )
                   => ( ord_less_nat @ X5 @ B3 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_nat @ B3 @ A8 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_8257_finite__linorder__max__induct,axiom,
    ! [A: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B3: int,A8: set_int] :
              ( ( finite_finite_int @ A8 )
             => ( ! [X5: int] :
                    ( ( member_int @ X5 @ A8 )
                   => ( ord_less_int @ X5 @ B3 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_int @ B3 @ A8 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_8258_abs__idempotent,axiom,
    ! [A2: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A2 ) )
      = ( abs_abs_real @ A2 ) ) ).

% abs_idempotent
thf(fact_8259_abs__idempotent,axiom,
    ! [A2: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A2 ) )
      = ( abs_abs_int @ A2 ) ) ).

% abs_idempotent
thf(fact_8260_abs__idempotent,axiom,
    ! [A2: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A2 ) )
      = ( abs_abs_Code_integer @ A2 ) ) ).

% abs_idempotent
thf(fact_8261_abs__idempotent,axiom,
    ! [A2: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A2 ) )
      = ( abs_abs_rat @ A2 ) ) ).

% abs_idempotent
thf(fact_8262_abs__abs,axiom,
    ! [A2: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A2 ) )
      = ( abs_abs_real @ A2 ) ) ).

% abs_abs
thf(fact_8263_abs__abs,axiom,
    ! [A2: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A2 ) )
      = ( abs_abs_int @ A2 ) ) ).

% abs_abs
thf(fact_8264_abs__abs,axiom,
    ! [A2: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A2 ) )
      = ( abs_abs_Code_integer @ A2 ) ) ).

% abs_abs
thf(fact_8265_abs__abs,axiom,
    ! [A2: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A2 ) )
      = ( abs_abs_rat @ A2 ) ) ).

% abs_abs
thf(fact_8266_abs__0__eq,axiom,
    ! [A2: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( abs_abs_Code_integer @ A2 ) )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
thf(fact_8267_abs__0__eq,axiom,
    ! [A2: real] :
      ( ( zero_zero_real
        = ( abs_abs_real @ A2 ) )
      = ( A2 = zero_zero_real ) ) ).

% abs_0_eq
thf(fact_8268_abs__0__eq,axiom,
    ! [A2: rat] :
      ( ( zero_zero_rat
        = ( abs_abs_rat @ A2 ) )
      = ( A2 = zero_zero_rat ) ) ).

% abs_0_eq
thf(fact_8269_abs__0__eq,axiom,
    ! [A2: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A2 ) )
      = ( A2 = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_8270_abs__eq__0,axiom,
    ! [A2: code_integer] :
      ( ( ( abs_abs_Code_integer @ A2 )
        = zero_z3403309356797280102nteger )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
thf(fact_8271_abs__eq__0,axiom,
    ! [A2: real] :
      ( ( ( abs_abs_real @ A2 )
        = zero_zero_real )
      = ( A2 = zero_zero_real ) ) ).

% abs_eq_0
thf(fact_8272_abs__eq__0,axiom,
    ! [A2: rat] :
      ( ( ( abs_abs_rat @ A2 )
        = zero_zero_rat )
      = ( A2 = zero_zero_rat ) ) ).

% abs_eq_0
thf(fact_8273_abs__eq__0,axiom,
    ! [A2: int] :
      ( ( ( abs_abs_int @ A2 )
        = zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_8274_abs__zero,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_zero
thf(fact_8275_abs__zero,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_zero
thf(fact_8276_abs__zero,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_zero
thf(fact_8277_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_8278_abs__0,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_0
thf(fact_8279_abs__0,axiom,
    ( ( abs_abs_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% abs_0
thf(fact_8280_abs__0,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_0
thf(fact_8281_abs__0,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_0
thf(fact_8282_abs__0,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_0
thf(fact_8283_abs__numeral,axiom,
    ! [N3: num] :
      ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N3 ) )
      = ( numera6620942414471956472nteger @ N3 ) ) ).

% abs_numeral
thf(fact_8284_abs__numeral,axiom,
    ! [N3: num] :
      ( ( abs_abs_real @ ( numeral_numeral_real @ N3 ) )
      = ( numeral_numeral_real @ N3 ) ) ).

% abs_numeral
thf(fact_8285_abs__numeral,axiom,
    ! [N3: num] :
      ( ( abs_abs_rat @ ( numeral_numeral_rat @ N3 ) )
      = ( numeral_numeral_rat @ N3 ) ) ).

% abs_numeral
thf(fact_8286_abs__numeral,axiom,
    ! [N3: num] :
      ( ( abs_abs_int @ ( numeral_numeral_int @ N3 ) )
      = ( numeral_numeral_int @ N3 ) ) ).

% abs_numeral
thf(fact_8287_abs__add__abs,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) ) )
      = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).

% abs_add_abs
thf(fact_8288_abs__add__abs,axiom,
    ! [A2: real,B2: real] :
      ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B2 ) ) )
      = ( plus_plus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B2 ) ) ) ).

% abs_add_abs
thf(fact_8289_abs__add__abs,axiom,
    ! [A2: rat,B2: rat] :
      ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) ) )
      = ( plus_plus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) ) ) ).

% abs_add_abs
thf(fact_8290_abs__add__abs,axiom,
    ! [A2: int,B2: int] :
      ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) )
      = ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) ) ).

% abs_add_abs
thf(fact_8291_abs__1,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_1
thf(fact_8292_abs__1,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_1
thf(fact_8293_abs__1,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_1
thf(fact_8294_abs__1,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_1
thf(fact_8295_abs__mult__self__eq,axiom,
    ! [A2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ A2 ) )
      = ( times_3573771949741848930nteger @ A2 @ A2 ) ) ).

% abs_mult_self_eq
thf(fact_8296_abs__mult__self__eq,axiom,
    ! [A2: real] :
      ( ( times_times_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ A2 ) )
      = ( times_times_real @ A2 @ A2 ) ) ).

% abs_mult_self_eq
thf(fact_8297_abs__mult__self__eq,axiom,
    ! [A2: rat] :
      ( ( times_times_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ A2 ) )
      = ( times_times_rat @ A2 @ A2 ) ) ).

% abs_mult_self_eq
thf(fact_8298_abs__mult__self__eq,axiom,
    ! [A2: int] :
      ( ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ A2 ) )
      = ( times_times_int @ A2 @ A2 ) ) ).

% abs_mult_self_eq
thf(fact_8299_abs__divide,axiom,
    ! [A2: complex,B2: complex] :
      ( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A2 @ B2 ) )
      = ( divide1717551699836669952omplex @ ( abs_abs_complex @ A2 ) @ ( abs_abs_complex @ B2 ) ) ) ).

% abs_divide
thf(fact_8300_abs__divide,axiom,
    ! [A2: real,B2: real] :
      ( ( abs_abs_real @ ( divide_divide_real @ A2 @ B2 ) )
      = ( divide_divide_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B2 ) ) ) ).

% abs_divide
thf(fact_8301_abs__divide,axiom,
    ! [A2: rat,B2: rat] :
      ( ( abs_abs_rat @ ( divide_divide_rat @ A2 @ B2 ) )
      = ( divide_divide_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) ) ) ).

% abs_divide
thf(fact_8302_abs__minus__cancel,axiom,
    ! [A2: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A2 ) )
      = ( abs_abs_real @ A2 ) ) ).

% abs_minus_cancel
thf(fact_8303_abs__minus__cancel,axiom,
    ! [A2: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A2 ) )
      = ( abs_abs_int @ A2 ) ) ).

% abs_minus_cancel
thf(fact_8304_abs__minus__cancel,axiom,
    ! [A2: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( abs_abs_Code_integer @ A2 ) ) ).

% abs_minus_cancel
thf(fact_8305_abs__minus__cancel,axiom,
    ! [A2: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A2 ) )
      = ( abs_abs_rat @ A2 ) ) ).

% abs_minus_cancel
thf(fact_8306_abs__minus,axiom,
    ! [A2: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A2 ) )
      = ( abs_abs_real @ A2 ) ) ).

% abs_minus
thf(fact_8307_abs__minus,axiom,
    ! [A2: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A2 ) )
      = ( abs_abs_int @ A2 ) ) ).

% abs_minus
thf(fact_8308_abs__minus,axiom,
    ! [A2: complex] :
      ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A2 ) )
      = ( abs_abs_complex @ A2 ) ) ).

% abs_minus
thf(fact_8309_abs__minus,axiom,
    ! [A2: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A2 ) )
      = ( abs_abs_Code_integer @ A2 ) ) ).

% abs_minus
thf(fact_8310_abs__minus,axiom,
    ! [A2: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A2 ) )
      = ( abs_abs_rat @ A2 ) ) ).

% abs_minus
thf(fact_8311_dvd__abs__iff,axiom,
    ! [M: real,K: real] :
      ( ( dvd_dvd_real @ M @ ( abs_abs_real @ K ) )
      = ( dvd_dvd_real @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_8312_dvd__abs__iff,axiom,
    ! [M: int,K: int] :
      ( ( dvd_dvd_int @ M @ ( abs_abs_int @ K ) )
      = ( dvd_dvd_int @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_8313_dvd__abs__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( ( dvd_dvd_Code_integer @ M @ ( abs_abs_Code_integer @ K ) )
      = ( dvd_dvd_Code_integer @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_8314_dvd__abs__iff,axiom,
    ! [M: rat,K: rat] :
      ( ( dvd_dvd_rat @ M @ ( abs_abs_rat @ K ) )
      = ( dvd_dvd_rat @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_8315_abs__dvd__iff,axiom,
    ! [M: real,K: real] :
      ( ( dvd_dvd_real @ ( abs_abs_real @ M ) @ K )
      = ( dvd_dvd_real @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_8316_abs__dvd__iff,axiom,
    ! [M: int,K: int] :
      ( ( dvd_dvd_int @ ( abs_abs_int @ M ) @ K )
      = ( dvd_dvd_int @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_8317_abs__dvd__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( abs_abs_Code_integer @ M ) @ K )
      = ( dvd_dvd_Code_integer @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_8318_abs__dvd__iff,axiom,
    ! [M: rat,K: rat] :
      ( ( dvd_dvd_rat @ ( abs_abs_rat @ M ) @ K )
      = ( dvd_dvd_rat @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_8319_abs__of__nat,axiom,
    ! [N3: nat] :
      ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N3 ) )
      = ( semiri681578069525770553at_rat @ N3 ) ) ).

% abs_of_nat
thf(fact_8320_abs__of__nat,axiom,
    ! [N3: nat] :
      ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N3 ) )
      = ( semiri5074537144036343181t_real @ N3 ) ) ).

% abs_of_nat
thf(fact_8321_abs__of__nat,axiom,
    ! [N3: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N3 ) )
      = ( semiri1314217659103216013at_int @ N3 ) ) ).

% abs_of_nat
thf(fact_8322_abs__of__nat,axiom,
    ! [N3: nat] :
      ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N3 ) )
      = ( semiri4939895301339042750nteger @ N3 ) ) ).

% abs_of_nat
thf(fact_8323_abs__of__nonneg,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( abs_abs_Code_integer @ A2 )
        = A2 ) ) ).

% abs_of_nonneg
thf(fact_8324_abs__of__nonneg,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( abs_abs_real @ A2 )
        = A2 ) ) ).

% abs_of_nonneg
thf(fact_8325_abs__of__nonneg,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
     => ( ( abs_abs_rat @ A2 )
        = A2 ) ) ).

% abs_of_nonneg
thf(fact_8326_abs__of__nonneg,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( abs_abs_int @ A2 )
        = A2 ) ) ).

% abs_of_nonneg
thf(fact_8327_abs__le__self__iff,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ A2 )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).

% abs_le_self_iff
thf(fact_8328_abs__le__self__iff,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ A2 )
      = ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).

% abs_le_self_iff
thf(fact_8329_abs__le__self__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ A2 )
      = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).

% abs_le_self_iff
thf(fact_8330_abs__le__self__iff,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ A2 )
      = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).

% abs_le_self_iff
thf(fact_8331_abs__le__zero__iff,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ zero_z3403309356797280102nteger )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
thf(fact_8332_abs__le__zero__iff,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ zero_zero_real )
      = ( A2 = zero_zero_real ) ) ).

% abs_le_zero_iff
thf(fact_8333_abs__le__zero__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ zero_zero_rat )
      = ( A2 = zero_zero_rat ) ) ).

% abs_le_zero_iff
thf(fact_8334_abs__le__zero__iff,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% abs_le_zero_iff
thf(fact_8335_zero__less__abs__iff,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A2 ) )
      = ( A2 != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
thf(fact_8336_zero__less__abs__iff,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A2 ) )
      = ( A2 != zero_zero_real ) ) ).

% zero_less_abs_iff
thf(fact_8337_zero__less__abs__iff,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A2 ) )
      = ( A2 != zero_zero_rat ) ) ).

% zero_less_abs_iff
thf(fact_8338_zero__less__abs__iff,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A2 ) )
      = ( A2 != zero_zero_int ) ) ).

% zero_less_abs_iff
thf(fact_8339_abs__neg__numeral,axiom,
    ! [N3: num] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
      = ( numeral_numeral_real @ N3 ) ) ).

% abs_neg_numeral
thf(fact_8340_abs__neg__numeral,axiom,
    ! [N3: num] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( numeral_numeral_int @ N3 ) ) ).

% abs_neg_numeral
thf(fact_8341_abs__neg__numeral,axiom,
    ! [N3: num] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N3 ) ) )
      = ( numera6620942414471956472nteger @ N3 ) ) ).

% abs_neg_numeral
thf(fact_8342_abs__neg__numeral,axiom,
    ! [N3: num] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N3 ) ) )
      = ( numeral_numeral_rat @ N3 ) ) ).

% abs_neg_numeral
thf(fact_8343_abs__neg__one,axiom,
    ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
    = one_one_real ) ).

% abs_neg_one
thf(fact_8344_abs__neg__one,axiom,
    ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
    = one_one_int ) ).

% abs_neg_one
thf(fact_8345_abs__neg__one,axiom,
    ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = one_one_Code_integer ) ).

% abs_neg_one
thf(fact_8346_abs__neg__one,axiom,
    ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = one_one_rat ) ).

% abs_neg_one
thf(fact_8347_abs__power__minus,axiom,
    ! [A2: real,N3: nat] :
      ( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A2 ) @ N3 ) )
      = ( abs_abs_real @ ( power_power_real @ A2 @ N3 ) ) ) ).

% abs_power_minus
thf(fact_8348_abs__power__minus,axiom,
    ! [A2: int,N3: nat] :
      ( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A2 ) @ N3 ) )
      = ( abs_abs_int @ ( power_power_int @ A2 @ N3 ) ) ) ).

% abs_power_minus
thf(fact_8349_abs__power__minus,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A2 ) @ N3 ) )
      = ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ).

% abs_power_minus
thf(fact_8350_abs__power__minus,axiom,
    ! [A2: rat,N3: nat] :
      ( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A2 ) @ N3 ) )
      = ( abs_abs_rat @ ( power_power_rat @ A2 @ N3 ) ) ) ).

% abs_power_minus
thf(fact_8351_zero__le__divide__abs__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A2 @ ( abs_abs_real @ B2 ) ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A2 )
        | ( B2 = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_8352_zero__le__divide__abs__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A2 @ ( abs_abs_rat @ B2 ) ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
        | ( B2 = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_8353_divide__le__0__abs__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A2 @ ( abs_abs_real @ B2 ) ) @ zero_zero_real )
      = ( ( ord_less_eq_real @ A2 @ zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
thf(fact_8354_divide__le__0__abs__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A2 @ ( abs_abs_rat @ B2 ) ) @ zero_zero_rat )
      = ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
thf(fact_8355_abs__of__nonpos,axiom,
    ! [A2: real] :
      ( ( ord_less_eq_real @ A2 @ zero_zero_real )
     => ( ( abs_abs_real @ A2 )
        = ( uminus_uminus_real @ A2 ) ) ) ).

% abs_of_nonpos
thf(fact_8356_abs__of__nonpos,axiom,
    ! [A2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A2 )
        = ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% abs_of_nonpos
thf(fact_8357_abs__of__nonpos,axiom,
    ! [A2: rat] :
      ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
     => ( ( abs_abs_rat @ A2 )
        = ( uminus_uminus_rat @ A2 ) ) ) ).

% abs_of_nonpos
thf(fact_8358_abs__of__nonpos,axiom,
    ! [A2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( abs_abs_int @ A2 )
        = ( uminus_uminus_int @ A2 ) ) ) ).

% abs_of_nonpos
thf(fact_8359_artanh__minus__real,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( artanh_real @ ( uminus_uminus_real @ X ) )
        = ( uminus_uminus_real @ ( artanh_real @ X ) ) ) ) ).

% artanh_minus_real
thf(fact_8360_zero__less__power__abs__iff,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A2 ) @ N3 ) )
      = ( ( A2 != zero_z3403309356797280102nteger )
        | ( N3 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_8361_zero__less__power__abs__iff,axiom,
    ! [A2: real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A2 ) @ N3 ) )
      = ( ( A2 != zero_zero_real )
        | ( N3 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_8362_zero__less__power__abs__iff,axiom,
    ! [A2: rat,N3: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A2 ) @ N3 ) )
      = ( ( A2 != zero_zero_rat )
        | ( N3 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_8363_zero__less__power__abs__iff,axiom,
    ! [A2: int,N3: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A2 ) @ N3 ) )
      = ( ( A2 != zero_zero_int )
        | ( N3 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_8364_power2__abs,axiom,
    ! [A2: rat] :
      ( ( power_power_rat @ ( abs_abs_rat @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_8365_power2__abs,axiom,
    ! [A2: real] :
      ( ( power_power_real @ ( abs_abs_real @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_8366_power2__abs,axiom,
    ! [A2: int] :
      ( ( power_power_int @ ( abs_abs_int @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_8367_power2__abs,axiom,
    ! [A2: code_integer] :
      ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_8368_abs__power2,axiom,
    ! [A2: rat] :
      ( ( abs_abs_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_8369_abs__power2,axiom,
    ! [A2: real] :
      ( ( abs_abs_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_8370_abs__power2,axiom,
    ! [A2: int] :
      ( ( abs_abs_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_8371_abs__power2,axiom,
    ! [A2: code_integer] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_8372_power__even__abs__numeral,axiom,
    ! [W: num,A2: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_power_rat @ ( abs_abs_rat @ A2 ) @ ( numeral_numeral_nat @ W ) )
        = ( power_power_rat @ A2 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_8373_power__even__abs__numeral,axiom,
    ! [W: num,A2: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_power_real @ ( abs_abs_real @ A2 ) @ ( numeral_numeral_nat @ W ) )
        = ( power_power_real @ A2 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_8374_power__even__abs__numeral,axiom,
    ! [W: num,A2: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_power_int @ ( abs_abs_int @ A2 ) @ ( numeral_numeral_nat @ W ) )
        = ( power_power_int @ A2 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_8375_power__even__abs__numeral,axiom,
    ! [W: num,A2: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A2 ) @ ( numeral_numeral_nat @ W ) )
        = ( power_8256067586552552935nteger @ A2 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_8376_abs__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( abs_abs_real @ X )
        = ( abs_abs_real @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_real @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_8377_abs__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( abs_abs_int @ X )
        = ( abs_abs_int @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_int @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_8378_abs__eq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( abs_abs_Code_integer @ X )
        = ( abs_abs_Code_integer @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus1351360451143612070nteger @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_8379_abs__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( abs_abs_rat @ X )
        = ( abs_abs_rat @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_rat @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_8380_pi__neq__zero,axiom,
    pi != zero_zero_real ).

% pi_neq_zero
thf(fact_8381_abs__ge__self,axiom,
    ! [A2: real] : ( ord_less_eq_real @ A2 @ ( abs_abs_real @ A2 ) ) ).

% abs_ge_self
thf(fact_8382_abs__ge__self,axiom,
    ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ A2 @ ( abs_abs_Code_integer @ A2 ) ) ).

% abs_ge_self
thf(fact_8383_abs__ge__self,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ A2 @ ( abs_abs_rat @ A2 ) ) ).

% abs_ge_self
thf(fact_8384_abs__ge__self,axiom,
    ! [A2: int] : ( ord_less_eq_int @ A2 @ ( abs_abs_int @ A2 ) ) ).

% abs_ge_self
thf(fact_8385_abs__le__D1,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ B2 )
     => ( ord_less_eq_real @ A2 @ B2 ) ) ).

% abs_le_D1
thf(fact_8386_abs__le__D1,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ B2 )
     => ( ord_le3102999989581377725nteger @ A2 @ B2 ) ) ).

% abs_le_D1
thf(fact_8387_abs__le__D1,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ B2 )
     => ( ord_less_eq_rat @ A2 @ B2 ) ) ).

% abs_le_D1
thf(fact_8388_abs__le__D1,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B2 )
     => ( ord_less_eq_int @ A2 @ B2 ) ) ).

% abs_le_D1
thf(fact_8389_abs__mult,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B2 ) )
      = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).

% abs_mult
thf(fact_8390_abs__mult,axiom,
    ! [A2: real,B2: real] :
      ( ( abs_abs_real @ ( times_times_real @ A2 @ B2 ) )
      = ( times_times_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B2 ) ) ) ).

% abs_mult
thf(fact_8391_abs__mult,axiom,
    ! [A2: rat,B2: rat] :
      ( ( abs_abs_rat @ ( times_times_rat @ A2 @ B2 ) )
      = ( times_times_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) ) ) ).

% abs_mult
thf(fact_8392_abs__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( abs_abs_int @ ( times_times_int @ A2 @ B2 ) )
      = ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) ) ).

% abs_mult
thf(fact_8393_abs__one,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_one
thf(fact_8394_abs__one,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_one
thf(fact_8395_abs__one,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_one
thf(fact_8396_abs__one,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_one
thf(fact_8397_abs__minus__commute,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ B2 ) )
      = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ A2 ) ) ) ).

% abs_minus_commute
thf(fact_8398_abs__minus__commute,axiom,
    ! [A2: real,B2: real] :
      ( ( abs_abs_real @ ( minus_minus_real @ A2 @ B2 ) )
      = ( abs_abs_real @ ( minus_minus_real @ B2 @ A2 ) ) ) ).

% abs_minus_commute
thf(fact_8399_abs__minus__commute,axiom,
    ! [A2: rat,B2: rat] :
      ( ( abs_abs_rat @ ( minus_minus_rat @ A2 @ B2 ) )
      = ( abs_abs_rat @ ( minus_minus_rat @ B2 @ A2 ) ) ) ).

% abs_minus_commute
thf(fact_8400_abs__minus__commute,axiom,
    ! [A2: int,B2: int] :
      ( ( abs_abs_int @ ( minus_minus_int @ A2 @ B2 ) )
      = ( abs_abs_int @ ( minus_minus_int @ B2 @ A2 ) ) ) ).

% abs_minus_commute
thf(fact_8401_abs__eq__0__iff,axiom,
    ! [A2: code_integer] :
      ( ( ( abs_abs_Code_integer @ A2 )
        = zero_z3403309356797280102nteger )
      = ( A2 = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
thf(fact_8402_abs__eq__0__iff,axiom,
    ! [A2: complex] :
      ( ( ( abs_abs_complex @ A2 )
        = zero_zero_complex )
      = ( A2 = zero_zero_complex ) ) ).

% abs_eq_0_iff
thf(fact_8403_abs__eq__0__iff,axiom,
    ! [A2: real] :
      ( ( ( abs_abs_real @ A2 )
        = zero_zero_real )
      = ( A2 = zero_zero_real ) ) ).

% abs_eq_0_iff
thf(fact_8404_abs__eq__0__iff,axiom,
    ! [A2: rat] :
      ( ( ( abs_abs_rat @ A2 )
        = zero_zero_rat )
      = ( A2 = zero_zero_rat ) ) ).

% abs_eq_0_iff
thf(fact_8405_abs__eq__0__iff,axiom,
    ! [A2: int] :
      ( ( ( abs_abs_int @ A2 )
        = zero_zero_int )
      = ( A2 = zero_zero_int ) ) ).

% abs_eq_0_iff
thf(fact_8406_dvd__if__abs__eq,axiom,
    ! [L2: real,K: real] :
      ( ( ( abs_abs_real @ L2 )
        = ( abs_abs_real @ K ) )
     => ( dvd_dvd_real @ L2 @ K ) ) ).

% dvd_if_abs_eq
thf(fact_8407_dvd__if__abs__eq,axiom,
    ! [L2: int,K: int] :
      ( ( ( abs_abs_int @ L2 )
        = ( abs_abs_int @ K ) )
     => ( dvd_dvd_int @ L2 @ K ) ) ).

% dvd_if_abs_eq
thf(fact_8408_dvd__if__abs__eq,axiom,
    ! [L2: code_integer,K: code_integer] :
      ( ( ( abs_abs_Code_integer @ L2 )
        = ( abs_abs_Code_integer @ K ) )
     => ( dvd_dvd_Code_integer @ L2 @ K ) ) ).

% dvd_if_abs_eq
thf(fact_8409_dvd__if__abs__eq,axiom,
    ! [L2: rat,K: rat] :
      ( ( ( abs_abs_rat @ L2 )
        = ( abs_abs_rat @ K ) )
     => ( dvd_dvd_rat @ L2 @ K ) ) ).

% dvd_if_abs_eq
thf(fact_8410_power__abs,axiom,
    ! [A2: rat,N3: nat] :
      ( ( abs_abs_rat @ ( power_power_rat @ A2 @ N3 ) )
      = ( power_power_rat @ ( abs_abs_rat @ A2 ) @ N3 ) ) ).

% power_abs
thf(fact_8411_power__abs,axiom,
    ! [A2: real,N3: nat] :
      ( ( abs_abs_real @ ( power_power_real @ A2 @ N3 ) )
      = ( power_power_real @ ( abs_abs_real @ A2 ) @ N3 ) ) ).

% power_abs
thf(fact_8412_power__abs,axiom,
    ! [A2: int,N3: nat] :
      ( ( abs_abs_int @ ( power_power_int @ A2 @ N3 ) )
      = ( power_power_int @ ( abs_abs_int @ A2 ) @ N3 ) ) ).

% power_abs
thf(fact_8413_power__abs,axiom,
    ! [A2: code_integer,N3: nat] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A2 @ N3 ) )
      = ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A2 ) @ N3 ) ) ).

% power_abs
thf(fact_8414_abs__ge__zero,axiom,
    ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A2 ) ) ).

% abs_ge_zero
thf(fact_8415_abs__ge__zero,axiom,
    ! [A2: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A2 ) ) ).

% abs_ge_zero
thf(fact_8416_abs__ge__zero,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A2 ) ) ).

% abs_ge_zero
thf(fact_8417_abs__ge__zero,axiom,
    ! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A2 ) ) ).

% abs_ge_zero
thf(fact_8418_abs__not__less__zero,axiom,
    ! [A2: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A2 ) @ zero_z3403309356797280102nteger ) ).

% abs_not_less_zero
thf(fact_8419_abs__not__less__zero,axiom,
    ! [A2: real] :
      ~ ( ord_less_real @ ( abs_abs_real @ A2 ) @ zero_zero_real ) ).

% abs_not_less_zero
thf(fact_8420_abs__not__less__zero,axiom,
    ! [A2: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A2 ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_8421_abs__not__less__zero,axiom,
    ! [A2: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A2 ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_8422_abs__of__pos,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
     => ( ( abs_abs_Code_integer @ A2 )
        = A2 ) ) ).

% abs_of_pos
thf(fact_8423_abs__of__pos,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( abs_abs_real @ A2 )
        = A2 ) ) ).

% abs_of_pos
thf(fact_8424_abs__of__pos,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A2 )
     => ( ( abs_abs_rat @ A2 )
        = A2 ) ) ).

% abs_of_pos
thf(fact_8425_abs__of__pos,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ zero_zero_int @ A2 )
     => ( ( abs_abs_int @ A2 )
        = A2 ) ) ).

% abs_of_pos
thf(fact_8426_abs__triangle__ineq,axiom,
    ! [A2: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).

% abs_triangle_ineq
thf(fact_8427_abs__triangle__ineq,axiom,
    ! [A2: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A2 @ B2 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B2 ) ) ) ).

% abs_triangle_ineq
thf(fact_8428_abs__triangle__ineq,axiom,
    ! [A2: rat,B2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A2 @ B2 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) ) ) ).

% abs_triangle_ineq
thf(fact_8429_abs__triangle__ineq,axiom,
    ! [A2: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A2 @ B2 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) ) ).

% abs_triangle_ineq
thf(fact_8430_abs__mult__less,axiom,
    ! [A2: code_integer,C2: code_integer,B2: code_integer,D2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A2 ) @ C2 )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B2 ) @ D2 )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( times_3573771949741848930nteger @ C2 @ D2 ) ) ) ) ).

% abs_mult_less
thf(fact_8431_abs__mult__less,axiom,
    ! [A2: real,C2: real,B2: real,D2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A2 ) @ C2 )
     => ( ( ord_less_real @ ( abs_abs_real @ B2 ) @ D2 )
       => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B2 ) ) @ ( times_times_real @ C2 @ D2 ) ) ) ) ).

% abs_mult_less
thf(fact_8432_abs__mult__less,axiom,
    ! [A2: rat,C2: rat,B2: rat,D2: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A2 ) @ C2 )
     => ( ( ord_less_rat @ ( abs_abs_rat @ B2 ) @ D2 )
       => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) ) @ ( times_times_rat @ C2 @ D2 ) ) ) ) ).

% abs_mult_less
thf(fact_8433_abs__mult__less,axiom,
    ! [A2: int,C2: int,B2: int,D2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A2 ) @ C2 )
     => ( ( ord_less_int @ ( abs_abs_int @ B2 ) @ D2 )
       => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) @ ( times_times_int @ C2 @ D2 ) ) ) ) ).

% abs_mult_less
thf(fact_8434_abs__triangle__ineq2,axiom,
    ! [A2: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ B2 ) ) ) ).

% abs_triangle_ineq2
thf(fact_8435_abs__triangle__ineq2,axiom,
    ! [A2: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B2 ) ) @ ( abs_abs_real @ ( minus_minus_real @ A2 @ B2 ) ) ) ).

% abs_triangle_ineq2
thf(fact_8436_abs__triangle__ineq2,axiom,
    ! [A2: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A2 @ B2 ) ) ) ).

% abs_triangle_ineq2
thf(fact_8437_abs__triangle__ineq2,axiom,
    ! [A2: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B2 ) ) ) ).

% abs_triangle_ineq2
thf(fact_8438_abs__triangle__ineq3,axiom,
    ! [A2: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ B2 ) ) ) ).

% abs_triangle_ineq3
thf(fact_8439_abs__triangle__ineq3,axiom,
    ! [A2: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B2 ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A2 @ B2 ) ) ) ).

% abs_triangle_ineq3
thf(fact_8440_abs__triangle__ineq3,axiom,
    ! [A2: rat,B2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A2 @ B2 ) ) ) ).

% abs_triangle_ineq3
thf(fact_8441_abs__triangle__ineq3,axiom,
    ! [A2: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B2 ) ) ) ).

% abs_triangle_ineq3
thf(fact_8442_abs__triangle__ineq2__sym,axiom,
    ! [A2: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ A2 ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_8443_abs__triangle__ineq2__sym,axiom,
    ! [A2: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B2 ) ) @ ( abs_abs_real @ ( minus_minus_real @ B2 @ A2 ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_8444_abs__triangle__ineq2__sym,axiom,
    ! [A2: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B2 @ A2 ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_8445_abs__triangle__ineq2__sym,axiom,
    ! [A2: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ A2 ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_8446_nonzero__abs__divide,axiom,
    ! [B2: real,A2: real] :
      ( ( B2 != zero_zero_real )
     => ( ( abs_abs_real @ ( divide_divide_real @ A2 @ B2 ) )
        = ( divide_divide_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B2 ) ) ) ) ).

% nonzero_abs_divide
thf(fact_8447_nonzero__abs__divide,axiom,
    ! [B2: rat,A2: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( abs_abs_rat @ ( divide_divide_rat @ A2 @ B2 ) )
        = ( divide_divide_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) ) ) ) ).

% nonzero_abs_divide
thf(fact_8448_abs__ge__minus__self,axiom,
    ! [A2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ ( abs_abs_real @ A2 ) ) ).

% abs_ge_minus_self
thf(fact_8449_abs__ge__minus__self,axiom,
    ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( abs_abs_Code_integer @ A2 ) ) ).

% abs_ge_minus_self
thf(fact_8450_abs__ge__minus__self,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ ( abs_abs_rat @ A2 ) ) ).

% abs_ge_minus_self
thf(fact_8451_abs__ge__minus__self,axiom,
    ! [A2: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ ( abs_abs_int @ A2 ) ) ).

% abs_ge_minus_self
thf(fact_8452_abs__le__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ B2 )
      = ( ( ord_less_eq_real @ A2 @ B2 )
        & ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ B2 ) ) ) ).

% abs_le_iff
thf(fact_8453_abs__le__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ B2 )
      = ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
        & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 ) ) ) ).

% abs_le_iff
thf(fact_8454_abs__le__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ B2 )
      = ( ( ord_less_eq_rat @ A2 @ B2 )
        & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ B2 ) ) ) ).

% abs_le_iff
thf(fact_8455_abs__le__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B2 )
      = ( ( ord_less_eq_int @ A2 @ B2 )
        & ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B2 ) ) ) ).

% abs_le_iff
thf(fact_8456_abs__le__D2,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ B2 )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ B2 ) ) ).

% abs_le_D2
thf(fact_8457_abs__le__D2,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ B2 )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 ) ) ).

% abs_le_D2
thf(fact_8458_abs__le__D2,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ B2 )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ B2 ) ) ).

% abs_le_D2
thf(fact_8459_abs__le__D2,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B2 )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B2 ) ) ).

% abs_le_D2
thf(fact_8460_abs__leI,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ B2 )
       => ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ B2 ) ) ) ).

% abs_leI
thf(fact_8461_abs__leI,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A2 @ B2 )
     => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 )
       => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ B2 ) ) ) ).

% abs_leI
thf(fact_8462_abs__leI,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_eq_rat @ A2 @ B2 )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ B2 )
       => ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ B2 ) ) ) ).

% abs_leI
thf(fact_8463_abs__leI,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B2 )
       => ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B2 ) ) ) ).

% abs_leI
thf(fact_8464_abs__less__iff,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A2 ) @ B2 )
      = ( ( ord_less_real @ A2 @ B2 )
        & ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ B2 ) ) ) ).

% abs_less_iff
thf(fact_8465_abs__less__iff,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A2 ) @ B2 )
      = ( ( ord_less_int @ A2 @ B2 )
        & ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B2 ) ) ) ).

% abs_less_iff
thf(fact_8466_abs__less__iff,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A2 ) @ B2 )
      = ( ( ord_le6747313008572928689nteger @ A2 @ B2 )
        & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B2 ) ) ) ).

% abs_less_iff
thf(fact_8467_abs__less__iff,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A2 ) @ B2 )
      = ( ( ord_less_rat @ A2 @ B2 )
        & ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ B2 ) ) ) ).

% abs_less_iff
thf(fact_8468_pi__not__less__zero,axiom,
    ~ ( ord_less_real @ pi @ zero_zero_real ) ).

% pi_not_less_zero
thf(fact_8469_pi__gt__zero,axiom,
    ord_less_real @ zero_zero_real @ pi ).

% pi_gt_zero
thf(fact_8470_pi__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ pi ).

% pi_ge_zero
thf(fact_8471_dense__eq0__I,axiom,
    ! [X: real] :
      ( ! [E2: real] :
          ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E2 ) )
     => ( X = zero_zero_real ) ) ).

% dense_eq0_I
thf(fact_8472_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E2 ) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
thf(fact_8473_abs__eq__mult,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
          | ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) )
        & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
          | ( ord_le3102999989581377725nteger @ B2 @ zero_z3403309356797280102nteger ) ) )
     => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B2 ) )
        = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) ) ) ) ).

% abs_eq_mult
thf(fact_8474_abs__eq__mult,axiom,
    ! [A2: real,B2: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
          | ( ord_less_eq_real @ A2 @ zero_zero_real ) )
        & ( ( ord_less_eq_real @ zero_zero_real @ B2 )
          | ( ord_less_eq_real @ B2 @ zero_zero_real ) ) )
     => ( ( abs_abs_real @ ( times_times_real @ A2 @ B2 ) )
        = ( times_times_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B2 ) ) ) ) ).

% abs_eq_mult
thf(fact_8475_abs__eq__mult,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
          | ( ord_less_eq_rat @ A2 @ zero_zero_rat ) )
        & ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
          | ( ord_less_eq_rat @ B2 @ zero_zero_rat ) ) )
     => ( ( abs_abs_rat @ ( times_times_rat @ A2 @ B2 ) )
        = ( times_times_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) ) ) ) ).

% abs_eq_mult
thf(fact_8476_abs__eq__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
          | ( ord_less_eq_int @ A2 @ zero_zero_int ) )
        & ( ( ord_less_eq_int @ zero_zero_int @ B2 )
          | ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
     => ( ( abs_abs_int @ ( times_times_int @ A2 @ B2 ) )
        = ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) ) ) ).

% abs_eq_mult
thf(fact_8477_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_8478_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_8479_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_8480_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_8481_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_8482_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_8483_zero__le__power__abs,axiom,
    ! [A2: real,N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A2 ) @ N3 ) ) ).

% zero_le_power_abs
thf(fact_8484_zero__le__power__abs,axiom,
    ! [A2: code_integer,N3: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A2 ) @ N3 ) ) ).

% zero_le_power_abs
thf(fact_8485_zero__le__power__abs,axiom,
    ! [A2: rat,N3: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A2 ) @ N3 ) ) ).

% zero_le_power_abs
thf(fact_8486_zero__le__power__abs,axiom,
    ! [A2: int,N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A2 ) @ N3 ) ) ).

% zero_le_power_abs
thf(fact_8487_abs__minus__le__zero,axiom,
    ! [A2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A2 ) ) @ zero_zero_real ) ).

% abs_minus_le_zero
thf(fact_8488_abs__minus__le__zero,axiom,
    ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A2 ) ) @ zero_z3403309356797280102nteger ) ).

% abs_minus_le_zero
thf(fact_8489_abs__minus__le__zero,axiom,
    ! [A2: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A2 ) ) @ zero_zero_rat ) ).

% abs_minus_le_zero
thf(fact_8490_abs__minus__le__zero,axiom,
    ! [A2: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A2 ) ) @ zero_zero_int ) ).

% abs_minus_le_zero
thf(fact_8491_eq__abs__iff_H,axiom,
    ! [A2: real,B2: real] :
      ( ( A2
        = ( abs_abs_real @ B2 ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A2 )
        & ( ( B2 = A2 )
          | ( B2
            = ( uminus_uminus_real @ A2 ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_8492_eq__abs__iff_H,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( A2
        = ( abs_abs_Code_integer @ B2 ) )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
        & ( ( B2 = A2 )
          | ( B2
            = ( uminus1351360451143612070nteger @ A2 ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_8493_eq__abs__iff_H,axiom,
    ! [A2: rat,B2: rat] :
      ( ( A2
        = ( abs_abs_rat @ B2 ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
        & ( ( B2 = A2 )
          | ( B2
            = ( uminus_uminus_rat @ A2 ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_8494_eq__abs__iff_H,axiom,
    ! [A2: int,B2: int] :
      ( ( A2
        = ( abs_abs_int @ B2 ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ A2 )
        & ( ( B2 = A2 )
          | ( B2
            = ( uminus_uminus_int @ A2 ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_8495_abs__eq__iff_H,axiom,
    ! [A2: real,B2: real] :
      ( ( ( abs_abs_real @ A2 )
        = B2 )
      = ( ( ord_less_eq_real @ zero_zero_real @ B2 )
        & ( ( A2 = B2 )
          | ( A2
            = ( uminus_uminus_real @ B2 ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_8496_abs__eq__iff_H,axiom,
    ! [A2: code_integer,B2: code_integer] :
      ( ( ( abs_abs_Code_integer @ A2 )
        = B2 )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B2 )
        & ( ( A2 = B2 )
          | ( A2
            = ( uminus1351360451143612070nteger @ B2 ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_8497_abs__eq__iff_H,axiom,
    ! [A2: rat,B2: rat] :
      ( ( ( abs_abs_rat @ A2 )
        = B2 )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ B2 )
        & ( ( A2 = B2 )
          | ( A2
            = ( uminus_uminus_rat @ B2 ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_8498_abs__eq__iff_H,axiom,
    ! [A2: int,B2: int] :
      ( ( ( abs_abs_int @ A2 )
        = B2 )
      = ( ( ord_less_eq_int @ zero_zero_int @ B2 )
        & ( ( A2 = B2 )
          | ( A2
            = ( uminus_uminus_int @ B2 ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_8499_abs__if__raw,axiom,
    ( abs_abs_real
    = ( ^ [A7: real] : ( if_real @ ( ord_less_real @ A7 @ zero_zero_real ) @ ( uminus_uminus_real @ A7 ) @ A7 ) ) ) ).

% abs_if_raw
thf(fact_8500_abs__if__raw,axiom,
    ( abs_abs_int
    = ( ^ [A7: int] : ( if_int @ ( ord_less_int @ A7 @ zero_zero_int ) @ ( uminus_uminus_int @ A7 ) @ A7 ) ) ) ).

% abs_if_raw
thf(fact_8501_abs__if__raw,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A7: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A7 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A7 ) @ A7 ) ) ) ).

% abs_if_raw
thf(fact_8502_abs__if__raw,axiom,
    ( abs_abs_rat
    = ( ^ [A7: rat] : ( if_rat @ ( ord_less_rat @ A7 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A7 ) @ A7 ) ) ) ).

% abs_if_raw
thf(fact_8503_abs__of__neg,axiom,
    ! [A2: real] :
      ( ( ord_less_real @ A2 @ zero_zero_real )
     => ( ( abs_abs_real @ A2 )
        = ( uminus_uminus_real @ A2 ) ) ) ).

% abs_of_neg
thf(fact_8504_abs__of__neg,axiom,
    ! [A2: int] :
      ( ( ord_less_int @ A2 @ zero_zero_int )
     => ( ( abs_abs_int @ A2 )
        = ( uminus_uminus_int @ A2 ) ) ) ).

% abs_of_neg
thf(fact_8505_abs__of__neg,axiom,
    ! [A2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A2 )
        = ( uminus1351360451143612070nteger @ A2 ) ) ) ).

% abs_of_neg
thf(fact_8506_abs__of__neg,axiom,
    ! [A2: rat] :
      ( ( ord_less_rat @ A2 @ zero_zero_rat )
     => ( ( abs_abs_rat @ A2 )
        = ( uminus_uminus_rat @ A2 ) ) ) ).

% abs_of_neg
thf(fact_8507_abs__if,axiom,
    ( abs_abs_real
    = ( ^ [A7: real] : ( if_real @ ( ord_less_real @ A7 @ zero_zero_real ) @ ( uminus_uminus_real @ A7 ) @ A7 ) ) ) ).

% abs_if
thf(fact_8508_abs__if,axiom,
    ( abs_abs_int
    = ( ^ [A7: int] : ( if_int @ ( ord_less_int @ A7 @ zero_zero_int ) @ ( uminus_uminus_int @ A7 ) @ A7 ) ) ) ).

% abs_if
thf(fact_8509_abs__if,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A7: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A7 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A7 ) @ A7 ) ) ) ).

% abs_if
thf(fact_8510_abs__if,axiom,
    ( abs_abs_rat
    = ( ^ [A7: rat] : ( if_rat @ ( ord_less_rat @ A7 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A7 ) @ A7 ) ) ) ).

% abs_if
thf(fact_8511_abs__diff__le__iff,axiom,
    ! [X: code_integer,A2: code_integer,R3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A2 ) ) @ R3 )
      = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A2 @ R3 ) @ X )
        & ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A2 @ R3 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_8512_abs__diff__le__iff,axiom,
    ! [X: real,A2: real,R3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A2 ) ) @ R3 )
      = ( ( ord_less_eq_real @ ( minus_minus_real @ A2 @ R3 ) @ X )
        & ( ord_less_eq_real @ X @ ( plus_plus_real @ A2 @ R3 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_8513_abs__diff__le__iff,axiom,
    ! [X: rat,A2: rat,R3: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A2 ) ) @ R3 )
      = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ R3 ) @ X )
        & ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A2 @ R3 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_8514_abs__diff__le__iff,axiom,
    ! [X: int,A2: int,R3: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A2 ) ) @ R3 )
      = ( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ R3 ) @ X )
        & ( ord_less_eq_int @ X @ ( plus_plus_int @ A2 @ R3 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_8515_abs__diff__triangle__ineq,axiom,
    ! [A2: code_integer,B2: code_integer,C2: code_integer,D2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ ( plus_p5714425477246183910nteger @ C2 @ D2 ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ C2 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B2 @ D2 ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_8516_abs__diff__triangle__ineq,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A2 @ B2 ) @ ( plus_plus_real @ C2 @ D2 ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A2 @ C2 ) ) @ ( abs_abs_real @ ( minus_minus_real @ B2 @ D2 ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_8517_abs__diff__triangle__ineq,axiom,
    ! [A2: rat,B2: rat,C2: rat,D2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B2 ) @ ( plus_plus_rat @ C2 @ D2 ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A2 @ C2 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B2 @ D2 ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_8518_abs__diff__triangle__ineq,axiom,
    ! [A2: int,B2: int,C2: int,D2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ ( plus_plus_int @ C2 @ D2 ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A2 @ C2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ D2 ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_8519_abs__triangle__ineq4,axiom,
    ! [A2: code_integer,B2: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ B2 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) ) ) ).

% abs_triangle_ineq4
thf(fact_8520_abs__triangle__ineq4,axiom,
    ! [A2: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A2 @ B2 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B2 ) ) ) ).

% abs_triangle_ineq4
thf(fact_8521_abs__triangle__ineq4,axiom,
    ! [A2: rat,B2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A2 @ B2 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) ) ) ).

% abs_triangle_ineq4
thf(fact_8522_abs__triangle__ineq4,axiom,
    ! [A2: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B2 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) ) ).

% abs_triangle_ineq4
thf(fact_8523_abs__diff__less__iff,axiom,
    ! [X: code_integer,A2: code_integer,R3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A2 ) ) @ R3 )
      = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A2 @ R3 ) @ X )
        & ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A2 @ R3 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_8524_abs__diff__less__iff,axiom,
    ! [X: real,A2: real,R3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A2 ) ) @ R3 )
      = ( ( ord_less_real @ ( minus_minus_real @ A2 @ R3 ) @ X )
        & ( ord_less_real @ X @ ( plus_plus_real @ A2 @ R3 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_8525_abs__diff__less__iff,axiom,
    ! [X: rat,A2: rat,R3: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A2 ) ) @ R3 )
      = ( ( ord_less_rat @ ( minus_minus_rat @ A2 @ R3 ) @ X )
        & ( ord_less_rat @ X @ ( plus_plus_rat @ A2 @ R3 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_8526_abs__diff__less__iff,axiom,
    ! [X: int,A2: int,R3: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A2 ) ) @ R3 )
      = ( ( ord_less_int @ ( minus_minus_int @ A2 @ R3 ) @ X )
        & ( ord_less_int @ X @ ( plus_plus_int @ A2 @ R3 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_8527_abs__real__def,axiom,
    ( abs_abs_real
    = ( ^ [A7: real] : ( if_real @ ( ord_less_real @ A7 @ zero_zero_real ) @ ( uminus_uminus_real @ A7 ) @ A7 ) ) ) ).

% abs_real_def
thf(fact_8528_lemma__interval__lt,axiom,
    ! [A2: real,X: real,B2: real] :
      ( ( ord_less_real @ A2 @ X )
     => ( ( ord_less_real @ X @ B2 )
       => ? [D5: real] :
            ( ( ord_less_real @ zero_zero_real @ D5 )
            & ! [Y4: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D5 )
               => ( ( ord_less_real @ A2 @ Y4 )
                  & ( ord_less_real @ Y4 @ B2 ) ) ) ) ) ) ).

% lemma_interval_lt
thf(fact_8529_sin__bound__lemma,axiom,
    ! [X: real,Y: real,U: real,V: real] :
      ( ( X = Y )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y ) ) @ V ) ) ) ).

% sin_bound_lemma
thf(fact_8530_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_8531_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_8532_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_8533_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_8534_of__int__leD,axiom,
    ! [N3: int,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N3 ) ) @ X )
     => ( ( N3 = zero_zero_int )
        | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).

% of_int_leD
thf(fact_8535_of__int__leD,axiom,
    ! [N3: int,X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N3 ) ) @ X )
     => ( ( N3 = zero_zero_int )
        | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% of_int_leD
thf(fact_8536_of__int__leD,axiom,
    ! [N3: int,X: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N3 ) ) @ X )
     => ( ( N3 = zero_zero_int )
        | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).

% of_int_leD
thf(fact_8537_of__int__leD,axiom,
    ! [N3: int,X: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N3 ) ) @ X )
     => ( ( N3 = zero_zero_int )
        | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).

% of_int_leD
thf(fact_8538_of__int__lessD,axiom,
    ! [N3: int,X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N3 ) ) @ X )
     => ( ( N3 = zero_zero_int )
        | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).

% of_int_lessD
thf(fact_8539_of__int__lessD,axiom,
    ! [N3: int,X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N3 ) ) @ X )
     => ( ( N3 = zero_zero_int )
        | ( ord_less_real @ one_one_real @ X ) ) ) ).

% of_int_lessD
thf(fact_8540_of__int__lessD,axiom,
    ! [N3: int,X: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N3 ) ) @ X )
     => ( ( N3 = zero_zero_int )
        | ( ord_less_rat @ one_one_rat @ X ) ) ) ).

% of_int_lessD
thf(fact_8541_of__int__lessD,axiom,
    ! [N3: int,X: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N3 ) ) @ X )
     => ( ( N3 = zero_zero_int )
        | ( ord_less_int @ one_one_int @ X ) ) ) ).

% of_int_lessD
thf(fact_8542_lemma__interval,axiom,
    ! [A2: real,X: real,B2: real] :
      ( ( ord_less_real @ A2 @ X )
     => ( ( ord_less_real @ X @ B2 )
       => ? [D5: real] :
            ( ( ord_less_real @ zero_zero_real @ D5 )
            & ! [Y4: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D5 )
               => ( ( ord_less_eq_real @ A2 @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ B2 ) ) ) ) ) ) ).

% lemma_interval
thf(fact_8543_round__diff__minimal,axiom,
    ! [Z: real,M: int] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ Z ) ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ Z @ ( ring_1_of_int_real @ M ) ) ) ) ).

% round_diff_minimal
thf(fact_8544_round__diff__minimal,axiom,
    ! [Z: rat,M: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ M ) ) ) ) ).

% round_diff_minimal
thf(fact_8545_norm__triangle__ineq3,axiom,
    ! [A2: real,B2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( real_V7735802525324610683m_real @ B2 ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A2 @ B2 ) ) ) ).

% norm_triangle_ineq3
thf(fact_8546_norm__triangle__ineq3,axiom,
    ! [A2: complex,B2: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A2 ) @ ( real_V1022390504157884413omplex @ B2 ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A2 @ B2 ) ) ) ).

% norm_triangle_ineq3
thf(fact_8547_pi__less__4,axiom,
    ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).

% pi_less_4
thf(fact_8548_pi__ge__two,axiom,
    ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).

% pi_ge_two
thf(fact_8549_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_8550_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_8551_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_8552_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_8553_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_8554_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_8555_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_8556_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_8557_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_8558_power__even__abs,axiom,
    ! [N3: nat,A2: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_rat @ ( abs_abs_rat @ A2 ) @ N3 )
        = ( power_power_rat @ A2 @ N3 ) ) ) ).

% power_even_abs
thf(fact_8559_power__even__abs,axiom,
    ! [N3: nat,A2: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_real @ ( abs_abs_real @ A2 ) @ N3 )
        = ( power_power_real @ A2 @ N3 ) ) ) ).

% power_even_abs
thf(fact_8560_power__even__abs,axiom,
    ! [N3: nat,A2: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_int @ ( abs_abs_int @ A2 ) @ N3 )
        = ( power_power_int @ A2 @ N3 ) ) ) ).

% power_even_abs
thf(fact_8561_power__even__abs,axiom,
    ! [N3: nat,A2: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A2 ) @ N3 )
        = ( power_8256067586552552935nteger @ A2 @ N3 ) ) ) ).

% power_even_abs
thf(fact_8562_pi__half__neq__zero,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% pi_half_neq_zero
thf(fact_8563_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_8564_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_8565_abs__sqrt__wlog,axiom,
    ! [P: real > real > $o,X: real] :
      ( ! [X4: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X4 )
         => ( P @ X4 @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_8566_abs__sqrt__wlog,axiom,
    ! [P: code_integer > code_integer > $o,X: code_integer] :
      ( ! [X4: code_integer] :
          ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X4 )
         => ( P @ X4 @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_8567_abs__sqrt__wlog,axiom,
    ! [P: rat > rat > $o,X: rat] :
      ( ! [X4: rat] :
          ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
         => ( P @ X4 @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_8568_abs__sqrt__wlog,axiom,
    ! [P: int > int > $o,X: int] :
      ( ! [X4: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X4 )
         => ( P @ X4 @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_8569_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_8570_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_8571_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_8572_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_8573_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_8574_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_8575_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_8576_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_8577_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_8578_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_8579_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_8580_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_8581_power__mono__even,axiom,
    ! [N3: nat,A2: real,B2: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B2 ) )
       => ( ord_less_eq_real @ ( power_power_real @ A2 @ N3 ) @ ( power_power_real @ B2 @ N3 ) ) ) ) ).

% power_mono_even
thf(fact_8582_power__mono__even,axiom,
    ! [N3: nat,A2: code_integer,B2: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B2 ) )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A2 @ N3 ) @ ( power_8256067586552552935nteger @ B2 @ N3 ) ) ) ) ).

% power_mono_even
thf(fact_8583_power__mono__even,axiom,
    ! [N3: nat,A2: rat,B2: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B2 ) )
       => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N3 ) @ ( power_power_rat @ B2 @ N3 ) ) ) ) ).

% power_mono_even
thf(fact_8584_power__mono__even,axiom,
    ! [N3: nat,A2: int,B2: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) )
       => ( ord_less_eq_int @ ( power_power_int @ A2 @ N3 ) @ ( power_power_int @ B2 @ N3 ) ) ) ) ).

% power_mono_even
thf(fact_8585_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_8586_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_8587_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_8588_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_8589_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_8590_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_8591_round__unique_H,axiom,
    ! [X: real,N3: int] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ N3 ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( archim8280529875227126926d_real @ X )
        = N3 ) ) ).

% round_unique'
thf(fact_8592_round__unique_H,axiom,
    ! [X: rat,N3: int] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N3 ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
     => ( ( archim7778729529865785530nd_rat @ X )
        = N3 ) ) ).

% round_unique'
thf(fact_8593_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_8594_ex__min__if__finite,axiom,
    ! [S: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( S != bot_bo3990330152332043303nteger )
       => ? [X4: code_integer] :
            ( ( member_Code_integer @ X4 @ S )
            & ~ ? [Xa2: code_integer] :
                  ( ( member_Code_integer @ Xa2 @ S )
                  & ( ord_le6747313008572928689nteger @ Xa2 @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_8595_ex__min__if__finite,axiom,
    ! [S: set_o] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ? [X4: $o] :
            ( ( member_o @ X4 @ S )
            & ~ ? [Xa2: $o] :
                  ( ( member_o @ Xa2 @ S )
                  & ( ord_less_o @ Xa2 @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_8596_ex__min__if__finite,axiom,
    ! [S: set_real] :
      ( ( finite_finite_real @ S )
     => ( ( S != bot_bot_set_real )
       => ? [X4: real] :
            ( ( member_real @ X4 @ S )
            & ~ ? [Xa2: real] :
                  ( ( member_real @ Xa2 @ S )
                  & ( ord_less_real @ Xa2 @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_8597_ex__min__if__finite,axiom,
    ! [S: set_rat] :
      ( ( finite_finite_rat @ S )
     => ( ( S != bot_bot_set_rat )
       => ? [X4: rat] :
            ( ( member_rat @ X4 @ S )
            & ~ ? [Xa2: rat] :
                  ( ( member_rat @ Xa2 @ S )
                  & ( ord_less_rat @ Xa2 @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_8598_ex__min__if__finite,axiom,
    ! [S: set_num] :
      ( ( finite_finite_num @ S )
     => ( ( S != bot_bot_set_num )
       => ? [X4: num] :
            ( ( member_num @ X4 @ S )
            & ~ ? [Xa2: num] :
                  ( ( member_num @ Xa2 @ S )
                  & ( ord_less_num @ Xa2 @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_8599_ex__min__if__finite,axiom,
    ! [S: set_nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ S )
            & ~ ? [Xa2: nat] :
                  ( ( member_nat @ Xa2 @ S )
                  & ( ord_less_nat @ Xa2 @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_8600_ex__min__if__finite,axiom,
    ! [S: set_int] :
      ( ( finite_finite_int @ S )
     => ( ( S != bot_bot_set_int )
       => ? [X4: int] :
            ( ( member_int @ X4 @ S )
            & ~ ? [Xa2: int] :
                  ( ( member_int @ Xa2 @ S )
                  & ( ord_less_int @ Xa2 @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_8601_infinite__growing,axiom,
    ! [X2: set_Code_integer] :
      ( ( X2 != bot_bo3990330152332043303nteger )
     => ( ! [X4: code_integer] :
            ( ( member_Code_integer @ X4 @ X2 )
           => ? [Xa2: code_integer] :
                ( ( member_Code_integer @ Xa2 @ X2 )
                & ( ord_le6747313008572928689nteger @ X4 @ Xa2 ) ) )
       => ~ ( finite6017078050557962740nteger @ X2 ) ) ) ).

% infinite_growing
thf(fact_8602_infinite__growing,axiom,
    ! [X2: set_o] :
      ( ( X2 != bot_bot_set_o )
     => ( ! [X4: $o] :
            ( ( member_o @ X4 @ X2 )
           => ? [Xa2: $o] :
                ( ( member_o @ Xa2 @ X2 )
                & ( ord_less_o @ X4 @ Xa2 ) ) )
       => ~ ( finite_finite_o @ X2 ) ) ) ).

% infinite_growing
thf(fact_8603_infinite__growing,axiom,
    ! [X2: set_real] :
      ( ( X2 != bot_bot_set_real )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ X2 )
           => ? [Xa2: real] :
                ( ( member_real @ Xa2 @ X2 )
                & ( ord_less_real @ X4 @ Xa2 ) ) )
       => ~ ( finite_finite_real @ X2 ) ) ) ).

% infinite_growing
thf(fact_8604_infinite__growing,axiom,
    ! [X2: set_rat] :
      ( ( X2 != bot_bot_set_rat )
     => ( ! [X4: rat] :
            ( ( member_rat @ X4 @ X2 )
           => ? [Xa2: rat] :
                ( ( member_rat @ Xa2 @ X2 )
                & ( ord_less_rat @ X4 @ Xa2 ) ) )
       => ~ ( finite_finite_rat @ X2 ) ) ) ).

% infinite_growing
thf(fact_8605_infinite__growing,axiom,
    ! [X2: set_num] :
      ( ( X2 != bot_bot_set_num )
     => ( ! [X4: num] :
            ( ( member_num @ X4 @ X2 )
           => ? [Xa2: num] :
                ( ( member_num @ Xa2 @ X2 )
                & ( ord_less_num @ X4 @ Xa2 ) ) )
       => ~ ( finite_finite_num @ X2 ) ) ) ).

% infinite_growing
thf(fact_8606_infinite__growing,axiom,
    ! [X2: set_nat] :
      ( ( X2 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ X2 )
           => ? [Xa2: nat] :
                ( ( member_nat @ Xa2 @ X2 )
                & ( ord_less_nat @ X4 @ Xa2 ) ) )
       => ~ ( finite_finite_nat @ X2 ) ) ) ).

% infinite_growing
thf(fact_8607_infinite__growing,axiom,
    ! [X2: set_int] :
      ( ( X2 != bot_bot_set_int )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ X2 )
           => ? [Xa2: int] :
                ( ( member_int @ Xa2 @ X2 )
                & ( ord_less_int @ X4 @ Xa2 ) ) )
       => ~ ( finite_finite_int @ X2 ) ) ) ).

% infinite_growing
thf(fact_8608_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_8609_finite__ranking__induct,axiom,
    ! [S: set_VEBT_VEBT,P: set_VEBT_VEBT > $o,F2: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ S )
     => ( ( P @ bot_bo8194388402131092736T_VEBT )
       => ( ! [X4: vEBT_VEBT,S6: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ S6 )
             => ( ! [Y4: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ Y4 @ S6 )
                   => ( ord_less_eq_rat @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_VEBT_VEBT @ X4 @ S6 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_8610_finite__ranking__induct,axiom,
    ! [S: set_Code_integer,P: set_Code_integer > $o,F2: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [X4: code_integer,S6: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ S6 )
             => ( ! [Y4: code_integer] :
                    ( ( member_Code_integer @ Y4 @ S6 )
                   => ( ord_less_eq_rat @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_Code_integer @ X4 @ S6 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_8611_finite__ranking__induct,axiom,
    ! [S: set_complex,P: set_complex > $o,F2: complex > rat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X4: complex,S6: set_complex] :
              ( ( finite3207457112153483333omplex @ S6 )
             => ( ! [Y4: complex] :
                    ( ( member_complex @ Y4 @ S6 )
                   => ( ord_less_eq_rat @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_complex @ X4 @ S6 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_8612_finite__ranking__induct,axiom,
    ! [S: set_real,P: set_real > $o,F2: real > rat] :
      ( ( finite_finite_real @ S )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X4: real,S6: set_real] :
              ( ( finite_finite_real @ S6 )
             => ( ! [Y4: real] :
                    ( ( member_real @ Y4 @ S6 )
                   => ( ord_less_eq_rat @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_real @ X4 @ S6 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_8613_finite__ranking__induct,axiom,
    ! [S: set_o,P: set_o > $o,F2: $o > rat] :
      ( ( finite_finite_o @ S )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X4: $o,S6: set_o] :
              ( ( finite_finite_o @ S6 )
             => ( ! [Y4: $o] :
                    ( ( member_o @ Y4 @ S6 )
                   => ( ord_less_eq_rat @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_o @ X4 @ S6 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_8614_finite__ranking__induct,axiom,
    ! [S: set_nat,P: set_nat > $o,F2: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X4: nat,S6: set_nat] :
              ( ( finite_finite_nat @ S6 )
             => ( ! [Y4: nat] :
                    ( ( member_nat @ Y4 @ S6 )
                   => ( ord_less_eq_rat @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_nat @ X4 @ S6 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_8615_finite__ranking__induct,axiom,
    ! [S: set_int,P: set_int > $o,F2: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X4: int,S6: set_int] :
              ( ( finite_finite_int @ S6 )
             => ( ! [Y4: int] :
                    ( ( member_int @ Y4 @ S6 )
                   => ( ord_less_eq_rat @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_int @ X4 @ S6 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_8616_finite__ranking__induct,axiom,
    ! [S: set_VEBT_VEBT,P: set_VEBT_VEBT > $o,F2: vEBT_VEBT > num] :
      ( ( finite5795047828879050333T_VEBT @ S )
     => ( ( P @ bot_bo8194388402131092736T_VEBT )
       => ( ! [X4: vEBT_VEBT,S6: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ S6 )
             => ( ! [Y4: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ Y4 @ S6 )
                   => ( ord_less_eq_num @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_VEBT_VEBT @ X4 @ S6 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_8617_finite__ranking__induct,axiom,
    ! [S: set_Code_integer,P: set_Code_integer > $o,F2: code_integer > num] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [X4: code_integer,S6: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ S6 )
             => ( ! [Y4: code_integer] :
                    ( ( member_Code_integer @ Y4 @ S6 )
                   => ( ord_less_eq_num @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_Code_integer @ X4 @ S6 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_8618_finite__ranking__induct,axiom,
    ! [S: set_complex,P: set_complex > $o,F2: complex > num] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X4: complex,S6: set_complex] :
              ( ( finite3207457112153483333omplex @ S6 )
             => ( ! [Y4: complex] :
                    ( ( member_complex @ Y4 @ S6 )
                   => ( ord_less_eq_num @ ( F2 @ Y4 ) @ ( F2 @ X4 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_complex @ X4 @ S6 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_8619_finite__linorder__min__induct,axiom,
    ! [A: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [B3: code_integer,A8: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A8 )
             => ( ! [X5: code_integer] :
                    ( ( member_Code_integer @ X5 @ A8 )
                   => ( ord_le6747313008572928689nteger @ B3 @ X5 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_Code_integer @ B3 @ A8 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_8620_finite__linorder__min__induct,axiom,
    ! [A: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ A )
     => ( ( P @ bot_bot_set_o )
       => ( ! [B3: $o,A8: set_o] :
              ( ( finite_finite_o @ A8 )
             => ( ! [X5: $o] :
                    ( ( member_o @ X5 @ A8 )
                   => ( ord_less_o @ B3 @ X5 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_o @ B3 @ A8 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_8621_finite__linorder__min__induct,axiom,
    ! [A: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B3: real,A8: set_real] :
              ( ( finite_finite_real @ A8 )
             => ( ! [X5: real] :
                    ( ( member_real @ X5 @ A8 )
                   => ( ord_less_real @ B3 @ X5 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_real @ B3 @ A8 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_8622_finite__linorder__min__induct,axiom,
    ! [A: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B3: rat,A8: set_rat] :
              ( ( finite_finite_rat @ A8 )
             => ( ! [X5: rat] :
                    ( ( member_rat @ X5 @ A8 )
                   => ( ord_less_rat @ B3 @ X5 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_rat @ B3 @ A8 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_8623_finite__linorder__min__induct,axiom,
    ! [A: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B3: num,A8: set_num] :
              ( ( finite_finite_num @ A8 )
             => ( ! [X5: num] :
                    ( ( member_num @ X5 @ A8 )
                   => ( ord_less_num @ B3 @ X5 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_num @ B3 @ A8 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_8624_finite__linorder__min__induct,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B3: nat,A8: set_nat] :
              ( ( finite_finite_nat @ A8 )
             => ( ! [X5: nat] :
                    ( ( member_nat @ X5 @ A8 )
                   => ( ord_less_nat @ B3 @ X5 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_nat @ B3 @ A8 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_8625_finite__linorder__min__induct,axiom,
    ! [A: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B3: int,A8: set_int] :
              ( ( finite_finite_int @ A8 )
             => ( ! [X5: int] :
                    ( ( member_int @ X5 @ A8 )
                   => ( ord_less_int @ B3 @ X5 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_int @ B3 @ A8 ) ) ) ) )
         => ( P @ A ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_8626_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( topolo6980174941875973593q_real
        @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% monoseq_arctan_series
thf(fact_8627_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_8628_sin__cos__npi,axiom,
    ! [N3: nat] :
      ( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) ) ).

% sin_cos_npi
thf(fact_8629_signed__take__bit__rec,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N2: nat,A7: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A7 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A7 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A7 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_8630_signed__take__bit__rec,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N2: nat,A7: int] : ( if_int @ ( N2 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A7 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A7 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A7 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_8631_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_8632_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_8633_dbl__simps_I4_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_8634_dbl__simps_I4_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_8635_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_8636_signed__take__bit__of__0,axiom,
    ! [N3: nat] :
      ( ( bit_ri631733984087533419it_int @ N3 @ zero_zero_int )
      = zero_zero_int ) ).

% signed_take_bit_of_0
thf(fact_8637_sin__zero,axiom,
    ( ( sin_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% sin_zero
thf(fact_8638_sin__zero,axiom,
    ( ( sin_real @ zero_zero_real )
    = zero_zero_real ) ).

% sin_zero
thf(fact_8639_arctan__zero__zero,axiom,
    ( ( arctan @ zero_zero_real )
    = zero_zero_real ) ).

% arctan_zero_zero
thf(fact_8640_arctan__eq__zero__iff,axiom,
    ! [X: real] :
      ( ( ( arctan @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% arctan_eq_zero_iff
thf(fact_8641_dbl__simps_I2_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% dbl_simps(2)
thf(fact_8642_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_real @ zero_zero_real )
    = zero_zero_real ) ).

% dbl_simps(2)
thf(fact_8643_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% dbl_simps(2)
thf(fact_8644_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_int @ zero_zero_int )
    = zero_zero_int ) ).

% dbl_simps(2)
thf(fact_8645_signed__take__bit__Suc__1,axiom,
    ! [N3: nat] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N3 ) @ one_one_int )
      = one_one_int ) ).

% signed_take_bit_Suc_1
thf(fact_8646_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_8647_signed__take__bit__of__minus__1,axiom,
    ! [N3: nat] :
      ( ( bit_ri6519982836138164636nteger @ N3 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% signed_take_bit_of_minus_1
thf(fact_8648_signed__take__bit__of__minus__1,axiom,
    ! [N3: nat] :
      ( ( bit_ri631733984087533419it_int @ N3 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% signed_take_bit_of_minus_1
thf(fact_8649_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
      = ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
thf(fact_8650_sin__pi,axiom,
    ( ( sin_real @ pi )
    = zero_zero_real ) ).

% sin_pi
thf(fact_8651_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_8652_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_8653_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_8654_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_8655_sin__pi__minus,axiom,
    ! [X: real] :
      ( ( sin_real @ ( minus_minus_real @ pi @ X ) )
      = ( sin_real @ X ) ) ).

% sin_pi_minus
thf(fact_8656_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
      = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_8657_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_8658_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_8659_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_8660_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_8661_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_8662_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_8663_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_8664_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_8665_sin__periodic__pi2,axiom,
    ! [X: real] :
      ( ( sin_real @ ( plus_plus_real @ pi @ X ) )
      = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).

% sin_periodic_pi2
thf(fact_8666_sin__periodic__pi,axiom,
    ! [X: real] :
      ( ( sin_real @ ( plus_plus_real @ X @ pi ) )
      = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).

% sin_periodic_pi
thf(fact_8667_sin__minus__pi,axiom,
    ! [X: real] :
      ( ( sin_real @ ( minus_minus_real @ X @ pi ) )
      = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).

% sin_minus_pi
thf(fact_8668_signed__take__bit__Suc__bit0,axiom,
    ! [N3: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N3 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N3 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_bit0
thf(fact_8669_sin__npi,axiom,
    ! [N3: nat] :
      ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ pi ) )
      = zero_zero_real ) ).

% sin_npi
thf(fact_8670_sin__npi2,axiom,
    ! [N3: nat] :
      ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N3 ) ) )
      = zero_zero_real ) ).

% sin_npi2
thf(fact_8671_dbl__simps_I3_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_8672_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_8673_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_8674_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_8675_sin__npi__int,axiom,
    ! [N3: int] :
      ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N3 ) ) )
      = zero_zero_real ) ).

% sin_npi_int
thf(fact_8676_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N3: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N3 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_minus_bit0
thf(fact_8677_signed__take__bit__0,axiom,
    ! [A2: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A2 )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% signed_take_bit_0
thf(fact_8678_signed__take__bit__0,axiom,
    ! [A2: int] :
      ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A2 )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% signed_take_bit_0
thf(fact_8679_sin__two__pi,axiom,
    ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = zero_zero_real ) ).

% sin_two_pi
thf(fact_8680_sin__pi__half,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = one_one_real ) ).

% sin_pi_half
thf(fact_8681_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_8682_signed__take__bit__Suc__bit1,axiom,
    ! [N3: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N3 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N3 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_bit1
thf(fact_8683_sin__2npi,axiom,
    ! [N3: nat] :
      ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) @ pi ) )
      = zero_zero_real ) ).

% sin_2npi
thf(fact_8684_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_8685_sin__int__2pin,axiom,
    ! [N3: int] :
      ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N3 ) ) )
      = zero_zero_real ) ).

% sin_int_2pin
thf(fact_8686_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N3: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N3 @ ( 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_8687_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_8688_signed__take__bit__add,axiom,
    ! [N3: nat,K: int,L2: int] :
      ( ( bit_ri631733984087533419it_int @ N3 @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N3 @ K ) @ ( bit_ri631733984087533419it_int @ N3 @ L2 ) ) )
      = ( bit_ri631733984087533419it_int @ N3 @ ( plus_plus_int @ K @ L2 ) ) ) ).

% signed_take_bit_add
thf(fact_8689_signed__take__bit__diff,axiom,
    ! [N3: nat,K: int,L2: int] :
      ( ( bit_ri631733984087533419it_int @ N3 @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N3 @ K ) @ ( bit_ri631733984087533419it_int @ N3 @ L2 ) ) )
      = ( bit_ri631733984087533419it_int @ N3 @ ( minus_minus_int @ K @ L2 ) ) ) ).

% signed_take_bit_diff
thf(fact_8690_signed__take__bit__mult,axiom,
    ! [N3: nat,K: int,L2: int] :
      ( ( bit_ri631733984087533419it_int @ N3 @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N3 @ K ) @ ( bit_ri631733984087533419it_int @ N3 @ L2 ) ) )
      = ( bit_ri631733984087533419it_int @ N3 @ ( times_times_int @ K @ L2 ) ) ) ).

% signed_take_bit_mult
thf(fact_8691_arctan__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% arctan_le_iff
thf(fact_8692_arctan__monotone_H,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).

% arctan_monotone'
thf(fact_8693_arctan__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% arctan_less_iff
thf(fact_8694_arctan__monotone,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).

% arctan_monotone
thf(fact_8695_signed__take__bit__minus,axiom,
    ! [N3: nat,K: int] :
      ( ( bit_ri631733984087533419it_int @ N3 @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N3 @ K ) ) )
      = ( bit_ri631733984087533419it_int @ N3 @ ( uminus_uminus_int @ K ) ) ) ).

% signed_take_bit_minus
thf(fact_8696_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_8697_sin__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( sin_real @ X ) @ one_one_real ) ).

% sin_le_one
thf(fact_8698_abs__sin__x__le__abs__x,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ ( abs_abs_real @ X ) ) ).

% abs_sin_x_le_abs_x
thf(fact_8699_dbl__def,axiom,
    ( neg_numeral_dbl_real
    = ( ^ [X3: real] : ( plus_plus_real @ X3 @ X3 ) ) ) ).

% dbl_def
thf(fact_8700_dbl__def,axiom,
    ( neg_numeral_dbl_rat
    = ( ^ [X3: rat] : ( plus_plus_rat @ X3 @ X3 ) ) ) ).

% dbl_def
thf(fact_8701_dbl__def,axiom,
    ( neg_numeral_dbl_int
    = ( ^ [X3: int] : ( plus_plus_int @ X3 @ X3 ) ) ) ).

% dbl_def
thf(fact_8702_abs__div,axiom,
    ! [Y: int,X: int] :
      ( ( dvd_dvd_int @ Y @ X )
     => ( ( abs_abs_int @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) ) ) ) ).

% abs_div
thf(fact_8703_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_8704_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_8705_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_8706_sin__ge__minus__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X ) ) ).

% sin_ge_minus_one
thf(fact_8707_abs__sin__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ one_one_real ) ).

% abs_sin_le_one
thf(fact_8708_zabs__def,axiom,
    ( abs_abs_int
    = ( ^ [I2: int] : ( if_int @ ( ord_less_int @ I2 @ zero_zero_int ) @ ( uminus_uminus_int @ I2 ) @ I2 ) ) ) ).

% zabs_def
thf(fact_8709_abs__mod__less,axiom,
    ! [L2: int,K: int] :
      ( ( L2 != zero_zero_int )
     => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L2 ) ) @ ( abs_abs_int @ L2 ) ) ) ).

% abs_mod_less
thf(fact_8710_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] :
      ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
      = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).

% nat_abs_mult_distrib
thf(fact_8711_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_8712_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_8713_nat__abs__triangle__ineq,axiom,
    ! [K: int,L2: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L2 ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ).

% nat_abs_triangle_ineq
thf(fact_8714_sin__zero__iff__int2,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [I2: int] :
            ( X
            = ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ pi ) ) ) ) ).

% sin_zero_iff_int2
thf(fact_8715_div__abs__eq__div__nat,axiom,
    ! [K: int,L2: int] :
      ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ).

% div_abs_eq_div_nat
thf(fact_8716_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_8717_signed__take__bit__int__less__exp,axiom,
    ! [N3: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N3 @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ).

% signed_take_bit_int_less_exp
thf(fact_8718_even__signed__take__bit__iff,axiom,
    ! [M: nat,A2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A2 ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) ).

% even_signed_take_bit_iff
thf(fact_8719_even__add__abs__iff,axiom,
    ! [K: int,L2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L2 ) ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).

% even_add_abs_iff
thf(fact_8720_even__abs__add__iff,axiom,
    ! [K: int,L2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L2 ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).

% even_abs_add_iff
thf(fact_8721_nat__abs__int__diff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
          = ( minus_minus_nat @ B2 @ A2 ) ) )
      & ( ~ ( ord_less_eq_nat @ A2 @ B2 )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
          = ( minus_minus_nat @ A2 @ B2 ) ) ) ) ).

% nat_abs_int_diff
thf(fact_8722_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_8723_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N3: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N3 @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% signed_take_bit_int_greater_eq_self_iff
thf(fact_8724_signed__take__bit__int__less__self__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N3 @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ K ) ) ).

% signed_take_bit_int_less_self_iff
thf(fact_8725_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N3 @ K ) @ K )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) @ K ) ) ).

% signed_take_bit_int_less_eq_self_iff
thf(fact_8726_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N3: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) @ ( bit_ri631733984087533419it_int @ N3 @ K ) ) ).

% signed_take_bit_int_greater_eq_minus_exp
thf(fact_8727_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N3: nat] :
      ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N3 @ K ) )
      = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% signed_take_bit_int_greater_self_iff
thf(fact_8728_sin__pi__divide__n__ge__0,axiom,
    ! [N3: nat] :
      ( ( N3 != zero_zero_nat )
     => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ).

% sin_pi_divide_n_ge_0
thf(fact_8729_nat__intermed__int__val,axiom,
    ! [M: nat,N3: nat,F2: nat > int,K: int] :
      ( ! [I5: nat] :
          ( ( ( ord_less_eq_nat @ M @ I5 )
            & ( ord_less_nat @ I5 @ N3 ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F2 @ ( suc @ I5 ) ) @ ( F2 @ I5 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M @ N3 )
       => ( ( ord_less_eq_int @ ( F2 @ M ) @ K )
         => ( ( ord_less_eq_int @ K @ ( F2 @ N3 ) )
           => ? [I5: nat] :
                ( ( ord_less_eq_nat @ M @ I5 )
                & ( ord_less_eq_nat @ I5 @ N3 )
                & ( ( F2 @ I5 )
                  = K ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_8730_decr__lemma,axiom,
    ! [D2: int,X: int,Z: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D2 ) ) @ Z ) ) ).

% decr_lemma
thf(fact_8731_incr__lemma,axiom,
    ! [D2: int,Z: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ D2 )
     => ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D2 ) ) ) ) ).

% incr_lemma
thf(fact_8732_arctan__ubound,axiom,
    ! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arctan_ubound
thf(fact_8733_arctan__one,axiom,
    ( ( arctan @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% arctan_one
thf(fact_8734_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_8735_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_8736_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_8737_signed__take__bit__int__less__eq,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ K )
     => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N3 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) ) ) ) ).

% signed_take_bit_int_less_eq
thf(fact_8738_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_8739_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_8740_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_8741_signed__take__bit__int__eq__self,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
       => ( ( bit_ri631733984087533419it_int @ N3 @ K )
          = K ) ) ) ).

% signed_take_bit_int_eq_self
thf(fact_8742_signed__take__bit__int__eq__self__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ( bit_ri631733984087533419it_int @ N3 @ K )
        = K )
      = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% signed_take_bit_int_eq_self_iff
thf(fact_8743_nat__ivt__aux,axiom,
    ! [N3: nat,F2: nat > int,K: int] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ N3 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F2 @ ( suc @ I5 ) ) @ ( F2 @ I5 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F2 @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F2 @ N3 ) )
         => ? [I5: nat] :
              ( ( ord_less_eq_nat @ I5 @ N3 )
              & ( ( F2 @ I5 )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_8744_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_8745_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_8746_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_8747_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_8748_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_8749_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_8750_sin__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ? [X4: real] :
            ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
            & ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( sin_real @ X4 )
              = Y )
            & ! [Y4: real] :
                ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
                  & ( ( sin_real @ Y4 )
                    = Y ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% sin_total
thf(fact_8751_nat0__intermed__int__val,axiom,
    ! [N3: nat,F2: nat > int,K: int] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ N3 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F2 @ ( plus_plus_nat @ I5 @ one_one_nat ) ) @ ( F2 @ I5 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F2 @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F2 @ N3 ) )
         => ? [I5: nat] :
              ( ( ord_less_eq_nat @ I5 @ N3 )
              & ( ( F2 @ I5 )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_8752_sin__pi__divide__n__gt__0,axiom,
    ! [N3: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ).

% sin_pi_divide_n_gt_0
thf(fact_8753_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N3: nat] :
      ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) ) @ ( bit_ri631733984087533419it_int @ N3 @ K ) ) ) ).

% signed_take_bit_int_greater_eq
thf(fact_8754_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_8755_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_8756_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_8757_signed__take__bit__Suc,axiom,
    ! [N3: nat,A2: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ ( suc @ N3 ) @ A2 )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N3 @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_8758_signed__take__bit__Suc,axiom,
    ! [N3: nat,A2: int] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N3 ) @ A2 )
      = ( plus_plus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N3 @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_8759_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [I2: int] :
            ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I2 )
            & ( X
              = ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_iff_int
thf(fact_8760_sin__zero__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( sin_real @ X )
          = zero_zero_real )
       => ? [N: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_lemma
thf(fact_8761_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( ( 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 ) ) ) ) ) )
        | ? [N2: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% sin_zero_iff
thf(fact_8762_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_8763_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_8764_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L2: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( 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_8765_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_8766_summable__single,axiom,
    ! [I: nat,F2: nat > complex] :
      ( summable_complex
      @ ^ [R5: nat] : ( if_complex @ ( R5 = I ) @ ( F2 @ R5 ) @ zero_zero_complex ) ) ).

% summable_single
thf(fact_8767_summable__single,axiom,
    ! [I: nat,F2: nat > real] :
      ( summable_real
      @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F2 @ R5 ) @ zero_zero_real ) ) ).

% summable_single
thf(fact_8768_summable__single,axiom,
    ! [I: nat,F2: nat > nat] :
      ( summable_nat
      @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F2 @ R5 ) @ zero_zero_nat ) ) ).

% summable_single
thf(fact_8769_summable__single,axiom,
    ! [I: nat,F2: nat > int] :
      ( summable_int
      @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F2 @ R5 ) @ zero_zero_int ) ) ).

% summable_single
thf(fact_8770_summable__zero,axiom,
    ( summable_complex
    @ ^ [N2: nat] : zero_zero_complex ) ).

% summable_zero
thf(fact_8771_summable__zero,axiom,
    ( summable_real
    @ ^ [N2: nat] : zero_zero_real ) ).

% summable_zero
thf(fact_8772_summable__zero,axiom,
    ( summable_nat
    @ ^ [N2: nat] : zero_zero_nat ) ).

% summable_zero
thf(fact_8773_summable__zero,axiom,
    ( summable_int
    @ ^ [N2: nat] : zero_zero_int ) ).

% summable_zero
thf(fact_8774_summable__iff__shift,axiom,
    ! [F2: nat > real,K: nat] :
      ( ( summable_real
        @ ^ [N2: nat] : ( F2 @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( summable_real @ F2 ) ) ).

% summable_iff_shift
thf(fact_8775_cos__zero,axiom,
    ( ( cos_complex @ zero_zero_complex )
    = one_one_complex ) ).

% cos_zero
thf(fact_8776_cos__zero,axiom,
    ( ( cos_real @ zero_zero_real )
    = one_one_real ) ).

% cos_zero
thf(fact_8777_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one )
    = zero_zero_nat ) ).

% pred_numeral_simps(1)
thf(fact_8778_eq__numeral__Suc,axiom,
    ! [K: num,N3: nat] :
      ( ( ( numeral_numeral_nat @ K )
        = ( suc @ N3 ) )
      = ( ( pred_numeral @ K )
        = N3 ) ) ).

% eq_numeral_Suc
thf(fact_8779_Suc__eq__numeral,axiom,
    ! [N3: nat,K: num] :
      ( ( ( suc @ N3 )
        = ( numeral_numeral_nat @ K ) )
      = ( N3
        = ( pred_numeral @ K ) ) ) ).

% Suc_eq_numeral
thf(fact_8780_summable__cmult__iff,axiom,
    ! [C2: complex,F2: nat > complex] :
      ( ( summable_complex
        @ ^ [N2: nat] : ( times_times_complex @ C2 @ ( F2 @ N2 ) ) )
      = ( ( C2 = zero_zero_complex )
        | ( summable_complex @ F2 ) ) ) ).

% summable_cmult_iff
thf(fact_8781_summable__cmult__iff,axiom,
    ! [C2: real,F2: nat > real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ C2 @ ( F2 @ N2 ) ) )
      = ( ( C2 = zero_zero_real )
        | ( summable_real @ F2 ) ) ) ).

% summable_cmult_iff
thf(fact_8782_summable__divide__iff,axiom,
    ! [F2: nat > complex,C2: complex] :
      ( ( summable_complex
        @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F2 @ N2 ) @ C2 ) )
      = ( ( C2 = zero_zero_complex )
        | ( summable_complex @ F2 ) ) ) ).

% summable_divide_iff
thf(fact_8783_summable__divide__iff,axiom,
    ! [F2: nat > real,C2: real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( divide_divide_real @ ( F2 @ N2 ) @ C2 ) )
      = ( ( C2 = zero_zero_real )
        | ( summable_real @ F2 ) ) ) ).

% summable_divide_iff
thf(fact_8784_summable__If__finite,axiom,
    ! [P: nat > $o,F2: nat > complex] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_complex
        @ ^ [R5: nat] : ( if_complex @ ( P @ R5 ) @ ( F2 @ R5 ) @ zero_zero_complex ) ) ) ).

% summable_If_finite
thf(fact_8785_summable__If__finite,axiom,
    ! [P: nat > $o,F2: nat > real] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_real
        @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F2 @ R5 ) @ zero_zero_real ) ) ) ).

% summable_If_finite
thf(fact_8786_summable__If__finite,axiom,
    ! [P: nat > $o,F2: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_nat
        @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F2 @ R5 ) @ zero_zero_nat ) ) ) ).

% summable_If_finite
thf(fact_8787_summable__If__finite,axiom,
    ! [P: nat > $o,F2: nat > int] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_int
        @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F2 @ R5 ) @ zero_zero_int ) ) ) ).

% summable_If_finite
thf(fact_8788_summable__If__finite__set,axiom,
    ! [A: set_nat,F2: nat > complex] :
      ( ( finite_finite_nat @ A )
     => ( summable_complex
        @ ^ [R5: nat] : ( if_complex @ ( member_nat @ R5 @ A ) @ ( F2 @ R5 ) @ zero_zero_complex ) ) ) ).

% summable_If_finite_set
thf(fact_8789_summable__If__finite__set,axiom,
    ! [A: set_nat,F2: nat > real] :
      ( ( finite_finite_nat @ A )
     => ( summable_real
        @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A ) @ ( F2 @ R5 ) @ zero_zero_real ) ) ) ).

% summable_If_finite_set
thf(fact_8790_summable__If__finite__set,axiom,
    ! [A: set_nat,F2: nat > nat] :
      ( ( finite_finite_nat @ A )
     => ( summable_nat
        @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A ) @ ( F2 @ R5 ) @ zero_zero_nat ) ) ) ).

% summable_If_finite_set
thf(fact_8791_summable__If__finite__set,axiom,
    ! [A: set_nat,F2: nat > int] :
      ( ( finite_finite_nat @ A )
     => ( summable_int
        @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A ) @ ( F2 @ R5 ) @ zero_zero_int ) ) ) ).

% summable_If_finite_set
thf(fact_8792_pred__numeral__simps_I3_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit1 @ K ) )
      = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).

% pred_numeral_simps(3)
thf(fact_8793_less__numeral__Suc,axiom,
    ! [K: num,N3: nat] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N3 ) )
      = ( ord_less_nat @ ( pred_numeral @ K ) @ N3 ) ) ).

% less_numeral_Suc
thf(fact_8794_less__Suc__numeral,axiom,
    ! [N3: nat,K: num] :
      ( ( ord_less_nat @ ( suc @ N3 ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_nat @ N3 @ ( pred_numeral @ K ) ) ) ).

% less_Suc_numeral
thf(fact_8795_le__numeral__Suc,axiom,
    ! [K: num,N3: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N3 ) )
      = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N3 ) ) ).

% le_numeral_Suc
thf(fact_8796_le__Suc__numeral,axiom,
    ! [N3: nat,K: num] :
      ( ( ord_less_eq_nat @ ( suc @ N3 ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_eq_nat @ N3 @ ( pred_numeral @ K ) ) ) ).

% le_Suc_numeral
thf(fact_8797_diff__numeral__Suc,axiom,
    ! [K: num,N3: nat] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N3 ) )
      = ( minus_minus_nat @ ( pred_numeral @ K ) @ N3 ) ) ).

% diff_numeral_Suc
thf(fact_8798_diff__Suc__numeral,axiom,
    ! [N3: nat,K: num] :
      ( ( minus_minus_nat @ ( suc @ N3 ) @ ( numeral_numeral_nat @ K ) )
      = ( minus_minus_nat @ N3 @ ( pred_numeral @ K ) ) ) ).

% diff_Suc_numeral
thf(fact_8799_cos__periodic__pi,axiom,
    ! [X: real] :
      ( ( cos_real @ ( plus_plus_real @ X @ pi ) )
      = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).

% cos_periodic_pi
thf(fact_8800_cos__periodic__pi2,axiom,
    ! [X: real] :
      ( ( cos_real @ ( plus_plus_real @ pi @ X ) )
      = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).

% cos_periodic_pi2
thf(fact_8801_max__numeral__Suc,axiom,
    ! [K: num,N3: nat] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N3 ) )
      = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N3 ) ) ) ).

% max_numeral_Suc
thf(fact_8802_max__Suc__numeral,axiom,
    ! [N3: nat,K: num] :
      ( ( ord_max_nat @ ( suc @ N3 ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_max_nat @ N3 @ ( pred_numeral @ K ) ) ) ) ).

% max_Suc_numeral
thf(fact_8803_cos__pi__minus,axiom,
    ! [X: real] :
      ( ( cos_real @ ( minus_minus_real @ pi @ X ) )
      = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).

% cos_pi_minus
thf(fact_8804_cos__minus__pi,axiom,
    ! [X: real] :
      ( ( cos_real @ ( minus_minus_real @ X @ pi ) )
      = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).

% cos_minus_pi
thf(fact_8805_sin__cos__squared__add3,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ X ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ X ) ) )
      = one_one_real ) ).

% sin_cos_squared_add3
thf(fact_8806_summable__geometric__iff,axiom,
    ! [C2: real] :
      ( ( summable_real @ ( power_power_real @ C2 ) )
      = ( ord_less_real @ ( real_V7735802525324610683m_real @ C2 ) @ one_one_real ) ) ).

% summable_geometric_iff
thf(fact_8807_summable__geometric__iff,axiom,
    ! [C2: complex] :
      ( ( summable_complex @ ( power_power_complex @ C2 ) )
      = ( ord_less_real @ ( real_V1022390504157884413omplex @ C2 ) @ one_one_real ) ) ).

% summable_geometric_iff
thf(fact_8808_cos__pi__half,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = zero_zero_real ) ).

% cos_pi_half
thf(fact_8809_cos__two__pi,axiom,
    ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_real ) ).

% cos_two_pi
thf(fact_8810_signed__take__bit__numeral__bit0,axiom,
    ! [L2: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_numeral_bit0
thf(fact_8811_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_8812_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_8813_cos__npi2,axiom,
    ! [N3: nat] :
      ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N3 ) ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) ) ).

% cos_npi2
thf(fact_8814_cos__npi,axiom,
    ! [N3: nat] :
      ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ pi ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) ) ).

% cos_npi
thf(fact_8815_sin__cos__squared__add2,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% sin_cos_squared_add2
thf(fact_8816_sin__cos__squared__add2,axiom,
    ! [X: complex] :
      ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add2
thf(fact_8817_sin__cos__squared__add,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% sin_cos_squared_add
thf(fact_8818_sin__cos__squared__add,axiom,
    ! [X: complex] :
      ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add
thf(fact_8819_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L2: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_numeral_minus_bit0
thf(fact_8820_cos__2npi,axiom,
    ! [N3: nat] :
      ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) @ pi ) )
      = one_one_real ) ).

% cos_2npi
thf(fact_8821_cos__int__2pin,axiom,
    ! [N3: int] :
      ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N3 ) ) )
      = one_one_real ) ).

% cos_int_2pin
thf(fact_8822_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_8823_signed__take__bit__numeral__bit1,axiom,
    ! [L2: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_bit1
thf(fact_8824_cos__npi__int,axiom,
    ! [N3: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 )
       => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N3 ) ) )
          = one_one_real ) )
      & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 )
       => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N3 ) ) )
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% cos_npi_int
thf(fact_8825_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_8826_summable__comparison__test_H,axiom,
    ! [G: nat > real,N7: nat,F2: nat > real] :
      ( ( summable_real @ G )
     => ( ! [N: nat] :
            ( ( ord_less_eq_nat @ N7 @ N )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F2 @ N ) ) @ ( G @ N ) ) )
       => ( summable_real @ F2 ) ) ) ).

% summable_comparison_test'
thf(fact_8827_summable__comparison__test_H,axiom,
    ! [G: nat > real,N7: nat,F2: nat > complex] :
      ( ( summable_real @ G )
     => ( ! [N: nat] :
            ( ( ord_less_eq_nat @ N7 @ N )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F2 @ N ) ) @ ( G @ N ) ) )
       => ( summable_complex @ F2 ) ) ) ).

% summable_comparison_test'
thf(fact_8828_summable__comparison__test,axiom,
    ! [F2: nat > real,G: nat > real] :
      ( ? [N9: nat] :
        ! [N: nat] :
          ( ( ord_less_eq_nat @ N9 @ N )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F2 @ N ) ) @ ( G @ N ) ) )
     => ( ( summable_real @ G )
       => ( summable_real @ F2 ) ) ) ).

% summable_comparison_test
thf(fact_8829_summable__comparison__test,axiom,
    ! [F2: nat > complex,G: nat > real] :
      ( ? [N9: nat] :
        ! [N: nat] :
          ( ( ord_less_eq_nat @ N9 @ N )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F2 @ N ) ) @ ( G @ N ) ) )
     => ( ( summable_real @ G )
       => ( summable_complex @ F2 ) ) ) ).

% summable_comparison_test
thf(fact_8830_summable__const__iff,axiom,
    ! [C2: complex] :
      ( ( summable_complex
        @ ^ [Uu3: nat] : C2 )
      = ( C2 = zero_zero_complex ) ) ).

% summable_const_iff
thf(fact_8831_summable__const__iff,axiom,
    ! [C2: real] :
      ( ( summable_real
        @ ^ [Uu3: nat] : C2 )
      = ( C2 = zero_zero_real ) ) ).

% summable_const_iff
thf(fact_8832_summable__add,axiom,
    ! [F2: nat > real,G: nat > real] :
      ( ( summable_real @ F2 )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N2: nat] : ( plus_plus_real @ ( F2 @ N2 ) @ ( G @ N2 ) ) ) ) ) ).

% summable_add
thf(fact_8833_summable__add,axiom,
    ! [F2: nat > nat,G: nat > nat] :
      ( ( summable_nat @ F2 )
     => ( ( summable_nat @ G )
       => ( summable_nat
          @ ^ [N2: nat] : ( plus_plus_nat @ ( F2 @ N2 ) @ ( G @ N2 ) ) ) ) ) ).

% summable_add
thf(fact_8834_summable__add,axiom,
    ! [F2: nat > int,G: nat > int] :
      ( ( summable_int @ F2 )
     => ( ( summable_int @ G )
       => ( summable_int
          @ ^ [N2: nat] : ( plus_plus_int @ ( F2 @ N2 ) @ ( G @ N2 ) ) ) ) ) ).

% summable_add
thf(fact_8835_summable__mult2,axiom,
    ! [F2: nat > real,C2: real] :
      ( ( summable_real @ F2 )
     => ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ ( F2 @ N2 ) @ C2 ) ) ) ).

% summable_mult2
thf(fact_8836_summable__mult,axiom,
    ! [F2: nat > real,C2: real] :
      ( ( summable_real @ F2 )
     => ( summable_real
        @ ^ [N2: nat] : ( times_times_real @ C2 @ ( F2 @ N2 ) ) ) ) ).

% summable_mult
thf(fact_8837_summable__Suc__iff,axiom,
    ! [F2: nat > real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( F2 @ ( suc @ N2 ) ) )
      = ( summable_real @ F2 ) ) ).

% summable_Suc_iff
thf(fact_8838_cos__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( cos_real @ X ) @ one_one_real ) ).

% cos_le_one
thf(fact_8839_polar__Ex,axiom,
    ! [X: real,Y: real] :
    ? [R2: real,A3: real] :
      ( ( X
        = ( times_times_real @ R2 @ ( cos_real @ A3 ) ) )
      & ( Y
        = ( times_times_real @ R2 @ ( sin_real @ A3 ) ) ) ) ).

% polar_Ex
thf(fact_8840_cos__arctan__not__zero,axiom,
    ! [X: real] :
      ( ( cos_real @ ( arctan @ X ) )
     != zero_zero_real ) ).

% cos_arctan_not_zero
thf(fact_8841_numeral__eq__Suc,axiom,
    ( numeral_numeral_nat
    = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).

% numeral_eq_Suc
thf(fact_8842_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_8843_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_8844_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_8845_cos__ge__minus__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X ) ) ).

% cos_ge_minus_one
thf(fact_8846_abs__cos__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X ) ) @ one_one_real ) ).

% abs_cos_le_one
thf(fact_8847_summable__rabs__comparison__test,axiom,
    ! [F2: nat > real,G: nat > real] :
      ( ? [N9: nat] :
        ! [N: nat] :
          ( ( ord_less_eq_nat @ N9 @ N )
         => ( ord_less_eq_real @ ( abs_abs_real @ ( F2 @ N ) ) @ ( G @ N ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N2: nat] : ( abs_abs_real @ ( F2 @ N2 ) ) ) ) ) ).

% summable_rabs_comparison_test
thf(fact_8848_pred__numeral__def,axiom,
    ( pred_numeral
    = ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).

% pred_numeral_def
thf(fact_8849_summable__rabs,axiom,
    ! [F2: nat > real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( abs_abs_real @ ( F2 @ N2 ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F2 ) )
        @ ( suminf_real
          @ ^ [N2: nat] : ( abs_abs_real @ ( F2 @ N2 ) ) ) ) ) ).

% summable_rabs
thf(fact_8850_cos__two__neq__zero,axiom,
    ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% cos_two_neq_zero
thf(fact_8851_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_8852_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_8853_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_8854_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_8855_cos__two__less__zero,axiom,
    ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_less_zero
thf(fact_8856_cos__is__zero,axiom,
    ? [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
      & ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      & ( ( cos_real @ X4 )
        = zero_zero_real )
      & ! [Y4: real] :
          ( ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
            & ( ord_less_eq_real @ Y4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ Y4 )
              = zero_zero_real ) )
         => ( Y4 = X4 ) ) ) ).

% cos_is_zero
thf(fact_8857_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_8858_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_8859_cos__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ? [X4: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X4 )
            & ( ord_less_eq_real @ X4 @ pi )
            & ( ( cos_real @ X4 )
              = Y )
            & ! [Y4: real] :
                ( ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ pi )
                  & ( ( cos_real @ Y4 )
                    = Y ) )
               => ( Y4 = X4 ) ) ) ) ) ).

% cos_total
thf(fact_8860_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y3 )
      & ( ord_less_eq_real @ Y3 @ pi )
      & ( ( sin_real @ Y3 )
        = ( sin_real @ X ) )
      & ( ( cos_real @ Y3 )
        = ( cos_real @ X ) ) ) ).

% sincos_principal_value
thf(fact_8861_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_8862_summable__power__series,axiom,
    ! [F2: nat > real,Z: real] :
      ( ! [I5: nat] : ( ord_less_eq_real @ ( F2 @ I5 ) @ one_one_real )
     => ( ! [I5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F2 @ I5 ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ Z )
         => ( ( ord_less_real @ Z @ one_one_real )
           => ( summable_real
              @ ^ [I2: nat] : ( times_times_real @ ( F2 @ I2 ) @ ( power_power_real @ Z @ I2 ) ) ) ) ) ) ) ).

% summable_power_series
thf(fact_8863_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_8864_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_8865_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_8866_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_8867_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = one_one_real )
      = ( ? [X3: int] :
            ( X
            = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).

% cos_one_2pi_int
thf(fact_8868_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_8869_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_8870_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = one_one_real )
      = ( ? [X3: nat] :
            ( X
            = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
        | ? [X3: nat] :
            ( X
            = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).

% cos_one_2pi
thf(fact_8871_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_8872_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_8873_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = zero_zero_real )
      = ( ? [I2: int] :
            ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I2 )
            & ( X
              = ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_iff_int
thf(fact_8874_cos__zero__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( cos_real @ X )
          = zero_zero_real )
       => ? [N: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_lemma
thf(fact_8875_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( ( 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 ) ) ) ) ) )
        | ? [N2: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% cos_zero_iff
thf(fact_8876_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_8877_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_8878_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_8879_infinite__int__iff__unbounded,axiom,
    ! [S: set_int] :
      ( ( ~ ( finite_finite_int @ S ) )
      = ( ! [M5: int] :
          ? [N2: int] :
            ( ( ord_less_int @ M5 @ ( abs_abs_int @ N2 ) )
            & ( member_int @ N2 @ S ) ) ) ) ).

% infinite_int_iff_unbounded
thf(fact_8880_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( ( real_V1022390504157884413omplex @ Z )
        = one_one_real )
     => ~ ! [T6: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T6 )
           => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( Z
               != ( complex2 @ ( cos_real @ T6 ) @ ( sin_real @ T6 ) ) ) ) ) ) ).

% complex_unimodular_polar
thf(fact_8881_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 ) ) )
         => ~ ! [Va3: nat] :
                ( ( X
                  = ( suc @ ( suc @ Va3 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
thf(fact_8882_tan__pi,axiom,
    ( ( tan_real @ pi )
    = zero_zero_real ) ).

% tan_pi
thf(fact_8883_tan__periodic__pi,axiom,
    ! [X: real] :
      ( ( tan_real @ ( plus_plus_real @ X @ pi ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_pi
thf(fact_8884_tan__npi,axiom,
    ! [N3: nat] :
      ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ pi ) )
      = zero_zero_real ) ).

% tan_npi
thf(fact_8885_tan__periodic__n,axiom,
    ! [X: real,N3: num] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ N3 ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_n
thf(fact_8886_tan__periodic__nat,axiom,
    ! [X: real,N3: nat] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_nat
thf(fact_8887_tan__periodic__int,axiom,
    ! [X: real,I: int] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_int
thf(fact_8888_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_8889_complex__diff,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( minus_minus_complex @ ( complex2 @ A2 @ B2 ) @ ( complex2 @ C2 @ D2 ) )
      = ( complex2 @ ( minus_minus_real @ A2 @ C2 ) @ ( minus_minus_real @ B2 @ D2 ) ) ) ).

% complex_diff
thf(fact_8890_Complex__eq__0,axiom,
    ! [A2: real,B2: real] :
      ( ( ( complex2 @ A2 @ B2 )
        = zero_zero_complex )
      = ( ( A2 = zero_zero_real )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_0
thf(fact_8891_zero__complex_Ocode,axiom,
    ( zero_zero_complex
    = ( complex2 @ zero_zero_real @ zero_zero_real ) ) ).

% zero_complex.code
thf(fact_8892_one__complex_Ocode,axiom,
    ( one_one_complex
    = ( complex2 @ one_one_real @ zero_zero_real ) ) ).

% one_complex.code
thf(fact_8893_Complex__eq__1,axiom,
    ! [A2: real,B2: real] :
      ( ( ( complex2 @ A2 @ B2 )
        = one_one_complex )
      = ( ( A2 = one_one_real )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_1
thf(fact_8894_Complex__eq__numeral,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( ( complex2 @ A2 @ B2 )
        = ( numera6690914467698888265omplex @ W ) )
      = ( ( A2
          = ( numeral_numeral_real @ W ) )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_numeral
thf(fact_8895_complex__add,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( plus_plus_complex @ ( complex2 @ A2 @ B2 ) @ ( complex2 @ C2 @ D2 ) )
      = ( complex2 @ ( plus_plus_real @ A2 @ C2 ) @ ( plus_plus_real @ B2 @ D2 ) ) ) ).

% complex_add
thf(fact_8896_Complex__eq__neg__1,axiom,
    ! [A2: real,B2: real] :
      ( ( ( complex2 @ A2 @ B2 )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( ( A2
          = ( uminus_uminus_real @ one_one_real ) )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
thf(fact_8897_Complex__eq__neg__numeral,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( ( complex2 @ A2 @ B2 )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( A2
          = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
thf(fact_8898_complex__mult,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] :
      ( ( times_times_complex @ ( complex2 @ A2 @ B2 ) @ ( complex2 @ C2 @ D2 ) )
      = ( complex2 @ ( minus_minus_real @ ( times_times_real @ A2 @ C2 ) @ ( times_times_real @ B2 @ D2 ) ) @ ( plus_plus_real @ ( times_times_real @ A2 @ D2 ) @ ( times_times_real @ B2 @ C2 ) ) ) ) ).

% complex_mult
thf(fact_8899_tan__45,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = one_one_real ) ).

% tan_45
thf(fact_8900_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_8901_lemma__tan__total,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ? [X4: real] :
          ( ( ord_less_real @ zero_zero_real @ X4 )
          & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ord_less_real @ Y @ ( tan_real @ X4 ) ) ) ) ).

% lemma_tan_total
thf(fact_8902_tan__total,axiom,
    ! [Y: real] :
    ? [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
      & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X4 )
        = Y )
      & ! [Y4: real] :
          ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
            & ( ord_less_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( tan_real @ Y4 )
              = Y ) )
         => ( Y4 = X4 ) ) ) ).

% tan_total
thf(fact_8903_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_8904_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_8905_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_8906_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
      & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X4 )
        = Y ) ) ).

% lemma_tan_total1
thf(fact_8907_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_8908_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_8909_infinite__nat__iff__unbounded,axiom,
    ! [S: set_nat] :
      ( ( ~ ( finite_finite_nat @ S ) )
      = ( ! [M5: nat] :
          ? [N2: nat] :
            ( ( ord_less_nat @ M5 @ N2 )
            & ( member_nat @ N2 @ S ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_8910_unbounded__k__infinite,axiom,
    ! [K: nat,S: set_nat] :
      ( ! [M4: nat] :
          ( ( ord_less_nat @ K @ M4 )
         => ? [N10: nat] :
              ( ( ord_less_nat @ M4 @ N10 )
              & ( member_nat @ N10 @ S ) ) )
     => ~ ( finite_finite_nat @ S ) ) ).

% unbounded_k_infinite
thf(fact_8911_infinite__nat__iff__unbounded__le,axiom,
    ! [S: set_nat] :
      ( ( ~ ( finite_finite_nat @ S ) )
      = ( ! [M5: nat] :
          ? [N2: nat] :
            ( ( ord_less_eq_nat @ M5 @ N2 )
            & ( member_nat @ N2 @ S ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_8912_tan__total__pos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ? [X4: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X4 )
          & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( tan_real @ X4 )
            = Y ) ) ) ).

% tan_total_pos
thf(fact_8913_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_8914_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_8915_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_8916_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_8917_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_8918_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_8919_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_8920_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_8921_tan__total__pi4,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ? [Z2: real] :
          ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z2 )
          & ( ord_less_real @ Z2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
          & ( ( tan_real @ Z2 )
            = X ) ) ) ).

% tan_total_pi4
thf(fact_8922_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.simps(3)
thf(fact_8923_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B2: real,K: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ one_one_real @ B2 )
       => ( ( ( archim7802044766580827645g_real @ ( log @ B2 @ X ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
          = ( ( ord_less_real @ ( powr_real @ B2 @ ( semiri5074537144036343181t_real @ K ) ) @ X )
            & ( ord_less_eq_real @ X @ ( powr_real @ B2 @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).

% ceiling_log_eq_powr_iff
thf(fact_8924_powr__gt__zero,axiom,
    ! [X: real,A2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X @ A2 ) )
      = ( X != zero_zero_real ) ) ).

% powr_gt_zero
thf(fact_8925_powr__nonneg__iff,axiom,
    ! [A2: real,X: real] :
      ( ( ord_less_eq_real @ ( powr_real @ A2 @ X ) @ zero_zero_real )
      = ( A2 = zero_zero_real ) ) ).

% powr_nonneg_iff
thf(fact_8926_powr__less__cancel__iff,axiom,
    ! [X: real,A2: real,B2: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_real @ ( powr_real @ X @ A2 ) @ ( powr_real @ X @ B2 ) )
        = ( ord_less_real @ A2 @ B2 ) ) ) ).

% powr_less_cancel_iff
thf(fact_8927_powr__eq__one__iff,axiom,
    ! [A2: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A2 )
     => ( ( ( powr_real @ A2 @ X )
          = one_one_real )
        = ( X = zero_zero_real ) ) ) ).

% powr_eq_one_iff
thf(fact_8928_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_8929_powr__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ one_one_real )
        = X ) ) ).

% powr_one
thf(fact_8930_powr__le__cancel__iff,axiom,
    ! [X: real,A2: real,B2: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_eq_real @ ( powr_real @ X @ A2 ) @ ( powr_real @ X @ B2 ) )
        = ( ord_less_eq_real @ A2 @ B2 ) ) ) ).

% powr_le_cancel_iff
thf(fact_8931_numeral__powr__numeral__real,axiom,
    ! [M: num,N3: num] :
      ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N3 ) )
      = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N3 ) ) ) ).

% numeral_powr_numeral_real
thf(fact_8932_log__powr__cancel,axiom,
    ! [A2: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( A2 != one_one_real )
       => ( ( log @ A2 @ ( powr_real @ A2 @ Y ) )
          = Y ) ) ) ).

% log_powr_cancel
thf(fact_8933_powr__log__cancel,axiom,
    ! [A2: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( A2 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( powr_real @ A2 @ ( log @ A2 @ X ) )
            = X ) ) ) ) ).

% powr_log_cancel
thf(fact_8934_powr__numeral,axiom,
    ! [X: real,N3: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( numeral_numeral_real @ N3 ) )
        = ( power_power_real @ X @ ( numeral_numeral_nat @ N3 ) ) ) ) ).

% powr_numeral
thf(fact_8935_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_8936_powr__powr,axiom,
    ! [X: real,A2: real,B2: real] :
      ( ( powr_real @ ( powr_real @ X @ A2 ) @ B2 )
      = ( powr_real @ X @ ( times_times_real @ A2 @ B2 ) ) ) ).

% powr_powr
thf(fact_8937_powr__less__mono2__neg,axiom,
    ! [A2: real,X: real,Y: real] :
      ( ( ord_less_real @ A2 @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ X @ Y )
         => ( ord_less_real @ ( powr_real @ Y @ A2 ) @ ( powr_real @ X @ A2 ) ) ) ) ) ).

% powr_less_mono2_neg
thf(fact_8938_powr__non__neg,axiom,
    ! [A2: real,X: real] :
      ~ ( ord_less_real @ ( powr_real @ A2 @ X ) @ zero_zero_real ) ).

% powr_non_neg
thf(fact_8939_powr__ge__pzero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y ) ) ).

% powr_ge_pzero
thf(fact_8940_powr__mono2,axiom,
    ! [A2: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ Y )
         => ( ord_less_eq_real @ ( powr_real @ X @ A2 ) @ ( powr_real @ Y @ A2 ) ) ) ) ) ).

% powr_mono2
thf(fact_8941_powr__less__mono,axiom,
    ! [A2: real,B2: real,X: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ord_less_real @ one_one_real @ X )
       => ( ord_less_real @ ( powr_real @ X @ A2 ) @ ( powr_real @ X @ B2 ) ) ) ) ).

% powr_less_mono
thf(fact_8942_powr__less__cancel,axiom,
    ! [X: real,A2: real,B2: real] :
      ( ( ord_less_real @ ( powr_real @ X @ A2 ) @ ( powr_real @ X @ B2 ) )
     => ( ( ord_less_real @ one_one_real @ X )
       => ( ord_less_real @ A2 @ B2 ) ) ) ).

% powr_less_cancel
thf(fact_8943_powr__mono,axiom,
    ! [A2: real,B2: real,X: real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( ord_less_eq_real @ one_one_real @ X )
       => ( ord_less_eq_real @ ( powr_real @ X @ A2 ) @ ( powr_real @ X @ B2 ) ) ) ) ).

% powr_mono
thf(fact_8944_powr__less__mono2,axiom,
    ! [A2: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ X @ Y )
         => ( ord_less_real @ ( powr_real @ X @ A2 ) @ ( powr_real @ Y @ A2 ) ) ) ) ) ).

% powr_less_mono2
thf(fact_8945_powr__mono2_H,axiom,
    ! [A2: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ A2 @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ Y )
         => ( ord_less_eq_real @ ( powr_real @ Y @ A2 ) @ ( powr_real @ X @ A2 ) ) ) ) ) ).

% powr_mono2'
thf(fact_8946_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_8947_powr__inj,axiom,
    ! [A2: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( A2 != one_one_real )
       => ( ( ( powr_real @ A2 @ X )
            = ( powr_real @ A2 @ Y ) )
          = ( X = Y ) ) ) ) ).

% powr_inj
thf(fact_8948_ge__one__powr__ge__zero,axiom,
    ! [X: real,A2: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
       => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A2 ) ) ) ) ).

% ge_one_powr_ge_zero
thf(fact_8949_powr__mono__both,axiom,
    ! [A2: real,B2: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ A2 @ B2 )
       => ( ( ord_less_eq_real @ one_one_real @ X )
         => ( ( ord_less_eq_real @ X @ Y )
           => ( ord_less_eq_real @ ( powr_real @ X @ A2 ) @ ( powr_real @ Y @ B2 ) ) ) ) ) ) ).

% powr_mono_both
thf(fact_8950_powr__le1,axiom,
    ! [A2: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ one_one_real )
         => ( ord_less_eq_real @ ( powr_real @ X @ A2 ) @ one_one_real ) ) ) ) ).

% powr_le1
thf(fact_8951_powr__divide,axiom,
    ! [X: real,Y: real,A2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( powr_real @ ( divide_divide_real @ X @ Y ) @ A2 )
          = ( divide_divide_real @ ( powr_real @ X @ A2 ) @ ( powr_real @ Y @ A2 ) ) ) ) ) ).

% powr_divide
thf(fact_8952_powr__mult,axiom,
    ! [X: real,Y: real,A2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( powr_real @ ( times_times_real @ X @ Y ) @ A2 )
          = ( times_times_real @ ( powr_real @ X @ A2 ) @ ( powr_real @ Y @ A2 ) ) ) ) ) ).

% powr_mult
thf(fact_8953_divide__powr__uminus,axiom,
    ! [A2: real,B2: real,C2: real] :
      ( ( divide_divide_real @ A2 @ ( powr_real @ B2 @ C2 ) )
      = ( times_times_real @ A2 @ ( powr_real @ B2 @ ( uminus_uminus_real @ C2 ) ) ) ) ).

% divide_powr_uminus
thf(fact_8954_log__base__powr,axiom,
    ! [A2: real,B2: real,X: real] :
      ( ( A2 != zero_zero_real )
     => ( ( log @ ( powr_real @ A2 @ B2 ) @ X )
        = ( divide_divide_real @ ( log @ A2 @ X ) @ B2 ) ) ) ).

% log_base_powr
thf(fact_8955_log__powr,axiom,
    ! [X: real,B2: real,Y: real] :
      ( ( X != zero_zero_real )
     => ( ( log @ B2 @ ( powr_real @ X @ Y ) )
        = ( times_times_real @ Y @ ( log @ B2 @ X ) ) ) ) ).

% log_powr
thf(fact_8956_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_8957_powr__realpow,axiom,
    ! [X: real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N3 ) )
        = ( power_power_real @ X @ N3 ) ) ) ).

% powr_realpow
thf(fact_8958_less__log__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ Y @ ( log @ B2 @ X ) )
          = ( ord_less_real @ ( powr_real @ B2 @ Y ) @ X ) ) ) ) ).

% less_log_iff
thf(fact_8959_log__less__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( log @ B2 @ X ) @ Y )
          = ( ord_less_real @ X @ ( powr_real @ B2 @ Y ) ) ) ) ) ).

% log_less_iff
thf(fact_8960_less__powr__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ X @ ( powr_real @ B2 @ Y ) )
          = ( ord_less_real @ ( log @ B2 @ X ) @ Y ) ) ) ) ).

% less_powr_iff
thf(fact_8961_powr__less__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( powr_real @ B2 @ Y ) @ X )
          = ( ord_less_real @ Y @ ( log @ B2 @ X ) ) ) ) ) ).

% powr_less_iff
thf(fact_8962_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_8963_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_8964_powr__le__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( powr_real @ B2 @ Y ) @ X )
          = ( ord_less_eq_real @ Y @ ( log @ B2 @ X ) ) ) ) ) ).

% powr_le_iff
thf(fact_8965_le__powr__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ ( powr_real @ B2 @ Y ) )
          = ( ord_less_eq_real @ ( log @ B2 @ X ) @ Y ) ) ) ) ).

% le_powr_iff
thf(fact_8966_log__le__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( log @ B2 @ X ) @ Y )
          = ( ord_less_eq_real @ X @ ( powr_real @ B2 @ Y ) ) ) ) ) ).

% log_le_iff
thf(fact_8967_le__log__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ Y @ ( log @ B2 @ X ) )
          = ( ord_less_eq_real @ ( powr_real @ B2 @ Y ) @ X ) ) ) ) ).

% le_log_iff
thf(fact_8968_set__encode__def,axiom,
    ( nat_set_encode
    = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% set_encode_def
thf(fact_8969_ln__powr__bound,axiom,
    ! [X: real,A2: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ A2 )
       => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A2 ) @ A2 ) ) ) ) ).

% ln_powr_bound
thf(fact_8970_ln__powr__bound2,axiom,
    ! [X: real,A2: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ A2 )
       => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A2 ) @ ( times_times_real @ ( powr_real @ A2 @ A2 ) @ X ) ) ) ) ).

% ln_powr_bound2
thf(fact_8971_log__add__eq__powr,axiom,
    ! [B2: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B2 )
     => ( ( B2 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( plus_plus_real @ ( log @ B2 @ X ) @ Y )
            = ( log @ B2 @ ( times_times_real @ X @ ( powr_real @ B2 @ Y ) ) ) ) ) ) ) ).

% log_add_eq_powr
thf(fact_8972_add__log__eq__powr,axiom,
    ! [B2: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B2 )
     => ( ( B2 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( plus_plus_real @ Y @ ( log @ B2 @ X ) )
            = ( log @ B2 @ ( times_times_real @ ( powr_real @ B2 @ Y ) @ X ) ) ) ) ) ) ).

% add_log_eq_powr
thf(fact_8973_minus__log__eq__powr,axiom,
    ! [B2: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B2 )
     => ( ( B2 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( minus_minus_real @ Y @ ( log @ B2 @ X ) )
            = ( log @ B2 @ ( divide_divide_real @ ( powr_real @ B2 @ Y ) @ X ) ) ) ) ) ) ).

% minus_log_eq_powr
thf(fact_8974_log__minus__eq__powr,axiom,
    ! [B2: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B2 )
     => ( ( B2 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( minus_minus_real @ ( log @ B2 @ X ) @ Y )
            = ( log @ B2 @ ( times_times_real @ X @ ( powr_real @ B2 @ ( uminus_uminus_real @ Y ) ) ) ) ) ) ) ) ).

% log_minus_eq_powr
thf(fact_8975_powr__neg__numeral,axiom,
    ! [X: real,N3: num] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N3 ) ) )
        = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N3 ) ) ) ) ) ).

% powr_neg_numeral
thf(fact_8976_mask__eq__sum__exp__nat,axiom,
    ! [N3: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( suc @ zero_zero_nat ) )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N3 ) ) ) ) ).

% mask_eq_sum_exp_nat
thf(fact_8977_gauss__sum__nat,axiom,
    ! [N3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
      = ( divide_divide_nat @ ( times_times_nat @ N3 @ ( suc @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_nat
thf(fact_8978_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_8979_Sum__Ico__nat,axiom,
    ! [M: nat,N3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ ( set_or4665077453230672383an_nat @ M @ N3 ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N3 @ ( minus_minus_nat @ N3 @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Ico_nat
thf(fact_8980_arith__series__nat,axiom,
    ! [A2: nat,D2: nat,N3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( plus_plus_nat @ A2 @ ( times_times_nat @ I2 @ D2 ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N3 ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_nat @ N3 @ D2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series_nat
thf(fact_8981_Sum__Icc__nat,axiom,
    ! [M: nat,N3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ ( set_or1269000886237332187st_nat @ M @ N3 ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N3 @ ( plus_plus_nat @ N3 @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Icc_nat
thf(fact_8982_powr__int,axiom,
    ! [X: real,I: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
            = ( power_power_real @ X @ ( nat2 @ I ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
            = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).

% powr_int
thf(fact_8983_Chebyshev__sum__upper__nat,axiom,
    ! [N3: nat,A2: nat > nat,B2: nat > nat] :
      ( ! [I5: nat,J3: nat] :
          ( ( ord_less_eq_nat @ I5 @ J3 )
         => ( ( ord_less_nat @ J3 @ N3 )
           => ( ord_less_eq_nat @ ( A2 @ I5 ) @ ( A2 @ J3 ) ) ) )
     => ( ! [I5: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I5 @ J3 )
           => ( ( ord_less_nat @ J3 @ N3 )
             => ( ord_less_eq_nat @ ( B2 @ J3 ) @ ( B2 @ I5 ) ) ) )
       => ( ord_less_eq_nat
          @ ( times_times_nat @ N3
            @ ( groups3542108847815614940at_nat
              @ ^ [I2: nat] : ( times_times_nat @ ( A2 @ I2 ) @ ( B2 @ I2 ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) )
          @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) @ ( groups3542108847815614940at_nat @ B2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) ) ) ) ) ).

% Chebyshev_sum_upper_nat
thf(fact_8984_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_8985_real__sqrt__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( sqrt @ X )
        = ( sqrt @ Y ) )
      = ( X = Y ) ) ).

% real_sqrt_eq_iff
thf(fact_8986_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_8987_real__sqrt__zero,axiom,
    ( ( sqrt @ zero_zero_real )
    = zero_zero_real ) ).

% real_sqrt_zero
thf(fact_8988_real__sqrt__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% real_sqrt_less_iff
thf(fact_8989_real__sqrt__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% real_sqrt_le_iff
thf(fact_8990_real__sqrt__one,axiom,
    ( ( sqrt @ one_one_real )
    = one_one_real ) ).

% real_sqrt_one
thf(fact_8991_real__sqrt__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( sqrt @ X )
        = one_one_real )
      = ( X = one_one_real ) ) ).

% real_sqrt_eq_1_iff
thf(fact_8992_finite__lessThan,axiom,
    ! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).

% finite_lessThan
thf(fact_8993_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_8994_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_8995_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_8996_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_8997_real__sqrt__lt__1__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( sqrt @ X ) @ one_one_real )
      = ( ord_less_real @ X @ one_one_real ) ) ).

% real_sqrt_lt_1_iff
thf(fact_8998_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ one_one_real @ ( sqrt @ Y ) )
      = ( ord_less_real @ one_one_real @ Y ) ) ).

% real_sqrt_gt_1_iff
thf(fact_8999_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ one_one_real @ Y ) ) ).

% real_sqrt_ge_1_iff
thf(fact_9000_real__sqrt__le__1__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ one_one_real )
      = ( ord_less_eq_real @ X @ one_one_real ) ) ).

% real_sqrt_le_1_iff
thf(fact_9001_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_9002_real__sqrt__mult__self,axiom,
    ! [A2: real] :
      ( ( times_times_real @ ( sqrt @ A2 ) @ ( sqrt @ A2 ) )
      = ( abs_abs_real @ A2 ) ) ).

% real_sqrt_mult_self
thf(fact_9003_real__sqrt__abs2,axiom,
    ! [X: real] :
      ( ( sqrt @ ( times_times_real @ X @ X ) )
      = ( abs_abs_real @ X ) ) ).

% real_sqrt_abs2
thf(fact_9004_real__sqrt__four,axiom,
    ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% real_sqrt_four
thf(fact_9005_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_9006_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_9007_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_9008_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X: real,Y: real,Xa: 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 @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_squared_eq
thf(fact_9009_real__sqrt__minus,axiom,
    ! [X: real] :
      ( ( sqrt @ ( uminus_uminus_real @ X ) )
      = ( uminus_uminus_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_minus
thf(fact_9010_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_9011_real__sqrt__le__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_le_mono
thf(fact_9012_real__sqrt__power,axiom,
    ! [X: real,K: nat] :
      ( ( sqrt @ ( power_power_real @ X @ K ) )
      = ( power_power_real @ ( sqrt @ X ) @ K ) ) ).

% real_sqrt_power
thf(fact_9013_real__sqrt__divide,axiom,
    ! [X: real,Y: real] :
      ( ( sqrt @ ( divide_divide_real @ X @ Y ) )
      = ( divide_divide_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_divide
thf(fact_9014_real__sqrt__less__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_less_mono
thf(fact_9015_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_9016_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_9017_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_9018_real__sqrt__ge__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_ge_one
thf(fact_9019_lessThan__atLeast0,axiom,
    ( set_ord_lessThan_nat
    = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).

% lessThan_atLeast0
thf(fact_9020_lessThan__empty__iff,axiom,
    ! [N3: nat] :
      ( ( ( set_ord_lessThan_nat @ N3 )
        = bot_bot_set_nat )
      = ( N3 = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_9021_lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).

% lessThan_Suc
thf(fact_9022_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_9023_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_9024_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_9025_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_9026_sqrt2__less__2,axiom,
    ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% sqrt2_less_2
thf(fact_9027_sum__nth__roots,axiom,
    ! [N3: nat,C2: complex] :
      ( ( ord_less_nat @ one_one_nat @ N3 )
     => ( ( groups7754918857620584856omplex
          @ ^ [X3: complex] : X3
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N3 )
                = C2 ) ) )
        = zero_zero_complex ) ) ).

% sum_nth_roots
thf(fact_9028_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_9029_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_9030_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_9031_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_9032_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_9033_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_9034_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_9035_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_9036_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_9037_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_9038_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A2: real,C2: real,B2: real,D2: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A2 @ C2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B2 @ D2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_triangle_ineq
thf(fact_9039_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_9040_sum__roots__unity,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ one_one_nat @ N3 )
     => ( ( groups7754918857620584856omplex
          @ ^ [X3: complex] : X3
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N3 )
                = one_one_complex ) ) )
        = zero_zero_complex ) ) ).

% sum_roots_unity
thf(fact_9041_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_9042_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_9043_tan__60,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).

% tan_60
thf(fact_9044_atLeast1__lessThan__eq__remove0,axiom,
    ! [N3: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N3 )
      = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N3 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_lessThan_eq_remove0
thf(fact_9045_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_9046_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_9047_sqrt__even__pow2,axiom,
    ! [N3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N3 ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% sqrt_even_pow2
thf(fact_9048_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_9049_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_9050_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_9051_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_9052_arsinh__real__def,axiom,
    ( arsinh_real
    = ( ^ [X3: real] : ( ln_ln_real @ ( plus_plus_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arsinh_real_def
thf(fact_9053_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_9054_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_9055_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_9056_real__sqrt__power__even,axiom,
    ! [N3: nat,X: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( sqrt @ X ) @ N3 )
          = ( power_power_real @ X @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_power_even
thf(fact_9057_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_9058_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa: 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 @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_ge_zero
thf(fact_9059_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_9060_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_9061_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_9062_sum__split__even__odd,axiom,
    ! [F2: nat > real,G: nat > real,N3: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) @ ( F2 @ I2 ) @ ( G @ I2 ) )
        @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [I2: nat] : ( F2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) )
          @ ( set_ord_lessThan_nat @ N3 ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) @ one_one_nat ) )
          @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% sum_split_even_odd
thf(fact_9063_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_9064_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_9065_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_9066_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_9067_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_9068_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_9069_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_9070_arctan__half,axiom,
    ( arctan
    = ( ^ [X3: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X3 @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% arctan_half
thf(fact_9071_Sum__Icc__int,axiom,
    ! [M: int,N3: int] :
      ( ( ord_less_eq_int @ M @ N3 )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X3: int] : X3
          @ ( set_or1266510415728281911st_int @ M @ N3 ) )
        = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N3 @ ( plus_plus_int @ N3 @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% Sum_Icc_int
thf(fact_9072_sum__pos__lt__pair,axiom,
    ! [F2: nat > real,K: nat] :
      ( ( summable_real @ F2 )
     => ( ! [D5: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F2 @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D5 ) ) ) @ ( F2 @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D5 ) @ one_one_nat ) ) ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F2 @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F2 ) ) ) ) ).

% sum_pos_lt_pair
thf(fact_9073_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_9074_sumr__cos__zero__one,axiom,
    ! [N3: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ zero_zero_real @ M5 ) )
        @ ( set_ord_lessThan_nat @ ( suc @ N3 ) ) )
      = one_one_real ) ).

% sumr_cos_zero_one
thf(fact_9075_cos__coeff__0,axiom,
    ( ( cos_coeff @ zero_zero_nat )
    = one_one_real ) ).

% cos_coeff_0
thf(fact_9076_complex__of__real__def,axiom,
    ( real_V4546457046886955230omplex
    = ( ^ [R5: real] : ( complex2 @ R5 @ zero_zero_real ) ) ) ).

% complex_of_real_def
thf(fact_9077_complex__of__real__code,axiom,
    ( real_V4546457046886955230omplex
    = ( ^ [X3: real] : ( complex2 @ X3 @ zero_zero_real ) ) ) ).

% complex_of_real_code
thf(fact_9078_complex__eq__cancel__iff2,axiom,
    ! [X: real,Y: real,Xa: real] :
      ( ( ( complex2 @ X @ Y )
        = ( real_V4546457046886955230omplex @ Xa ) )
      = ( ( X = Xa )
        & ( Y = zero_zero_real ) ) ) ).

% complex_eq_cancel_iff2
thf(fact_9079_Complex__mult__complex__of__real,axiom,
    ! [X: real,Y: real,R3: real] :
      ( ( times_times_complex @ ( complex2 @ X @ Y ) @ ( real_V4546457046886955230omplex @ R3 ) )
      = ( complex2 @ ( times_times_real @ X @ R3 ) @ ( times_times_real @ Y @ R3 ) ) ) ).

% Complex_mult_complex_of_real
thf(fact_9080_complex__of__real__mult__Complex,axiom,
    ! [R3: real,X: real,Y: real] :
      ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( complex2 @ X @ Y ) )
      = ( complex2 @ ( times_times_real @ R3 @ X ) @ ( times_times_real @ R3 @ Y ) ) ) ).

% complex_of_real_mult_Complex
thf(fact_9081_complex__of__real__add__Complex,axiom,
    ! [R3: real,X: real,Y: real] :
      ( ( plus_plus_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( complex2 @ X @ Y ) )
      = ( complex2 @ ( plus_plus_real @ R3 @ X ) @ Y ) ) ).

% complex_of_real_add_Complex
thf(fact_9082_Complex__add__complex__of__real,axiom,
    ! [X: real,Y: real,R3: real] :
      ( ( plus_plus_complex @ ( complex2 @ X @ Y ) @ ( real_V4546457046886955230omplex @ R3 ) )
      = ( complex2 @ ( plus_plus_real @ X @ R3 ) @ Y ) ) ).

% Complex_add_complex_of_real
thf(fact_9083_Maclaurin__minus__cos__expansion,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( 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
                  @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N3 ) )
                @ ( 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 @ N3 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
thf(fact_9084_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ T6 )
            & ( ord_less_real @ T6 @ X )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N3 ) )
                @ ( 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 @ N3 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ) ) ).

% Maclaurin_cos_expansion2
thf(fact_9085_Maclaurin__cos__expansion,axiom,
    ! [X: real,N3: nat] :
    ? [T6: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
      & ( ( cos_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N3 ) )
          @ ( 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 @ N3 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ).

% Maclaurin_cos_expansion
thf(fact_9086_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_9087_arcsin__0,axiom,
    ( ( arcsin @ zero_zero_real )
    = zero_zero_real ) ).

% arcsin_0
thf(fact_9088_sin__arcsin,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( sin_real @ ( arcsin @ Y ) )
          = Y ) ) ) ).

% sin_arcsin
thf(fact_9089_arcsin__1,axiom,
    ( ( arcsin @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arcsin_1
thf(fact_9090_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_9091_arcsin__le__arcsin,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).

% arcsin_le_arcsin
thf(fact_9092_arcsin__minus,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( arcsin @ ( uminus_uminus_real @ X ) )
          = ( uminus_uminus_real @ ( arcsin @ X ) ) ) ) ) ).

% arcsin_minus
thf(fact_9093_arcsin__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ( arcsin @ X )
            = ( arcsin @ Y ) )
          = ( X = Y ) ) ) ) ).

% arcsin_eq_iff
thf(fact_9094_arcsin__le__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
          = ( ord_less_eq_real @ X @ Y ) ) ) ) ).

% arcsin_le_mono
thf(fact_9095_arcsin__less__arcsin,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).

% arcsin_less_arcsin
thf(fact_9096_arcsin__less__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
          = ( ord_less_real @ X @ Y ) ) ) ) ).

% arcsin_less_mono
thf(fact_9097_square__fact__le__2__fact,axiom,
    ! [N3: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N3 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% square_fact_le_2_fact
thf(fact_9098_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_9099_Maclaurin__lemma,axiom,
    ! [H2: real,F2: real > real,J2: nat > real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ? [B6: real] :
          ( ( F2 @ H2 )
          = ( plus_plus_real
            @ ( groups6591440286371151544t_real
              @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( J2 @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H2 @ M5 ) )
              @ ( set_ord_lessThan_nat @ N3 ) )
            @ ( times_times_real @ B6 @ ( divide_divide_real @ ( power_power_real @ H2 @ N3 ) @ ( semiri2265585572941072030t_real @ N3 ) ) ) ) ) ) ).

% Maclaurin_lemma
thf(fact_9100_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_9101_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_9102_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_9103_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_9104_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_9105_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_9106_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_9107_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_9108_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_9109_cos__coeff__def,axiom,
    ( cos_coeff
    = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ zero_zero_real ) ) ) ).

% cos_coeff_def
thf(fact_9110_Maclaurin__sin__expansion3,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( 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
                  @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N3 ) )
                @ ( 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 @ N3 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ) ) ).

% Maclaurin_sin_expansion3
thf(fact_9111_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N3: 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
                @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
                @ ( set_ord_lessThan_nat @ N3 ) )
              @ ( 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 @ N3 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ) ).

% Maclaurin_sin_expansion4
thf(fact_9112_Maclaurin__sin__expansion2,axiom,
    ! [X: real,N3: nat] :
    ? [T6: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
      & ( ( sin_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N3 ) )
          @ ( 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 @ N3 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ).

% Maclaurin_sin_expansion2
thf(fact_9113_Maclaurin__sin__expansion,axiom,
    ! [X: real,N3: nat] :
    ? [T6: real] :
      ( ( sin_real @ X )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
          @ ( set_ord_lessThan_nat @ N3 ) )
        @ ( 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 @ N3 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ).

% Maclaurin_sin_expansion
thf(fact_9114_sin__coeff__0,axiom,
    ( ( sin_coeff @ zero_zero_nat )
    = zero_zero_real ) ).

% sin_coeff_0
thf(fact_9115_fact__mono__nat,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N3 ) ) ) ).

% fact_mono_nat
thf(fact_9116_fact__ge__self,axiom,
    ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( semiri1408675320244567234ct_nat @ N3 ) ) ).

% fact_ge_self
thf(fact_9117_fact__less__mono__nat,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N3 )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N3 ) ) ) ) ).

% fact_less_mono_nat
thf(fact_9118_fact__ge__Suc__0__nat,axiom,
    ! [N3: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N3 ) ) ).

% fact_ge_Suc_0_nat
thf(fact_9119_dvd__fact,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N3 )
       => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N3 ) ) ) ) ).

% dvd_fact
thf(fact_9120_fact__diff__Suc,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ N3 @ ( suc @ M ) )
     => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N3 ) )
        = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N3 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N3 ) ) ) ) ) ).

% fact_diff_Suc
thf(fact_9121_fact__div__fact__le__pow,axiom,
    ! [R3: nat,N3: nat] :
      ( ( ord_less_eq_nat @ R3 @ N3 )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N3 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N3 @ R3 ) ) ) @ ( power_power_nat @ N3 @ R3 ) ) ) ).

% fact_div_fact_le_pow
thf(fact_9122_sin__coeff__Suc,axiom,
    ! [N3: nat] :
      ( ( sin_coeff @ ( suc @ N3 ) )
      = ( divide_divide_real @ ( cos_coeff @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) ).

% sin_coeff_Suc
thf(fact_9123_cos__coeff__Suc,axiom,
    ! [N3: nat] :
      ( ( cos_coeff @ ( suc @ N3 ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N3 ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) ).

% cos_coeff_Suc
thf(fact_9124_sin__coeff__def,axiom,
    ( sin_coeff
    = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ).

% sin_coeff_def
thf(fact_9125_Maclaurin__exp__lt,axiom,
    ! [X: real,N3: nat] :
      ( ( X != zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ? [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
                  @ ^ [M5: nat] : ( divide_divide_real @ ( power_power_real @ X @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N3 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_9126_sin__paired,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
      @ ( sin_real @ X ) ) ).

% sin_paired
thf(fact_9127_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_9128_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_9129_exp__less__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).

% exp_less_mono
thf(fact_9130_exp__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% exp_less_cancel_iff
thf(fact_9131_exp__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% exp_le_cancel_iff
thf(fact_9132_exp__eq__one__iff,axiom,
    ! [X: real] :
      ( ( ( exp_real @ X )
        = one_one_real )
      = ( X = zero_zero_real ) ) ).

% exp_eq_one_iff
thf(fact_9133_arccos__1,axiom,
    ( ( arccos @ one_one_real )
    = zero_zero_real ) ).

% arccos_1
thf(fact_9134_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_9135_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_9136_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_9137_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_9138_exp__ln,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( exp_real @ ( ln_ln_real @ X ) )
        = X ) ) ).

% exp_ln
thf(fact_9139_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_9140_cos__arccos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( cos_real @ ( arccos @ Y ) )
          = Y ) ) ) ).

% cos_arccos
thf(fact_9141_arccos__0,axiom,
    ( ( arccos @ zero_zero_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arccos_0
thf(fact_9142_exp__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
     => ( ord_less_real @ X @ Y ) ) ).

% exp_less_cancel
thf(fact_9143_exp__total,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ? [X4: real] :
          ( ( exp_real @ X4 )
          = Y ) ) ).

% exp_total
thf(fact_9144_exp__gt__zero,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).

% exp_gt_zero
thf(fact_9145_not__exp__less__zero,axiom,
    ! [X: real] :
      ~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).

% not_exp_less_zero
thf(fact_9146_exp__ge__zero,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).

% exp_ge_zero
thf(fact_9147_not__exp__le__zero,axiom,
    ! [X: real] :
      ~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).

% not_exp_le_zero
thf(fact_9148_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_9149_exp__ge__add__one__self,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ).

% exp_ge_add_one_self
thf(fact_9150_arccos__le__arccos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).

% arccos_le_arccos
thf(fact_9151_arccos__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
        & ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real ) )
     => ( ( ( arccos @ X )
          = ( arccos @ Y ) )
        = ( X = Y ) ) ) ).

% arccos_eq_iff
thf(fact_9152_arccos__le__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arccos @ X ) @ ( arccos @ Y ) )
          = ( ord_less_eq_real @ Y @ X ) ) ) ) ).

% arccos_le_mono
thf(fact_9153_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_9154_lemma__exp__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ one_one_real @ Y )
     => ? [X4: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X4 )
          & ( ord_less_eq_real @ X4 @ ( minus_minus_real @ Y @ one_one_real ) )
          & ( ( exp_real @ X4 )
            = Y ) ) ) ).

% lemma_exp_total
thf(fact_9155_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_9156_ln__x__over__x__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y ) @ Y ) @ ( divide_divide_real @ ( ln_ln_real @ X ) @ X ) ) ) ) ).

% ln_x_over_x_mono
thf(fact_9157_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_9158_arccos__less__arccos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).

% arccos_less_arccos
thf(fact_9159_arccos__less__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_real @ ( arccos @ X ) @ ( arccos @ Y ) )
          = ( ord_less_real @ Y @ X ) ) ) ) ).

% arccos_less_mono
thf(fact_9160_exp__le,axiom,
    ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).

% exp_le
thf(fact_9161_arccos__ubound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ).

% arccos_ubound
thf(fact_9162_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_9163_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
     => ( ( cos_real @ ( arccos @ Y ) )
        = Y ) ) ).

% cos_arccos_abs
thf(fact_9164_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
     => ( ( arccos @ ( cos_real @ Theta ) )
        = ( abs_abs_real @ Theta ) ) ) ).

% arccos_cos_eq_abs
thf(fact_9165_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_9166_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_9167_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_9168_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_9169_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_9170_arccos__minus,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( arccos @ ( uminus_uminus_real @ X ) )
          = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ) ).

% arccos_minus
thf(fact_9171_power__half__series,axiom,
    ( sums_real
    @ ^ [N2: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N2 ) )
    @ one_one_real ) ).

% power_half_series
thf(fact_9172_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_9173_arccos__minus__abs,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( arccos @ ( uminus_uminus_real @ X ) )
        = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ).

% arccos_minus_abs
thf(fact_9174_sums__if_H,axiom,
    ! [G: nat > real,X: real] :
      ( ( sums_real @ G @ X )
     => ( sums_real
        @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        @ X ) ) ).

% sums_if'
thf(fact_9175_sums__if,axiom,
    ! [G: nat > real,X: real,F2: nat > real,Y: real] :
      ( ( sums_real @ G @ X )
     => ( ( sums_real @ F2 @ Y )
       => ( sums_real
          @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( F2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          @ ( plus_plus_real @ X @ Y ) ) ) ) ).

% sums_if
thf(fact_9176_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_9177_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_9178_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ) @ N3 ) @ ( exp_real @ X ) ) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
thf(fact_9179_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N3: nat] :
      ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N3 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ) @ N3 ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
thf(fact_9180_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_9181_Maclaurin__exp__le,axiom,
    ! [X: real,N3: nat] :
    ? [T6: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
      & ( ( exp_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( divide_divide_real @ ( power_power_real @ X @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) )
            @ ( set_ord_lessThan_nat @ N3 ) )
          @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ).

% Maclaurin_exp_le
thf(fact_9182_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_9183_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_9184_tanh__real__altdef,axiom,
    ( tanh_real
    = ( ^ [X3: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ) ) ).

% tanh_real_altdef
thf(fact_9185_cos__paired,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_real @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      @ ( cos_real @ X ) ) ).

% cos_paired
thf(fact_9186_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_9187_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_9188_vebt__assn__raw_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa: vEBT_VEBTi,Y: assn] :
      ( ( ( vEBT_vebt_assn_raw @ X @ Xa )
        = Y )
     => ( ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ X @ Xa ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ! [Ai: $o,Bi: $o] :
                  ( ( Xa
                    = ( vEBT_Leafi @ Ai @ Bi ) )
                 => ( ( Y
                      = ( pure_assn
                        @ ( ( Ai = A3 )
                          & ( Bi = B3 ) ) ) )
                   => ~ ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ ( vEBT_Leaf @ A3 @ B3 ) @ ( vEBT_Leafi @ Ai @ Bi ) ) ) ) ) )
         => ( ! [Mmo: option4927543243414619207at_nat,Deg2: nat,Tree_list: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mmo @ Deg2 @ Tree_list @ Summary2 ) )
               => ! [Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] :
                    ( ( Xa
                      = ( vEBT_Nodei @ Mmoi @ Degi @ Tree_array @ Summaryi ) )
                   => ( ( Y
                        = ( times_times_assn
                          @ ( times_times_assn
                            @ ( pure_assn
                              @ ( ( Mmoi = Mmo )
                                & ( Degi = Deg2 ) ) )
                            @ ( vEBT_vebt_assn_raw @ Summary2 @ Summaryi ) )
                          @ ( ex_ass463751140784270563_VEBTi
                            @ ^ [Tree_is2: list_VEBT_VEBTi] : ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Tree_array @ Tree_is2 ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Tree_list @ Tree_is2 ) ) ) ) )
                     => ~ ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ ( vEBT_Node @ Mmo @ Deg2 @ Tree_list @ Summary2 ) @ ( vEBT_Nodei @ Mmoi @ Degi @ Tree_array @ Summaryi ) ) ) ) ) )
           => ( ! [V2: option4927543243414619207at_nat,Va3: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ V2 @ Va3 @ Vb3 @ Vc3 ) )
                 => ! [Vd3: $o,Ve3: $o] :
                      ( ( Xa
                        = ( vEBT_Leafi @ Vd3 @ Ve3 ) )
                     => ( ( Y = bot_bot_assn )
                       => ~ ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ ( vEBT_Node @ V2 @ Va3 @ Vb3 @ Vc3 ) @ ( vEBT_Leafi @ Vd3 @ Ve3 ) ) ) ) ) )
             => ~ ! [Vd3: $o,Ve3: $o] :
                    ( ( X
                      = ( vEBT_Leaf @ Vd3 @ Ve3 ) )
                   => ! [V2: option4927543243414619207at_nat,Va3: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] :
                        ( ( Xa
                          = ( vEBT_Nodei @ V2 @ Va3 @ Vb3 @ Vc3 ) )
                       => ( ( Y = bot_bot_assn )
                         => ~ ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ ( vEBT_Leaf @ Vd3 @ Ve3 ) @ ( vEBT_Nodei @ V2 @ Va3 @ Vb3 @ Vc3 ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_assn_raw.pelims
thf(fact_9189_VEBT__internal_Oheight_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      ( ( vEBT_VEBT_height @ ( vEBT_Leaf @ A2 @ B2 ) )
      = zero_zero_nat ) ).

% VEBT_internal.height.simps(1)
thf(fact_9190_merge__true__star,axiom,
    ( ( times_times_assn @ top_top_assn @ top_top_assn )
    = top_top_assn ) ).

% merge_true_star
thf(fact_9191_assn__basic__inequalities_I1_J,axiom,
    top_top_assn != one_one_assn ).

% assn_basic_inequalities(1)
thf(fact_9192_assn__basic__inequalities_I5_J,axiom,
    top_top_assn != bot_bot_assn ).

% assn_basic_inequalities(5)
thf(fact_9193_merge__true__star__ctx,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ top_top_assn @ ( times_times_assn @ top_top_assn @ P ) )
      = ( times_times_assn @ top_top_assn @ P ) ) ).

% merge_true_star_ctx
thf(fact_9194_ent__true,axiom,
    ! [P: assn] : ( entails @ P @ top_top_assn ) ).

% ent_true
thf(fact_9195_ent__true__drop_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( ( entails @ P @ Q )
     => ( entails @ P @ ( times_times_assn @ Q @ top_top_assn ) ) ) ).

% ent_true_drop(2)
thf(fact_9196_ent__true__drop_I1_J,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( ( entails @ P @ ( times_times_assn @ Q @ top_top_assn ) )
     => ( entails @ ( times_times_assn @ P @ R ) @ ( times_times_assn @ Q @ top_top_assn ) ) ) ).

% ent_true_drop(1)
thf(fact_9197_ent__refl__true,axiom,
    ! [A: assn] : ( entails @ A @ ( times_times_assn @ A @ top_top_assn ) ) ).

% ent_refl_true
thf(fact_9198_ent__star__mono__true,axiom,
    ! [A: assn,A5: assn,B: assn,B10: assn] :
      ( ( entails @ A @ ( times_times_assn @ A5 @ top_top_assn ) )
     => ( ( entails @ B @ ( times_times_assn @ B10 @ top_top_assn ) )
       => ( entails @ ( times_times_assn @ ( times_times_assn @ A @ B ) @ top_top_assn ) @ ( times_times_assn @ ( times_times_assn @ A5 @ B10 ) @ top_top_assn ) ) ) ) ).

% ent_star_mono_true
thf(fact_9199_mod__star__trueI,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ P @ H2 )
     => ( rep_assn @ ( times_times_assn @ P @ top_top_assn ) @ H2 ) ) ).

% mod_star_trueI
thf(fact_9200_mod__star__trueE,axiom,
    ! [P: assn,H2: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P @ top_top_assn ) @ H2 )
     => ~ ! [H5: produc3658429121746597890et_nat] :
            ~ ( rep_assn @ P @ H5 ) ) ).

% mod_star_trueE
thf(fact_9201_mod__h__bot__iff_I2_J,axiom,
    ! [H2: heap_e7401611519738050253t_unit] : ( rep_assn @ top_top_assn @ ( produc7507926704131184380et_nat @ H2 @ bot_bot_set_nat ) ) ).

% mod_h_bot_iff(2)
thf(fact_9202_VEBT__internal_Oheight_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A3: $o,B3: $o] :
          ( X
         != ( vEBT_Leaf @ A3 @ B3 ) )
     => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
            ( X
           != ( vEBT_Node @ Uu2 @ Deg2 @ TreeList3 @ Summary2 ) ) ) ).

% VEBT_internal.height.cases
thf(fact_9203_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N3 ) )
          = ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N3 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
thf(fact_9204_Maclaurin__sin__bound,axiom,
    ! [X: real,N3: nat] :
      ( ord_less_eq_real
      @ ( abs_abs_real
        @ ( minus_minus_real @ ( sin_real @ X )
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X @ M5 ) )
            @ ( set_ord_lessThan_nat @ N3 ) ) ) )
      @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N3 ) ) ) ).

% Maclaurin_sin_bound
thf(fact_9205_floor__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A2 ) @ ( numeral_numeral_real @ B2 ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ A2 ) @ ( numeral_numeral_int @ B2 ) ) ) ).

% floor_divide_eq_div_numeral
thf(fact_9206_floor__one__divide__eq__div__numeral,axiom,
    ! [B2: num] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B2 ) ) )
      = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B2 ) ) ) ).

% floor_one_divide_eq_div_numeral
thf(fact_9207_floor__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A2 ) @ ( numeral_numeral_real @ B2 ) ) ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A2 ) ) @ ( numeral_numeral_int @ B2 ) ) ) ).

% floor_minus_divide_eq_div_numeral
thf(fact_9208_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B2: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B2 ) ) ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B2 ) ) ) ).

% floor_minus_one_divide_eq_div_numeral
thf(fact_9209_real__sqrt__inverse,axiom,
    ! [X: real] :
      ( ( sqrt @ ( inverse_inverse_real @ X ) )
      = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_inverse
thf(fact_9210_divide__real__def,axiom,
    ( divide_divide_real
    = ( ^ [X3: real,Y2: real] : ( times_times_real @ X3 @ ( inverse_inverse_real @ Y2 ) ) ) ) ).

% divide_real_def
thf(fact_9211_inverse__powr,axiom,
    ! [Y: real,A2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( powr_real @ ( inverse_inverse_real @ Y ) @ A2 )
        = ( inverse_inverse_real @ ( powr_real @ Y @ A2 ) ) ) ) ).

% inverse_powr
thf(fact_9212_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_9213_floor__eq3,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N3 ) ) ) ).

% floor_eq3
thf(fact_9214_le__nat__floor,axiom,
    ! [X: nat,A2: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A2 )
     => ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A2 ) ) ) ) ).

% le_nat_floor
thf(fact_9215_real__of__int__floor__add__one__gt,axiom,
    ! [R3: real] : ( ord_less_real @ R3 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R3 ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_gt
thf(fact_9216_floor__eq,axiom,
    ! [N3: int,X: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N3 ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N3 ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = N3 ) ) ) ).

% floor_eq
thf(fact_9217_real__of__int__floor__add__one__ge,axiom,
    ! [R3: real] : ( ord_less_eq_real @ R3 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R3 ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_ge
thf(fact_9218_forall__pos__mono__1,axiom,
    ! [P: real > $o,E: real] :
      ( ! [D5: real,E2: real] :
          ( ( ord_less_real @ D5 @ E2 )
         => ( ( P @ D5 )
           => ( P @ E2 ) ) )
     => ( ! [N: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E )
         => ( P @ E ) ) ) ) ).

% forall_pos_mono_1
thf(fact_9219_real__arch__inverse,axiom,
    ! [E: real] :
      ( ( ord_less_real @ zero_zero_real @ E )
      = ( ? [N2: nat] :
            ( ( N2 != zero_zero_nat )
            & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) )
            & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ E ) ) ) ) ).

% real_arch_inverse
thf(fact_9220_forall__pos__mono,axiom,
    ! [P: real > $o,E: real] :
      ( ! [D5: real,E2: real] :
          ( ( ord_less_real @ D5 @ E2 )
         => ( ( P @ D5 )
           => ( P @ E2 ) ) )
     => ( ! [N: nat] :
            ( ( N != zero_zero_nat )
           => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E )
         => ( P @ E ) ) ) ) ).

% forall_pos_mono
thf(fact_9221_real__of__int__floor__gt__diff__one,axiom,
    ! [R3: real] : ( ord_less_real @ ( minus_minus_real @ R3 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R3 ) ) ) ).

% real_of_int_floor_gt_diff_one
thf(fact_9222_real__of__int__floor__ge__diff__one,axiom,
    ! [R3: real] : ( ord_less_eq_real @ ( minus_minus_real @ R3 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R3 ) ) ) ).

% real_of_int_floor_ge_diff_one
thf(fact_9223_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_9224_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_9225_floor__eq4,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N3 ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N3 ) ) ) ).

% floor_eq4
thf(fact_9226_floor__eq2,axiom,
    ! [N3: int,X: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N3 ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N3 ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = N3 ) ) ) ).

% floor_eq2
thf(fact_9227_floor__divide__real__eq__div,axiom,
    ! [B2: int,A2: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ B2 )
     => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A2 @ ( ring_1_of_int_real @ B2 ) ) )
        = ( divide_divide_int @ ( archim6058952711729229775r_real @ A2 ) @ B2 ) ) ) ).

% floor_divide_real_eq_div
thf(fact_9228_log__inverse,axiom,
    ! [A2: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A2 )
     => ( ( A2 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( log @ A2 @ ( inverse_inverse_real @ X ) )
            = ( uminus_uminus_real @ ( log @ A2 @ X ) ) ) ) ) ) ).

% log_inverse
thf(fact_9229_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_9230_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_9231_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_9232_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_9233_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_9234_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_9235_floor__log__eq__powr__iff,axiom,
    ! [X: real,B2: real,K: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ one_one_real @ B2 )
       => ( ( ( archim6058952711729229775r_real @ ( log @ B2 @ X ) )
            = K )
          = ( ( ord_less_eq_real @ ( powr_real @ B2 @ ( ring_1_of_int_real @ K ) ) @ X )
            & ( ord_less_real @ X @ ( powr_real @ B2 @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).

% floor_log_eq_powr_iff
thf(fact_9236_powr__real__of__int,axiom,
    ! [X: real,N3: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ N3 )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ N3 ) )
            = ( power_power_real @ X @ ( nat2 @ N3 ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ N3 )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ N3 ) )
            = ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N3 ) ) ) ) ) ) ) ) ).

% powr_real_of_int
thf(fact_9237_floor__log2__div2,axiom,
    ! [N3: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
        = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).

% floor_log2_div2
thf(fact_9238_floor__log__nat__eq__if,axiom,
    ! [B2: nat,N3: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N3 ) @ K )
     => ( ( ord_less_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N3 @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
         => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B2 ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% floor_log_nat_eq_if
thf(fact_9239_divide__complex__def,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [X3: complex,Y2: complex] : ( times_times_complex @ X3 @ ( invers8013647133539491842omplex @ Y2 ) ) ) ) ).

% divide_complex_def
thf(fact_9240_real__scaleR__def,axiom,
    real_V1485227260804924795R_real = times_times_real ).

% real_scaleR_def
thf(fact_9241_complex__scaleR,axiom,
    ! [R3: real,A2: real,B2: real] :
      ( ( real_V2046097035970521341omplex @ R3 @ ( complex2 @ A2 @ B2 ) )
      = ( complex2 @ ( times_times_real @ R3 @ A2 ) @ ( times_times_real @ R3 @ B2 ) ) ) ).

% complex_scaleR
thf(fact_9242_prod__Suc__Suc__fact,axiom,
    ! [N3: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N3 ) )
      = ( semiri1408675320244567234ct_nat @ N3 ) ) ).

% prod_Suc_Suc_fact
thf(fact_9243_prod__Suc__fact,axiom,
    ! [N3: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) )
      = ( semiri1408675320244567234ct_nat @ N3 ) ) ).

% prod_Suc_fact
thf(fact_9244_fact__eq__fact__times,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( semiri1408675320244567234ct_nat @ M )
        = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N3 )
          @ ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ ( suc @ N3 ) @ M ) ) ) ) ) ).

% fact_eq_fact_times
thf(fact_9245_fact__div__fact,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_eq_nat @ N3 @ M )
     => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N3 ) )
        = ( groups708209901874060359at_nat
          @ ^ [X3: nat] : X3
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N3 @ one_one_nat ) @ M ) ) ) ) ).

% fact_div_fact
thf(fact_9246_complex__inverse,axiom,
    ! [A2: real,B2: real] :
      ( ( invers8013647133539491842omplex @ ( complex2 @ A2 @ B2 ) )
      = ( complex2 @ ( divide_divide_real @ A2 @ ( plus_plus_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ B2 ) @ ( plus_plus_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% complex_inverse
thf(fact_9247_sinh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( ( sinh_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% sinh_real_zero_iff
thf(fact_9248_sinh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% sinh_real_less_iff
thf(fact_9249_sinh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% sinh_real_le_iff
thf(fact_9250_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_9251_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_9252_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_9253_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_9254_sinh__less__cosh__real,axiom,
    ! [X: real] : ( ord_less_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).

% sinh_less_cosh_real
thf(fact_9255_sinh__le__cosh__real,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).

% sinh_le_cosh_real
thf(fact_9256_cosh__real__nonzero,axiom,
    ! [X: real] :
      ( ( cosh_real @ X )
     != zero_zero_real ) ).

% cosh_real_nonzero
thf(fact_9257_cosh__real__pos,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X ) ) ).

% cosh_real_pos
thf(fact_9258_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_9259_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_9260_cosh__real__nonneg,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X ) ) ).

% cosh_real_nonneg
thf(fact_9261_cosh__real__ge__1,axiom,
    ! [X: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X ) ) ).

% cosh_real_ge_1
thf(fact_9262_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_9263_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_9264_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_9265_prod__int__eq,axiom,
    ! [I: nat,J2: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J2 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X3: int] : X3
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J2 ) ) ) ) ).

% prod_int_eq
thf(fact_9266_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_9267_prod__int__plus__eq,axiom,
    ! [I: nat,J2: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J2 ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X3: int] : X3
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J2 ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_9268_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_9269_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_9270_binomial__code,axiom,
    ( binomial
    = ( ^ [N2: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N2 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N2 @ K3 ) @ one_one_nat ) @ N2 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).

% binomial_code
thf(fact_9271_binomial__Suc__n,axiom,
    ! [N3: nat] :
      ( ( binomial @ ( suc @ N3 ) @ N3 )
      = ( suc @ N3 ) ) ).

% binomial_Suc_n
thf(fact_9272_finite__atMost,axiom,
    ! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).

% finite_atMost
thf(fact_9273_binomial__1,axiom,
    ! [N3: nat] :
      ( ( binomial @ N3 @ ( suc @ zero_zero_nat ) )
      = N3 ) ).

% binomial_1
thf(fact_9274_binomial__0__Suc,axiom,
    ! [K: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_9275_binomial__eq__0__iff,axiom,
    ! [N3: nat,K: nat] :
      ( ( ( binomial @ N3 @ K )
        = zero_zero_nat )
      = ( ord_less_nat @ N3 @ K ) ) ).

% binomial_eq_0_iff
thf(fact_9276_binomial__Suc__Suc,axiom,
    ! [N3: nat,K: nat] :
      ( ( binomial @ ( suc @ N3 ) @ ( suc @ K ) )
      = ( plus_plus_nat @ ( binomial @ N3 @ K ) @ ( binomial @ N3 @ ( suc @ K ) ) ) ) ).

% binomial_Suc_Suc
thf(fact_9277_binomial__n__0,axiom,
    ! [N3: nat] :
      ( ( binomial @ N3 @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_9278_zero__less__binomial__iff,axiom,
    ! [N3: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N3 @ K ) )
      = ( ord_less_eq_nat @ K @ N3 ) ) ).

% zero_less_binomial_iff
thf(fact_9279_atMost__0,axiom,
    ( ( set_ord_atMost_nat @ zero_zero_nat )
    = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).

% atMost_0
thf(fact_9280_sum__choose__lower,axiom,
    ! [R3: nat,N3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R3 @ K3 ) @ K3 )
        @ ( set_ord_atMost_nat @ N3 ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R3 @ N3 ) ) @ N3 ) ) ).

% sum_choose_lower
thf(fact_9281_choose__rising__sum_I2_J,axiom,
    ! [N3: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J: nat] : ( binomial @ ( plus_plus_nat @ N3 @ J ) @ N3 )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N3 @ M ) @ one_one_nat ) @ M ) ) ).

% choose_rising_sum(2)
thf(fact_9282_choose__rising__sum_I1_J,axiom,
    ! [N3: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J: nat] : ( binomial @ ( plus_plus_nat @ N3 @ J ) @ N3 )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N3 @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).

% choose_rising_sum(1)
thf(fact_9283_sum__choose__upper,axiom,
    ! [M: nat,N3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ K3 @ M )
        @ ( set_ord_atMost_nat @ N3 ) )
      = ( binomial @ ( suc @ N3 ) @ ( suc @ M ) ) ) ).

% sum_choose_upper
thf(fact_9284_binomial__eq__0,axiom,
    ! [N3: nat,K: nat] :
      ( ( ord_less_nat @ N3 @ K )
     => ( ( binomial @ N3 @ K )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_9285_Suc__times__binomial,axiom,
    ! [K: nat,N3: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N3 ) @ ( suc @ K ) ) )
      = ( times_times_nat @ ( suc @ N3 ) @ ( binomial @ N3 @ K ) ) ) ).

% Suc_times_binomial
thf(fact_9286_Suc__times__binomial__eq,axiom,
    ! [N3: nat,K: nat] :
      ( ( times_times_nat @ ( suc @ N3 ) @ ( binomial @ N3 @ K ) )
      = ( times_times_nat @ ( binomial @ ( suc @ N3 ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).

% Suc_times_binomial_eq
thf(fact_9287_binomial__symmetric,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( binomial @ N3 @ K )
        = ( binomial @ N3 @ ( minus_minus_nat @ N3 @ K ) ) ) ) ).

% binomial_symmetric
thf(fact_9288_choose__mult__lemma,axiom,
    ! [M: nat,R3: nat,K: nat] :
      ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R3 ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
      = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R3 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R3 ) @ M ) ) ) ).

% choose_mult_lemma
thf(fact_9289_binomial__le__pow,axiom,
    ! [R3: nat,N3: nat] :
      ( ( ord_less_eq_nat @ R3 @ N3 )
     => ( ord_less_eq_nat @ ( binomial @ N3 @ R3 ) @ ( power_power_nat @ N3 @ R3 ) ) ) ).

% binomial_le_pow
thf(fact_9290_atMost__atLeast0,axiom,
    ( set_ord_atMost_nat
    = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).

% atMost_atLeast0
thf(fact_9291_lessThan__Suc__atMost,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( set_ord_atMost_nat @ K ) ) ).

% lessThan_Suc_atMost
thf(fact_9292_sum__choose__diagonal,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ( groups3542108847815614940at_nat
          @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N3 @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
          @ ( set_ord_atMost_nat @ M ) )
        = ( binomial @ ( suc @ N3 ) @ M ) ) ) ).

% sum_choose_diagonal
thf(fact_9293_vandermonde,axiom,
    ! [M: nat,N3: nat,R3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N3 @ ( minus_minus_nat @ R3 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ R3 ) )
      = ( binomial @ ( plus_plus_nat @ M @ N3 ) @ R3 ) ) ).

% vandermonde
thf(fact_9294_atMost__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K ) )
      = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).

% atMost_Suc
thf(fact_9295_choose__row__sum,axiom,
    ! [N3: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ N3 ) @ ( set_ord_atMost_nat @ N3 ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% choose_row_sum
thf(fact_9296_binomial,axiom,
    ! [A2: nat,B2: nat,N3: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A2 @ B2 ) @ N3 )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N3 @ K3 ) ) @ ( power_power_nat @ A2 @ K3 ) ) @ ( power_power_nat @ B2 @ ( minus_minus_nat @ N3 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N3 ) ) ) ).

% binomial
thf(fact_9297_zero__less__binomial,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N3 @ K ) ) ) ).

% zero_less_binomial
thf(fact_9298_Suc__times__binomial__add,axiom,
    ! [A2: nat,B2: nat] :
      ( ( times_times_nat @ ( suc @ A2 ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A2 @ B2 ) ) @ ( suc @ A2 ) ) )
      = ( times_times_nat @ ( suc @ B2 ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A2 @ B2 ) ) @ A2 ) ) ) ).

% Suc_times_binomial_add
thf(fact_9299_binomial__Suc__Suc__eq__times,axiom,
    ! [N3: nat,K: nat] :
      ( ( binomial @ ( suc @ N3 ) @ ( suc @ K ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N3 ) @ ( binomial @ N3 @ K ) ) @ ( suc @ K ) ) ) ).

% binomial_Suc_Suc_eq_times
thf(fact_9300_choose__mult,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ M @ N3 )
       => ( ( times_times_nat @ ( binomial @ N3 @ M ) @ ( binomial @ M @ K ) )
          = ( times_times_nat @ ( binomial @ N3 @ K ) @ ( binomial @ ( minus_minus_nat @ N3 @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).

% choose_mult
thf(fact_9301_binomial__absorb__comp,axiom,
    ! [N3: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ N3 @ K ) @ ( binomial @ N3 @ K ) )
      = ( times_times_nat @ N3 @ ( binomial @ ( minus_minus_nat @ N3 @ one_one_nat ) @ K ) ) ) ).

% binomial_absorb_comp
thf(fact_9302_choose__square__sum,axiom,
    ! [N3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( power_power_nat @ ( binomial @ N3 @ K3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( set_ord_atMost_nat @ N3 ) )
      = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ N3 ) ) ).

% choose_square_sum
thf(fact_9303_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_9304_binomial__absorption,axiom,
    ! [K: nat,N3: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N3 @ ( suc @ K ) ) )
      = ( times_times_nat @ N3 @ ( binomial @ ( minus_minus_nat @ N3 @ one_one_nat ) @ K ) ) ) ).

% binomial_absorption
thf(fact_9305_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_9306_choose__linear__sum,axiom,
    ! [N3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( times_times_nat @ I2 @ ( binomial @ N3 @ I2 ) )
        @ ( set_ord_atMost_nat @ N3 ) )
      = ( times_times_nat @ N3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ).

% choose_linear_sum
thf(fact_9307_binomial__fact__lemma,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N3 @ K ) ) ) @ ( binomial @ N3 @ K ) )
        = ( semiri1408675320244567234ct_nat @ N3 ) ) ) ).

% binomial_fact_lemma
thf(fact_9308_binomial__maximum,axiom,
    ! [N3: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N3 @ K ) @ ( binomial @ N3 @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% binomial_maximum
thf(fact_9309_binomial__antimono,axiom,
    ! [K: nat,K4: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ K4 )
     => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
       => ( ( ord_less_eq_nat @ K4 @ N3 )
         => ( ord_less_eq_nat @ ( binomial @ N3 @ K4 ) @ ( binomial @ N3 @ K ) ) ) ) ) ).

% binomial_antimono
thf(fact_9310_binomial__mono,axiom,
    ! [K: nat,K4: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ K4 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K4 ) @ N3 )
       => ( ord_less_eq_nat @ ( binomial @ N3 @ K ) @ ( binomial @ N3 @ K4 ) ) ) ) ).

% binomial_mono
thf(fact_9311_binomial__maximum_H,axiom,
    ! [N3: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ N3 ) ) ).

% binomial_maximum'
thf(fact_9312_binomial__le__pow2,axiom,
    ! [N3: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N3 @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% binomial_le_pow2
thf(fact_9313_choose__reduce__nat,axiom,
    ! [N3: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( binomial @ N3 @ K )
          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N3 @ one_one_nat ) @ K ) ) ) ) ) ).

% choose_reduce_nat
thf(fact_9314_times__binomial__minus1__eq,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( times_times_nat @ K @ ( binomial @ N3 @ K ) )
        = ( times_times_nat @ N3 @ ( binomial @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% times_binomial_minus1_eq
thf(fact_9315_binomial__altdef__nat,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_eq_nat @ K @ N3 )
     => ( ( binomial @ N3 @ K )
        = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N3 ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N3 @ K ) ) ) ) ) ) ).

% binomial_altdef_nat
thf(fact_9316_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N3: nat] :
      ( ( ord_less_nat @ K @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ord_less_nat @ ( binomial @ N3 @ K ) @ ( binomial @ N3 @ ( suc @ K ) ) ) ) ).

% binomial_less_binomial_Suc
thf(fact_9317_binomial__strict__mono,axiom,
    ! [K: nat,K4: nat,N3: nat] :
      ( ( ord_less_nat @ K @ K4 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K4 ) @ N3 )
       => ( ord_less_nat @ ( binomial @ N3 @ K ) @ ( binomial @ N3 @ K4 ) ) ) ) ).

% binomial_strict_mono
thf(fact_9318_binomial__strict__antimono,axiom,
    ! [K: nat,K4: nat,N3: nat] :
      ( ( ord_less_nat @ K @ K4 )
     => ( ( ord_less_eq_nat @ N3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
       => ( ( ord_less_eq_nat @ K4 @ N3 )
         => ( ord_less_nat @ ( binomial @ N3 @ K4 ) @ ( binomial @ N3 @ K ) ) ) ) ) ).

% binomial_strict_antimono
thf(fact_9319_central__binomial__odd,axiom,
    ! [N3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( binomial @ N3 @ ( suc @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = ( binomial @ N3 @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% central_binomial_odd
thf(fact_9320_binomial__addition__formula,axiom,
    ! [N3: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( binomial @ N3 @ ( suc @ K ) )
        = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N3 @ one_one_nat ) @ K ) ) ) ) ).

% binomial_addition_formula
thf(fact_9321_atLeast1__atMost__eq__remove0,axiom,
    ! [N3: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N3 )
      = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N3 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_atMost_eq_remove0
thf(fact_9322_choose__two,axiom,
    ! [N3: nat] :
      ( ( binomial @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( divide_divide_nat @ ( times_times_nat @ N3 @ ( minus_minus_nat @ N3 @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% choose_two
thf(fact_9323_polynomial__product__nat,axiom,
    ! [M: nat,A2: nat > nat,N3: nat,B2: nat > nat,X: nat] :
      ( ! [I5: nat] :
          ( ( ord_less_nat @ M @ I5 )
         => ( ( A2 @ I5 )
            = zero_zero_nat ) )
     => ( ! [J3: nat] :
            ( ( ord_less_nat @ N3 @ J3 )
           => ( ( B2 @ J3 )
              = zero_zero_nat ) )
       => ( ( times_times_nat
            @ ( groups3542108847815614940at_nat
              @ ^ [I2: nat] : ( times_times_nat @ ( A2 @ I2 ) @ ( power_power_nat @ X @ I2 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups3542108847815614940at_nat
              @ ^ [J: nat] : ( times_times_nat @ ( B2 @ J ) @ ( power_power_nat @ X @ J ) )
              @ ( set_ord_atMost_nat @ N3 ) ) )
          = ( groups3542108847815614940at_nat
            @ ^ [R5: nat] :
                ( times_times_nat
                @ ( groups3542108847815614940at_nat
                  @ ^ [K3: nat] : ( times_times_nat @ ( A2 @ K3 ) @ ( B2 @ ( minus_minus_nat @ R5 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_nat @ X @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N3 ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_9324_central__binomial__lower__bound,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N3 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ N3 ) ) ) ) ).

% central_binomial_lower_bound
thf(fact_9325_of__nat__id,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N2: nat] : N2 ) ) ).

% of_nat_id
thf(fact_9326_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_9327_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_9328_divide__i,axiom,
    ! [X: complex] :
      ( ( divide1717551699836669952omplex @ X @ imaginary_unit )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X ) ) ).

% divide_i
thf(fact_9329_divide__numeral__i,axiom,
    ! [Z: complex,N3: num] :
      ( ( divide1717551699836669952omplex @ Z @ ( times_times_complex @ ( numera6690914467698888265omplex @ N3 ) @ imaginary_unit ) )
      = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) @ ( numera6690914467698888265omplex @ N3 ) ) ) ).

% divide_numeral_i
thf(fact_9330_power2__i,axiom,
    ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power2_i
thf(fact_9331_i__even__power,axiom,
    ! [N3: nat] :
      ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N3 ) ) ).

% i_even_power
thf(fact_9332_imaginary__unit_Ocode,axiom,
    ( imaginary_unit
    = ( complex2 @ zero_zero_real @ one_one_real ) ) ).

% imaginary_unit.code
thf(fact_9333_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_9334_i__complex__of__real,axiom,
    ! [R3: real] :
      ( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R3 ) )
      = ( complex2 @ zero_zero_real @ R3 ) ) ).

% i_complex_of_real
thf(fact_9335_complex__of__real__i,axiom,
    ! [R3: real] :
      ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ imaginary_unit )
      = ( complex2 @ zero_zero_real @ R3 ) ) ).

% complex_of_real_i
thf(fact_9336_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_9337_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_9338_Arg__ii,axiom,
    ( ( arg @ imaginary_unit )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% Arg_ii
thf(fact_9339_power2__csqrt,axiom,
    ! [Z: complex] :
      ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = Z ) ).

% power2_csqrt
thf(fact_9340_Arg__zero,axiom,
    ( ( arg @ zero_zero_complex )
    = zero_zero_real ) ).

% Arg_zero
thf(fact_9341_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_9342_Arg__bounded,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
      & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).

% Arg_bounded
thf(fact_9343_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_9344_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_9345_cis__zero,axiom,
    ( ( cis @ zero_zero_real )
    = one_one_complex ) ).

% cis_zero
thf(fact_9346_cot__pi,axiom,
    ( ( cot_real @ pi )
    = zero_zero_real ) ).

% cot_pi
thf(fact_9347_cot__npi,axiom,
    ! [N3: nat] :
      ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ pi ) )
      = zero_zero_real ) ).

% cot_npi
thf(fact_9348_cis__pi__half,axiom,
    ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = imaginary_unit ) ).

% cis_pi_half
thf(fact_9349_cis__2pi,axiom,
    ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_complex ) ).

% cis_2pi
thf(fact_9350_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_9351_cis__mult,axiom,
    ! [A2: real,B2: real] :
      ( ( times_times_complex @ ( cis @ A2 ) @ ( cis @ B2 ) )
      = ( cis @ ( plus_plus_real @ A2 @ B2 ) ) ) ).

% cis_mult
thf(fact_9352_cis__divide,axiom,
    ! [A2: real,B2: real] :
      ( ( divide1717551699836669952omplex @ ( cis @ A2 ) @ ( cis @ B2 ) )
      = ( cis @ ( minus_minus_real @ A2 @ B2 ) ) ) ).

% cis_divide
thf(fact_9353_DeMoivre,axiom,
    ! [A2: real,N3: nat] :
      ( ( power_power_complex @ ( cis @ A2 ) @ N3 )
      = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ A2 ) ) ) ).

% DeMoivre
thf(fact_9354_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_9355_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_9356_bij__betw__roots__unity,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( 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 @ N3 ) ) )
        @ ( set_ord_lessThan_nat @ N3 )
        @ ( collect_complex
          @ ^ [Z5: complex] :
              ( ( power_power_complex @ Z5 @ N3 )
              = one_one_complex ) ) ) ) ).

% bij_betw_roots_unity
thf(fact_9357_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_9358_pred__numeral__simps_I2_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit0 @ K ) )
      = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).

% pred_numeral_simps(2)
thf(fact_9359_semiring__norm_I26_J,axiom,
    ( ( bitM @ one )
    = one ) ).

% semiring_norm(26)
thf(fact_9360_semiring__norm_I28_J,axiom,
    ! [N3: num] :
      ( ( bitM @ ( bit1 @ N3 ) )
      = ( bit1 @ ( bit0 @ N3 ) ) ) ).

% semiring_norm(28)
thf(fact_9361_semiring__norm_I27_J,axiom,
    ! [N3: num] :
      ( ( bitM @ ( bit0 @ N3 ) )
      = ( bit1 @ ( bitM @ N3 ) ) ) ).

% semiring_norm(27)
thf(fact_9362_eval__nat__numeral_I2_J,axiom,
    ! [N3: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N3 ) )
      = ( suc @ ( numeral_numeral_nat @ ( bitM @ N3 ) ) ) ) ).

% eval_nat_numeral(2)
thf(fact_9363_BitM__plus__one,axiom,
    ! [N3: num] :
      ( ( plus_plus_num @ ( bitM @ N3 ) @ one )
      = ( bit0 @ N3 ) ) ).

% BitM_plus_one
thf(fact_9364_one__plus__BitM,axiom,
    ! [N3: num] :
      ( ( plus_plus_num @ one @ ( bitM @ N3 ) )
      = ( bit0 @ N3 ) ) ).

% one_plus_BitM
thf(fact_9365_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_9366_Suc__0__mod__eq,axiom,
    ! [N3: nat] :
      ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N3 )
      = ( zero_n2687167440665602831ol_nat
        @ ( N3
         != ( suc @ zero_zero_nat ) ) ) ) ).

% Suc_0_mod_eq
thf(fact_9367_set__decode__zero,axiom,
    ( ( nat_set_decode @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% set_decode_zero
thf(fact_9368_set__decode__Suc,axiom,
    ! [N3: nat,X: nat] :
      ( ( member_nat @ ( suc @ N3 ) @ ( nat_set_decode @ X ) )
      = ( member_nat @ N3 @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_decode_Suc
thf(fact_9369_subset__decode__imp__le,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N3 ) )
     => ( ord_less_eq_nat @ M @ N3 ) ) ).

% subset_decode_imp_le
thf(fact_9370_set__decode__plus__power__2,axiom,
    ! [N3: nat,Z: nat] :
      ( ~ ( member_nat @ N3 @ ( nat_set_decode @ Z ) )
     => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ Z ) )
        = ( insert_nat @ N3 @ ( nat_set_decode @ Z ) ) ) ) ).

% set_decode_plus_power_2
thf(fact_9371_set__decode__def,axiom,
    ( nat_set_decode
    = ( ^ [X3: nat] :
          ( collect_nat
          @ ^ [N2: nat] :
              ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% set_decode_def
thf(fact_9372_bij__betw__nth__root__unity,axiom,
    ! [C2: complex,N3: nat] :
      ( ( C2 != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N3 @ ( real_V1022390504157884413omplex @ C2 ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C2 ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) )
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N3 )
                = one_one_complex ) )
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N3 )
                = C2 ) ) ) ) ) ).

% bij_betw_nth_root_unity
thf(fact_9373_and__int_Osimps,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L: 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 @ L @ ( 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 ) ) @ L ) ) ) )
          @ ( plus_plus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_int.simps
thf(fact_9374_and__int_Oelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
        = Y )
     => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
        & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.elims
thf(fact_9375_real__root__zero,axiom,
    ! [N3: nat] :
      ( ( root @ N3 @ zero_zero_real )
      = zero_zero_real ) ).

% real_root_zero
thf(fact_9376_real__root__Suc__0,axiom,
    ! [X: real] :
      ( ( root @ ( suc @ zero_zero_nat ) @ X )
      = X ) ).

% real_root_Suc_0
thf(fact_9377_real__root__eq__iff,axiom,
    ! [N3: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ( root @ N3 @ X )
          = ( root @ N3 @ Y ) )
        = ( X = Y ) ) ) ).

% real_root_eq_iff
thf(fact_9378_root__0,axiom,
    ! [X: real] :
      ( ( root @ zero_zero_nat @ X )
      = zero_zero_real ) ).

% root_0
thf(fact_9379_and__nonnegative__int__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        | ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).

% and_nonnegative_int_iff
thf(fact_9380_and__negative__int__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        & ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).

% and_negative_int_iff
thf(fact_9381_real__root__eq__0__iff,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ( root @ N3 @ X )
          = zero_zero_real )
        = ( X = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
thf(fact_9382_real__root__less__iff,axiom,
    ! [N3: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ ( root @ N3 @ X ) @ ( root @ N3 @ Y ) )
        = ( ord_less_real @ X @ Y ) ) ) ).

% real_root_less_iff
thf(fact_9383_real__root__le__iff,axiom,
    ! [N3: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_real @ ( root @ N3 @ X ) @ ( root @ N3 @ Y ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% real_root_le_iff
thf(fact_9384_real__root__one,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( root @ N3 @ one_one_real )
        = one_one_real ) ) ).

% real_root_one
thf(fact_9385_real__root__eq__1__iff,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ( root @ N3 @ X )
          = one_one_real )
        = ( X = one_one_real ) ) ) ).

% real_root_eq_1_iff
thf(fact_9386_real__root__gt__0__iff,axiom,
    ! [N3: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ ( root @ N3 @ Y ) )
        = ( ord_less_real @ zero_zero_real @ Y ) ) ) ).

% real_root_gt_0_iff
thf(fact_9387_real__root__lt__0__iff,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ ( root @ N3 @ X ) @ zero_zero_real )
        = ( ord_less_real @ X @ zero_zero_real ) ) ) ).

% real_root_lt_0_iff
thf(fact_9388_real__root__ge__0__iff,axiom,
    ! [N3: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N3 @ Y ) )
        = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).

% real_root_ge_0_iff
thf(fact_9389_real__root__le__0__iff,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_real @ ( root @ N3 @ X ) @ zero_zero_real )
        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).

% real_root_le_0_iff
thf(fact_9390_real__root__gt__1__iff,axiom,
    ! [N3: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ one_one_real @ ( root @ N3 @ Y ) )
        = ( ord_less_real @ one_one_real @ Y ) ) ) ).

% real_root_gt_1_iff
thf(fact_9391_real__root__lt__1__iff,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ ( root @ N3 @ X ) @ one_one_real )
        = ( ord_less_real @ X @ one_one_real ) ) ) ).

% real_root_lt_1_iff
thf(fact_9392_real__root__ge__1__iff,axiom,
    ! [N3: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_real @ one_one_real @ ( root @ N3 @ Y ) )
        = ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).

% real_root_ge_1_iff
thf(fact_9393_real__root__le__1__iff,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_real @ ( root @ N3 @ X ) @ one_one_real )
        = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).

% real_root_le_1_iff
thf(fact_9394_and__minus__numerals_I2_J,axiom,
    ! [N3: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) )
      = one_one_int ) ).

% and_minus_numerals(2)
thf(fact_9395_and__minus__numerals_I6_J,axiom,
    ! [N3: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) @ one_one_int )
      = one_one_int ) ).

% and_minus_numerals(6)
thf(fact_9396_real__root__pow__pos2,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N3 @ X ) @ N3 )
          = X ) ) ) ).

% real_root_pow_pos2
thf(fact_9397_and__minus__numerals_I1_J,axiom,
    ! [N3: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) )
      = zero_zero_int ) ).

% and_minus_numerals(1)
thf(fact_9398_and__minus__numerals_I5_J,axiom,
    ! [N3: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_minus_numerals(5)
thf(fact_9399_real__root__inverse,axiom,
    ! [N3: nat,X: real] :
      ( ( root @ N3 @ ( inverse_inverse_real @ X ) )
      = ( inverse_inverse_real @ ( root @ N3 @ X ) ) ) ).

% real_root_inverse
thf(fact_9400_real__root__divide,axiom,
    ! [N3: nat,X: real,Y: real] :
      ( ( root @ N3 @ ( divide_divide_real @ X @ Y ) )
      = ( divide_divide_real @ ( root @ N3 @ X ) @ ( root @ N3 @ Y ) ) ) ).

% real_root_divide
thf(fact_9401_real__root__mult__exp,axiom,
    ! [M: nat,N3: nat,X: real] :
      ( ( root @ ( times_times_nat @ M @ N3 ) @ X )
      = ( root @ M @ ( root @ N3 @ X ) ) ) ).

% real_root_mult_exp
thf(fact_9402_real__root__mult,axiom,
    ! [N3: nat,X: real,Y: real] :
      ( ( root @ N3 @ ( times_times_real @ X @ Y ) )
      = ( times_times_real @ ( root @ N3 @ X ) @ ( root @ N3 @ Y ) ) ) ).

% real_root_mult
thf(fact_9403_real__root__commute,axiom,
    ! [M: nat,N3: nat,X: real] :
      ( ( root @ M @ ( root @ N3 @ X ) )
      = ( root @ N3 @ ( root @ M @ X ) ) ) ).

% real_root_commute
thf(fact_9404_real__root__minus,axiom,
    ! [N3: nat,X: real] :
      ( ( root @ N3 @ ( uminus_uminus_real @ X ) )
      = ( uminus_uminus_real @ ( root @ N3 @ X ) ) ) ).

% real_root_minus
thf(fact_9405_real__root__pos__pos__le,axiom,
    ! [X: real,N3: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( root @ N3 @ X ) ) ) ).

% real_root_pos_pos_le
thf(fact_9406_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_9407_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_9408_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_9409_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).

% AND_upper1'
thf(fact_9410_AND__upper2_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).

% AND_upper2'
thf(fact_9411_real__root__less__mono,axiom,
    ! [N3: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ X @ Y )
       => ( ord_less_real @ ( root @ N3 @ X ) @ ( root @ N3 @ Y ) ) ) ) ).

% real_root_less_mono
thf(fact_9412_real__root__le__mono,axiom,
    ! [N3: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ord_less_eq_real @ ( root @ N3 @ X ) @ ( root @ N3 @ Y ) ) ) ) ).

% real_root_le_mono
thf(fact_9413_real__root__power,axiom,
    ! [N3: nat,X: real,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( root @ N3 @ ( power_power_real @ X @ K ) )
        = ( power_power_real @ ( root @ N3 @ X ) @ K ) ) ) ).

% real_root_power
thf(fact_9414_real__root__abs,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( root @ N3 @ ( abs_abs_real @ X ) )
        = ( abs_abs_real @ ( root @ N3 @ X ) ) ) ) ).

% real_root_abs
thf(fact_9415_and__less__eq,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ L2 @ zero_zero_int )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ K ) ) ).

% and_less_eq
thf(fact_9416_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).

% AND_upper1''
thf(fact_9417_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).

% AND_upper2''
thf(fact_9418_real__root__gt__zero,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_real @ zero_zero_real @ ( root @ N3 @ X ) ) ) ) ).

% real_root_gt_zero
thf(fact_9419_real__root__strict__decreasing,axiom,
    ! [N3: nat,N7: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_nat @ N3 @ N7 )
       => ( ( ord_less_real @ one_one_real @ X )
         => ( ord_less_real @ ( root @ N7 @ X ) @ ( root @ N3 @ X ) ) ) ) ) ).

% real_root_strict_decreasing
thf(fact_9420_sqrt__def,axiom,
    ( sqrt
    = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% sqrt_def
thf(fact_9421_root__abs__power,axiom,
    ! [N3: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( abs_abs_real @ ( root @ N3 @ ( power_power_real @ Y @ N3 ) ) )
        = ( abs_abs_real @ Y ) ) ) ).

% root_abs_power
thf(fact_9422_even__and__iff__int,axiom,
    ! [K: int,L2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) ).

% even_and_iff_int
thf(fact_9423_real__root__pos__pos,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ zero_zero_real @ ( root @ N3 @ X ) ) ) ) ).

% real_root_pos_pos
thf(fact_9424_odd__real__root__pow,axiom,
    ! [N3: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( power_power_real @ ( root @ N3 @ X ) @ N3 )
        = X ) ) ).

% odd_real_root_pow
thf(fact_9425_odd__real__root__unique,axiom,
    ! [N3: nat,Y: real,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( ( power_power_real @ Y @ N3 )
          = X )
       => ( ( root @ N3 @ X )
          = Y ) ) ) ).

% odd_real_root_unique
thf(fact_9426_odd__real__root__power__cancel,axiom,
    ! [N3: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( root @ N3 @ ( power_power_real @ X @ N3 ) )
        = X ) ) ).

% odd_real_root_power_cancel
thf(fact_9427_real__root__strict__increasing,axiom,
    ! [N3: nat,N7: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_nat @ N3 @ N7 )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ X @ one_one_real )
           => ( ord_less_real @ ( root @ N3 @ X ) @ ( root @ N7 @ X ) ) ) ) ) ) ).

% real_root_strict_increasing
thf(fact_9428_real__root__decreasing,axiom,
    ! [N3: nat,N7: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_nat @ N3 @ N7 )
       => ( ( ord_less_eq_real @ one_one_real @ X )
         => ( ord_less_eq_real @ ( root @ N7 @ X ) @ ( root @ N3 @ X ) ) ) ) ) ).

% real_root_decreasing
thf(fact_9429_real__root__pow__pos,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N3 @ X ) @ N3 )
          = X ) ) ) ).

% real_root_pow_pos
thf(fact_9430_real__root__power__cancel,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( root @ N3 @ ( power_power_real @ X @ N3 ) )
          = X ) ) ) ).

% real_root_power_cancel
thf(fact_9431_real__root__pos__unique,axiom,
    ! [N3: nat,Y: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( power_power_real @ Y @ N3 )
            = X )
         => ( ( root @ N3 @ X )
            = Y ) ) ) ) ).

% real_root_pos_unique
thf(fact_9432_real__root__increasing,axiom,
    ! [N3: nat,N7: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_eq_nat @ N3 @ N7 )
       => ( ( ord_less_eq_real @ zero_zero_real @ X )
         => ( ( ord_less_eq_real @ X @ one_one_real )
           => ( ord_less_eq_real @ ( root @ N3 @ X ) @ ( root @ N7 @ X ) ) ) ) ) ) ).

% real_root_increasing
thf(fact_9433_log__root,axiom,
    ! [N3: nat,A2: real,B2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ A2 )
       => ( ( log @ B2 @ ( root @ N3 @ A2 ) )
          = ( divide_divide_real @ ( log @ B2 @ A2 ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ).

% log_root
thf(fact_9434_log__base__root,axiom,
    ! [N3: nat,B2: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ( log @ ( root @ N3 @ B2 ) @ X )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( log @ B2 @ X ) ) ) ) ) ).

% log_base_root
thf(fact_9435_ln__root,axiom,
    ! [N3: nat,B2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ B2 )
       => ( ( ln_ln_real @ ( root @ N3 @ B2 ) )
          = ( divide_divide_real @ ( ln_ln_real @ B2 ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ).

% ln_root
thf(fact_9436_and__int__rec,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
              & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_int_rec
thf(fact_9437_root__powr__inverse,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( root @ N3 @ X )
          = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) ) ) ).

% root_powr_inverse
thf(fact_9438_and__int__unfold,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L: int] :
          ( if_int
          @ ( ( K3 = zero_zero_int )
            | ( L = zero_zero_int ) )
          @ zero_zero_int
          @ ( if_int
            @ ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            @ L
            @ ( if_int
              @ ( L
                = ( 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 @ L @ ( 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 @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% and_int_unfold
thf(fact_9439_and__int_Opelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) )
       => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                  & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( uminus_uminus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
              & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( plus_plus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
                    @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).

% and_int.pelims
thf(fact_9440_and__int_Opsimps,axiom,
    ! [K: int,L2: int] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L2 ) )
     => ( ( ( ( member_int @ K @ ( 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 ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L2 )
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) ) ) )
        & ( ~ ( ( member_int @ K @ ( 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 ) ) ) )
         => ( ( bit_se725231765392027082nd_int @ K @ L2 )
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.psimps
thf(fact_9441_forall__finite_I3_J,axiom,
    ! [X: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ ( suc @ X ) ) )
           => ( P @ I2 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ X ) )
           => ( P @ ( suc @ I2 ) ) ) ) ) ).

% forall_finite(3)
thf(fact_9442_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_9443_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_9444_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_9445_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_9446_and__Suc__0__eq,axiom,
    ! [N3: nat] :
      ( ( bit_se727722235901077358nd_nat @ N3 @ ( suc @ zero_zero_nat ) )
      = ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% and_Suc_0_eq
thf(fact_9447_Suc__0__and__eq,axiom,
    ! [N3: nat] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N3 )
      = ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Suc_0_and_eq
thf(fact_9448_and__nat__def,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M5: nat,N2: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% and_nat_def
thf(fact_9449_forall__finite_I1_J,axiom,
    ! [P: nat > $o,I6: nat] :
      ( ( ord_less_nat @ I6 @ zero_zero_nat )
     => ( P @ I6 ) ) ).

% forall_finite(1)
thf(fact_9450_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M5: nat,N2: nat] :
          ( if_nat
          @ ( ( M5 = zero_zero_nat )
            | ( N2 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_9451_and__nat__rec,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M5: nat,N2: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 )
              & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_9452_and__int_Opinduct,axiom,
    ! [A0: int,A12: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
     => ( ! [K2: int,L4: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L4 ) )
           => ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K2 @ L4 ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% and_int.pinduct
thf(fact_9453_Comparator__Generator_OAll__less__Suc,axiom,
    ! [X: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ X ) )
           => ( P @ I2 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ X )
           => ( P @ ( suc @ I2 ) ) ) ) ) ).

% Comparator_Generator.All_less_Suc
thf(fact_9454_forall__finite_I2_J,axiom,
    ! [P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ zero_zero_nat ) )
           => ( P @ I2 ) ) )
      = ( P @ zero_zero_nat ) ) ).

% forall_finite(2)
thf(fact_9455_upto_Opinduct,axiom,
    ! [A0: int,A12: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
     => ( ! [I5: int,J3: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I5 @ J3 ) )
           => ( ( ( ord_less_eq_int @ I5 @ J3 )
               => ( P @ ( plus_plus_int @ I5 @ one_one_int ) @ J3 ) )
             => ( P @ I5 @ J3 ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% upto.pinduct
thf(fact_9456_uint32_Osize__eq,axiom,
    ( size_size_uint32
    = ( ^ [P4: uint32] : ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% uint32.size_eq
thf(fact_9457_pred__numeral__inc,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( inc @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% pred_numeral_inc
thf(fact_9458_take__bit__of__Suc__0,axiom,
    ! [N3: nat] :
      ( ( bit_se2925701944663578781it_nat @ N3 @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% take_bit_of_Suc_0
thf(fact_9459_take__bit__mult,axiom,
    ! [N3: nat,K: int,L2: int] :
      ( ( bit_se2923211474154528505it_int @ N3 @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N3 @ K ) @ ( bit_se2923211474154528505it_int @ N3 @ L2 ) ) )
      = ( bit_se2923211474154528505it_int @ N3 @ ( times_times_int @ K @ L2 ) ) ) ).

% take_bit_mult
thf(fact_9460_take__bit__diff,axiom,
    ! [N3: nat,K: int,L2: int] :
      ( ( bit_se2923211474154528505it_int @ N3 @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N3 @ K ) @ ( bit_se2923211474154528505it_int @ N3 @ L2 ) ) )
      = ( bit_se2923211474154528505it_int @ N3 @ ( minus_minus_int @ K @ L2 ) ) ) ).

% take_bit_diff
thf(fact_9461_take__bit__nat__eq,axiom,
    ! [K: int,N3: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( bit_se2925701944663578781it_nat @ N3 @ ( nat2 @ K ) )
        = ( nat2 @ ( bit_se2923211474154528505it_int @ N3 @ K ) ) ) ) ).

% take_bit_nat_eq
thf(fact_9462_nat__take__bit__eq,axiom,
    ! [K: int,N3: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N3 @ K ) )
        = ( bit_se2925701944663578781it_nat @ N3 @ ( nat2 @ K ) ) ) ) ).

% nat_take_bit_eq
thf(fact_9463_take__bit__nat__less__eq__self,axiom,
    ! [N3: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N3 @ M ) @ M ) ).

% take_bit_nat_less_eq_self
thf(fact_9464_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N3: nat,Q3: nat] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q3 ) @ ( bit_se2925701944663578781it_nat @ N3 @ Q3 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_9465_take__bit__minus,axiom,
    ! [N3: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ N3 @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N3 @ K ) ) )
      = ( bit_se2923211474154528505it_int @ N3 @ ( uminus_uminus_int @ K ) ) ) ).

% take_bit_minus
thf(fact_9466_add__inc,axiom,
    ! [X: num,Y: num] :
      ( ( plus_plus_num @ X @ ( inc @ Y ) )
      = ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).

% add_inc
thf(fact_9467_num__induct,axiom,
    ! [P: num > $o,X: num] :
      ( ( P @ one )
     => ( ! [X4: num] :
            ( ( P @ X4 )
           => ( P @ ( inc @ X4 ) ) )
       => ( P @ X ) ) ) ).

% num_induct
thf(fact_9468_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N3: nat,K: int] :
      ( ( ord_less_eq_nat @ M @ N3 )
     => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N3 @ K ) ) ) ).

% take_bit_tightened_less_eq_int
thf(fact_9469_take__bit__int__less__eq__self__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N3 @ K ) @ K )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% take_bit_int_less_eq_self_iff
thf(fact_9470_take__bit__nonnegative,axiom,
    ! [N3: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N3 @ K ) ) ).

% take_bit_nonnegative
thf(fact_9471_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N3: nat] :
      ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N3 @ K ) )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% take_bit_int_greater_self_iff
thf(fact_9472_not__take__bit__negative,axiom,
    ! [N3: nat,K: int] :
      ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N3 @ K ) @ zero_zero_int ) ).

% not_take_bit_negative
thf(fact_9473_inc_Osimps_I1_J,axiom,
    ( ( inc @ one )
    = ( bit0 @ one ) ) ).

% inc.simps(1)
thf(fact_9474_inc_Osimps_I2_J,axiom,
    ! [X: num] :
      ( ( inc @ ( bit0 @ X ) )
      = ( bit1 @ X ) ) ).

% inc.simps(2)
thf(fact_9475_inc_Osimps_I3_J,axiom,
    ! [X: num] :
      ( ( inc @ ( bit1 @ X ) )
      = ( bit0 @ ( inc @ X ) ) ) ).

% inc.simps(3)
thf(fact_9476_add__One,axiom,
    ! [X: num] :
      ( ( plus_plus_num @ X @ one )
      = ( inc @ X ) ) ).

% add_One
thf(fact_9477_inc__BitM__eq,axiom,
    ! [N3: num] :
      ( ( inc @ ( bitM @ N3 ) )
      = ( bit0 @ N3 ) ) ).

% inc_BitM_eq
thf(fact_9478_BitM__inc__eq,axiom,
    ! [N3: num] :
      ( ( bitM @ ( inc @ N3 ) )
      = ( bit1 @ N3 ) ) ).

% BitM_inc_eq
thf(fact_9479_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_9480_take__bit__decr__eq,axiom,
    ! [N3: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N3 @ K )
       != zero_zero_int )
     => ( ( bit_se2923211474154528505it_int @ N3 @ ( minus_minus_int @ K @ one_one_int ) )
        = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N3 @ K ) @ one_one_int ) ) ) ).

% take_bit_decr_eq
thf(fact_9481_take__bit__nat__eq__self__iff,axiom,
    ! [N3: nat,M: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N3 @ M )
        = M )
      = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% take_bit_nat_eq_self_iff
thf(fact_9482_take__bit__nat__less__exp,axiom,
    ! [N3: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N3 @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% take_bit_nat_less_exp
thf(fact_9483_take__bit__nat__eq__self,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
     => ( ( bit_se2925701944663578781it_nat @ N3 @ M )
        = M ) ) ).

% take_bit_nat_eq_self
thf(fact_9484_take__bit__nat__def,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N2: nat,M5: nat] : ( modulo_modulo_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% take_bit_nat_def
thf(fact_9485_take__bit__Suc__minus__bit1,axiom,
    ! [N3: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N3 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_bit1
thf(fact_9486_take__bit__int__less__exp,axiom,
    ! [N3: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N3 @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ).

% take_bit_int_less_exp
thf(fact_9487_take__bit__int__def,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N2: nat,K3: int] : ( modulo_modulo_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% take_bit_int_def
thf(fact_9488_take__bit__numeral__minus__bit1,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_minus_bit1
thf(fact_9489_take__bit__nat__less__self__iff,axiom,
    ! [N3: nat,M: nat] :
      ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N3 @ M ) @ M )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ M ) ) ).

% take_bit_nat_less_self_iff
thf(fact_9490_take__bit__Suc__minus__bit0,axiom,
    ! [N3: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N3 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_minus_bit0
thf(fact_9491_take__bit__int__less__self__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N3 @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ K ) ) ).

% take_bit_int_less_self_iff
thf(fact_9492_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N3: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N3 @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% take_bit_int_greater_eq_self_iff
thf(fact_9493_take__bit__int__eq__self,axiom,
    ! [K: int,N3: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
       => ( ( bit_se2923211474154528505it_int @ N3 @ K )
          = K ) ) ) ).

% take_bit_int_eq_self
thf(fact_9494_take__bit__int__eq__self__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N3 @ K )
        = K )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% take_bit_int_eq_self_iff
thf(fact_9495_take__bit__incr__eq,axiom,
    ! [N3: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N3 @ K )
       != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) )
     => ( ( bit_se2923211474154528505it_int @ N3 @ ( plus_plus_int @ K @ one_one_int ) )
        = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N3 @ K ) ) ) ) ).

% take_bit_incr_eq
thf(fact_9496_take__bit__numeral__minus__bit0,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_minus_bit0
thf(fact_9497_take__bit__int__less__eq,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ K )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N3 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% take_bit_int_less_eq
thf(fact_9498_take__bit__int__greater__eq,axiom,
    ! [K: int,N3: nat] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) @ ( bit_se2923211474154528505it_int @ N3 @ K ) ) ) ).

% take_bit_int_greater_eq
thf(fact_9499_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ 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 @ L ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_9500_signed__take__bit__eq__take__bit__shift,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N2: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% signed_take_bit_eq_take_bit_shift
thf(fact_9501_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ 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 @ L ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_9502_take__bit__minus__small__eq,axiom,
    ! [K: int,N3: nat] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
       => ( ( bit_se2923211474154528505it_int @ N3 @ ( uminus_uminus_int @ K ) )
          = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ K ) ) ) ) ).

% take_bit_minus_small_eq
thf(fact_9503_divmod__step__integer__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_9504_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M5: nat,N2: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N2 = zero_zero_nat )
            | ( ord_less_nat @ M5 @ N2 ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M5 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M5 @ N2 ) @ N2 ) ) ) ) ) ).

% divmod_nat_if
thf(fact_9505_divmod__integer_H__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M5: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).

% divmod_integer'_def
thf(fact_9506_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
thf(fact_9507_divmod__nat__def,axiom,
    ( divmod_nat
    = ( ^ [M5: nat,N2: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M5 @ N2 ) @ ( modulo_modulo_nat @ M5 @ N2 ) ) ) ) ).

% divmod_nat_def
thf(fact_9508_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_9509_less__integer__code_I1_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).

% less_integer_code(1)
thf(fact_9510_minus__integer__code_I2_J,axiom,
    ! [L2: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L2 )
      = ( uminus1351360451143612070nteger @ L2 ) ) ).

% minus_integer_code(2)
thf(fact_9511_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% minus_integer_code(1)
thf(fact_9512_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_9513_int__ge__less__than2__def,axiom,
    ( int_ge_less_than2
    = ( ^ [D3: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z8: int,Z5: int] :
                ( ( ord_less_eq_int @ D3 @ Z5 )
                & ( ord_less_int @ Z8 @ Z5 ) ) ) ) ) ) ).

% int_ge_less_than2_def
thf(fact_9514_divide__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( ( divide6298287555418463151nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( divide_divide_int @ Xa @ X ) ) ) ).

% divide_integer.abs_eq
thf(fact_9515_less__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
      = ( ord_less_int @ Xa @ X ) ) ).

% less_integer.abs_eq
thf(fact_9516_minus__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( ( minus_8373710615458151222nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( minus_minus_int @ Xa @ X ) ) ) ).

% minus_integer.abs_eq
thf(fact_9517_int__ge__less__than__def,axiom,
    ( int_ge_less_than
    = ( ^ [D3: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z8: int,Z5: int] :
                ( ( ord_less_eq_int @ D3 @ Z8 )
                & ( ord_less_int @ Z8 @ Z5 ) ) ) ) ) ) ).

% int_ge_less_than_def
thf(fact_9518_arctan__def,axiom,
    ( arctan
    = ( ^ [Y2: real] :
          ( the_real
          @ ^ [X3: real] :
              ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
              & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( tan_real @ X3 )
                = Y2 ) ) ) ) ) ).

% arctan_def
thf(fact_9519_modulo__int__def,axiom,
    ( modulo_modulo_int
    = ( ^ [K3: int,L: int] :
          ( if_int @ ( L = zero_zero_int ) @ K3
          @ ( if_int
            @ ( ( sgn_sgn_int @ K3 )
              = ( sgn_sgn_int @ L ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L )
              @ ( minus_minus_int
                @ ( times_times_int @ ( abs_abs_int @ L )
                  @ ( zero_n2684676970156552555ol_int
                    @ ~ ( dvd_dvd_int @ L @ K3 ) ) )
                @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ) ) ) ) ) ).

% modulo_int_def
thf(fact_9520_arcsin__def,axiom,
    ( arcsin
    = ( ^ [Y2: real] :
          ( the_real
          @ ^ [X3: real] :
              ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
              & ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( sin_real @ X3 )
                = Y2 ) ) ) ) ) ).

% arcsin_def
thf(fact_9521_div__eq__sgn__abs,axiom,
    ! [K: int,L2: int] :
      ( ( ( sgn_sgn_int @ K )
        = ( sgn_sgn_int @ L2 ) )
     => ( ( divide_divide_int @ K @ L2 )
        = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ).

% div_eq_sgn_abs
thf(fact_9522_sgn__mod,axiom,
    ! [L2: int,K: int] :
      ( ( L2 != zero_zero_int )
     => ( ~ ( dvd_dvd_int @ L2 @ K )
       => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L2 ) )
          = ( sgn_sgn_int @ L2 ) ) ) ) ).

% sgn_mod
thf(fact_9523_ln__neg__is__const,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ln_ln_real @ X )
        = ( the_real
          @ ^ [X3: real] : $false ) ) ) ).

% ln_neg_is_const
thf(fact_9524_zsgn__def,axiom,
    ( sgn_sgn_int
    = ( ^ [I2: int] : ( if_int @ ( I2 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I2 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zsgn_def
thf(fact_9525_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L2: 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 @ L2 ) ) )
        = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ).

% div_sgn_abs_cancel
thf(fact_9526_div__dvd__sgn__abs,axiom,
    ! [L2: int,K: int] :
      ( ( dvd_dvd_int @ L2 @ K )
     => ( ( divide_divide_int @ K @ L2 )
        = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L2 ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ) ).

% div_dvd_sgn_abs
thf(fact_9527_arccos__def,axiom,
    ( arccos
    = ( ^ [Y2: real] :
          ( the_real
          @ ^ [X3: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ X3 )
              & ( ord_less_eq_real @ X3 @ pi )
              & ( ( cos_real @ X3 )
                = Y2 ) ) ) ) ) ).

% arccos_def
thf(fact_9528_eucl__rel__int__remainderI,axiom,
    ! [R3: int,L2: int,K: int,Q3: int] :
      ( ( ( sgn_sgn_int @ R3 )
        = ( sgn_sgn_int @ L2 ) )
     => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ L2 ) )
       => ( ( K
            = ( plus_plus_int @ ( times_times_int @ Q3 @ L2 ) @ R3 ) )
         => ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q3 @ R3 ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_9529_div__noneq__sgn__abs,axiom,
    ! [L2: int,K: int] :
      ( ( L2 != zero_zero_int )
     => ( ( ( sgn_sgn_int @ K )
         != ( sgn_sgn_int @ L2 ) )
       => ( ( divide_divide_int @ K @ L2 )
          = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) )
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( dvd_dvd_int @ L2 @ K ) ) ) ) ) ) ).

% div_noneq_sgn_abs
thf(fact_9530_eucl__rel__int_Ocases,axiom,
    ! [A12: int,A23: int,A32: product_prod_int_int] :
      ( ( eucl_rel_int @ A12 @ A23 @ A32 )
     => ( ( ( A23 = zero_zero_int )
         => ( A32
           != ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
       => ( ! [Q5: int] :
              ( ( A32
                = ( product_Pair_int_int @ Q5 @ zero_zero_int ) )
             => ( ( A23 != zero_zero_int )
               => ( A12
                 != ( times_times_int @ Q5 @ A23 ) ) ) )
         => ~ ! [R2: int,Q5: int] :
                ( ( A32
                  = ( product_Pair_int_int @ Q5 @ R2 ) )
               => ( ( ( sgn_sgn_int @ R2 )
                    = ( sgn_sgn_int @ A23 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ A23 ) )
                   => ( A12
                     != ( plus_plus_int @ ( times_times_int @ Q5 @ A23 ) @ R2 ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_9531_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A1: int,A22: int,A33: product_prod_int_int] :
          ( ? [K3: int] :
              ( ( A1 = K3 )
              & ( A22 = zero_zero_int )
              & ( A33
                = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
          | ? [L: int,K3: int,Q4: int] :
              ( ( A1 = K3 )
              & ( A22 = L )
              & ( A33
                = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
              & ( L != zero_zero_int )
              & ( K3
                = ( times_times_int @ Q4 @ L ) ) )
          | ? [R5: int,L: int,K3: int,Q4: int] :
              ( ( A1 = K3 )
              & ( A22 = L )
              & ( A33
                = ( product_Pair_int_int @ Q4 @ R5 ) )
              & ( ( sgn_sgn_int @ R5 )
                = ( sgn_sgn_int @ L ) )
              & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L ) )
              & ( K3
                = ( plus_plus_int @ ( times_times_int @ Q4 @ L ) @ R5 ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_9532_pi__half,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
    = ( the_real
      @ ^ [X3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X3 )
          & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
          & ( ( cos_real @ X3 )
            = zero_zero_real ) ) ) ) ).

% pi_half
thf(fact_9533_pi__def,axiom,
    ( pi
    = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
      @ ( the_real
        @ ^ [X3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X3 )
            & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ X3 )
              = zero_zero_real ) ) ) ) ) ).

% pi_def
thf(fact_9534_divide__int__unfold,axiom,
    ! [L2: int,K: int,N3: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L2 )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N3 = zero_zero_nat ) )
       => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N3 ) ) )
          = zero_zero_int ) )
      & ( ~ ( ( ( sgn_sgn_int @ L2 )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N3 = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L2 ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N3 ) ) )
              = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N3 ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L2 ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N3 ) ) )
              = ( uminus_uminus_int
                @ ( semiri1314217659103216013at_int
                  @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N3 )
                    @ ( zero_n2687167440665602831ol_nat
                      @ ~ ( dvd_dvd_nat @ N3 @ M ) ) ) ) ) ) ) ) ) ) ).

% divide_int_unfold
thf(fact_9535_modulo__int__unfold,axiom,
    ! [L2: int,K: int,N3: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L2 )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N3 = zero_zero_nat ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N3 ) ) )
          = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
      & ( ~ ( ( ( sgn_sgn_int @ L2 )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N3 = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L2 ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N3 ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N3 ) ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L2 ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N3 ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L2 )
                @ ( minus_minus_int
                  @ ( semiri1314217659103216013at_int
                    @ ( times_times_nat @ N3
                      @ ( zero_n2687167440665602831ol_nat
                        @ ~ ( dvd_dvd_nat @ N3 @ M ) ) ) )
                  @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N3 ) ) ) ) ) ) ) ) ) ).

% modulo_int_unfold
thf(fact_9536_divide__int__def,axiom,
    ( divide_divide_int
    = ( ^ [K3: int,L: int] :
          ( if_int @ ( L = zero_zero_int ) @ zero_zero_int
          @ ( if_int
            @ ( ( sgn_sgn_int @ K3 )
              = ( sgn_sgn_int @ L ) )
            @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) )
            @ ( uminus_uminus_int
              @ ( semiri1314217659103216013at_int
                @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) )
                  @ ( zero_n2687167440665602831ol_nat
                    @ ~ ( dvd_dvd_int @ L @ K3 ) ) ) ) ) ) ) ) ) ).

% divide_int_def
thf(fact_9537_sgn__div__eq__sgn__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( ( divide_divide_int @ A2 @ B2 )
       != zero_zero_int )
     => ( ( sgn_sgn_int @ ( divide_divide_int @ A2 @ B2 ) )
        = ( sgn_sgn_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).

% sgn_div_eq_sgn_mult
thf(fact_9538_signed__take__bit__eq__take__bit__minus,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N2: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N2 ) ) ) ) ) ) ).

% signed_take_bit_eq_take_bit_minus
thf(fact_9539_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_9540_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_9541_signed__take__bit__nonnegative__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N3 @ K ) )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) ).

% signed_take_bit_nonnegative_iff
thf(fact_9542_signed__take__bit__negative__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N3 @ K ) @ zero_zero_int )
      = ( bit_se1146084159140164899it_int @ K @ N3 ) ) ).

% signed_take_bit_negative_iff
thf(fact_9543_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,N3: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N3 ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N3 ) ) ).

% bit_minus_numeral_Bit0_Suc_iff
thf(fact_9544_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,N3: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N3 ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N3 ) ) ) ).

% bit_minus_numeral_Bit1_Suc_iff
thf(fact_9545_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,N3: num] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N3 ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N3 ) ) ) ).

% bit_minus_numeral_int(1)
thf(fact_9546_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,N3: num] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N3 ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N3 ) ) ) ) ).

% bit_minus_numeral_int(2)
thf(fact_9547_bin__nth__minus__Bit0,axiom,
    ! [N3: nat,W: num] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) @ N3 )
        = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ).

% bin_nth_minus_Bit0
thf(fact_9548_bin__nth__minus__Bit1,axiom,
    ! [N3: nat,W: num] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) @ N3 )
        = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ).

% bin_nth_minus_Bit1
thf(fact_9549_real__sgn__eq,axiom,
    ( sgn_sgn_real
    = ( ^ [X3: real] : ( divide_divide_real @ X3 @ ( abs_abs_real @ X3 ) ) ) ) ).

% real_sgn_eq
thf(fact_9550_bit__and__int__iff,axiom,
    ! [K: int,L2: int,N3: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ N3 )
      = ( ( bit_se1146084159140164899it_int @ K @ N3 )
        & ( bit_se1146084159140164899it_int @ L2 @ N3 ) ) ) ).

% bit_and_int_iff
thf(fact_9551_sgn__root,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( sgn_sgn_real @ ( root @ N3 @ X ) )
        = ( sgn_sgn_real @ X ) ) ) ).

% sgn_root
thf(fact_9552_bit__not__int__iff_H,axiom,
    ! [K: int,N3: nat] :
      ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N3 )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) ).

% bit_not_int_iff'
thf(fact_9553_sgn__eq,axiom,
    ( sgn_sgn_complex
    = ( ^ [Z5: complex] : ( divide1717551699836669952omplex @ Z5 @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ Z5 ) ) ) ) ) ).

% sgn_eq
thf(fact_9554_sgn__real__def,axiom,
    ( sgn_sgn_real
    = ( ^ [A7: real] : ( if_real @ ( A7 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A7 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_real_def
thf(fact_9555_bit__imp__take__bit__positive,axiom,
    ! [N3: nat,M: nat,K: int] :
      ( ( ord_less_nat @ N3 @ M )
     => ( ( bit_se1146084159140164899it_int @ K @ N3 )
       => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).

% bit_imp_take_bit_positive
thf(fact_9556_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_9557_sgn__power__injE,axiom,
    ! [A2: real,N3: nat,X: real,B2: real] :
      ( ( ( times_times_real @ ( sgn_sgn_real @ A2 ) @ ( power_power_real @ ( abs_abs_real @ A2 ) @ N3 ) )
        = X )
     => ( ( X
          = ( times_times_real @ ( sgn_sgn_real @ B2 ) @ ( power_power_real @ ( abs_abs_real @ B2 ) @ N3 ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ N3 )
         => ( A2 = B2 ) ) ) ) ).

% sgn_power_injE
thf(fact_9558_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ N @ M2 )
             => ( ( bit_se1146084159140164899it_int @ K @ M2 )
                = ( bit_se1146084159140164899it_int @ K @ N ) ) )
         => ~ ( ( ord_less_nat @ zero_zero_nat @ N )
             => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ one_one_nat ) )
                = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ) ) ).

% int_bit_bound
thf(fact_9559_sgn__power__root,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N3 @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N3 @ X ) ) @ N3 ) )
        = X ) ) ).

% sgn_power_root
thf(fact_9560_root__sgn__power,axiom,
    ! [N3: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( root @ N3 @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N3 ) ) )
        = Y ) ) ).

% root_sgn_power
thf(fact_9561_bit__int__def,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [K3: int,N2: nat] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% bit_int_def
thf(fact_9562_cis__Arg__unique,axiom,
    ! [Z: complex,X: real] :
      ( ( ( sgn_sgn_complex @ Z )
        = ( cis @ X ) )
     => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
       => ( ( ord_less_eq_real @ X @ pi )
         => ( ( arg @ Z )
            = X ) ) ) ) ).

% cis_Arg_unique
thf(fact_9563_split__root,axiom,
    ! [P: real > $o,N3: nat,X: real] :
      ( ( P @ ( root @ N3 @ X ) )
      = ( ( ( N3 = zero_zero_nat )
         => ( P @ zero_zero_real ) )
        & ( ( ord_less_nat @ zero_zero_nat @ N3 )
         => ! [Y2: real] :
              ( ( ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N3 ) )
                = X )
             => ( P @ Y2 ) ) ) ) ) ).

% split_root
thf(fact_9564_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( ( sgn_sgn_complex @ Z )
          = ( cis @ ( arg @ Z ) ) )
        & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
        & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).

% Arg_correct
thf(fact_9565_Bit__Operations_Oset__bit__eq,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N2: nat,K3: int] :
          ( plus_plus_int @ K3
          @ ( times_times_int
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( bit_se1146084159140164899it_int @ K3 @ N2 ) )
            @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% Bit_Operations.set_bit_eq
thf(fact_9566_unset__bit__eq,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N2: nat,K3: int] : ( minus_minus_int @ K3 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% unset_bit_eq
thf(fact_9567_take__bit__Suc__from__most,axiom,
    ! [N3: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ K )
      = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) @ ( bit_se2923211474154528505it_int @ N3 @ K ) ) ) ).

% take_bit_Suc_from_most
thf(fact_9568_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_9569_num_Osize__gen_I3_J,axiom,
    ! [X33: num] :
      ( ( size_num @ ( bit1 @ X33 ) )
      = ( plus_plus_nat @ ( size_num @ X33 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(3)
thf(fact_9570_mask__nat__positive__iff,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N3 ) )
      = ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).

% mask_nat_positive_iff
thf(fact_9571_not__bit__Suc__0__Suc,axiom,
    ! [N3: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N3 ) ) ).

% not_bit_Suc_0_Suc
thf(fact_9572_bit__Suc__0__iff,axiom,
    ! [N3: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N3 )
      = ( N3 = zero_zero_nat ) ) ).

% bit_Suc_0_iff
thf(fact_9573_nat__mask__eq,axiom,
    ! [N3: nat] :
      ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N3 ) )
      = ( bit_se2002935070580805687sk_nat @ N3 ) ) ).

% nat_mask_eq
thf(fact_9574_less__eq__mask,axiom,
    ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( bit_se2002935070580805687sk_nat @ N3 ) ) ).

% less_eq_mask
thf(fact_9575_mask__nonnegative__int,axiom,
    ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N3 ) ) ).

% mask_nonnegative_int
thf(fact_9576_not__mask__negative__int,axiom,
    ! [N3: nat] :
      ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N3 ) @ zero_zero_int ) ).

% not_mask_negative_int
thf(fact_9577_not__bit__Suc__0__numeral,axiom,
    ! [N3: num] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N3 ) ) ).

% not_bit_Suc_0_numeral
thf(fact_9578_less__mask,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N3 )
     => ( ord_less_nat @ N3 @ ( bit_se2002935070580805687sk_nat @ N3 ) ) ) ).

% less_mask
thf(fact_9579_bit__nat__iff,axiom,
    ! [K: int,N3: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N3 )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) ).

% bit_nat_iff
thf(fact_9580_take__bit__eq__mask__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N3 @ K )
        = ( bit_se2000444600071755411sk_int @ N3 ) )
      = ( ( bit_se2923211474154528505it_int @ N3 @ ( plus_plus_int @ K @ one_one_int ) )
        = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
thf(fact_9581_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_9582_Suc__mask__eq__exp,axiom,
    ! [N3: nat] :
      ( ( suc @ ( bit_se2002935070580805687sk_nat @ N3 ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% Suc_mask_eq_exp
thf(fact_9583_mask__nat__less__exp,axiom,
    ! [N3: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% mask_nat_less_exp
thf(fact_9584_bit__nat__def,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [M5: nat,N2: nat] :
          ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% bit_nat_def
thf(fact_9585_mask__half__int,axiom,
    ! [N3: nat] :
      ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N3 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ).

% mask_half_int
thf(fact_9586_mask__int__def,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N2: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ).

% mask_int_def
thf(fact_9587_mask__nat__def,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N2: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ).

% mask_nat_def
thf(fact_9588_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N3: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N3 @ K )
        = ( bit_se2000444600071755411sk_int @ N3 ) )
      = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).

% take_bit_eq_mask_iff_exp_dvd
thf(fact_9589_num_Osize__gen_I2_J,axiom,
    ! [X22: num] :
      ( ( size_num @ ( bit0 @ X22 ) )
      = ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(2)
thf(fact_9590_root__def,axiom,
    ( root
    = ( ^ [N2: nat,X3: real] :
          ( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
          @ ( the_in5290026491893676941l_real @ top_top_set_real
            @ ^ [Y2: real] : ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N2 ) )
            @ X3 ) ) ) ) ).

% root_def
thf(fact_9591_Arg__def,axiom,
    ( arg
    = ( ^ [Z5: complex] :
          ( if_real @ ( Z5 = zero_zero_complex ) @ zero_zero_real
          @ ( fChoice_real
            @ ^ [A7: real] :
                ( ( ( sgn_sgn_complex @ Z5 )
                  = ( cis @ A7 ) )
                & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A7 )
                & ( ord_less_eq_real @ A7 @ pi ) ) ) ) ) ) ).

% Arg_def
thf(fact_9592_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_9593_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_9594_bit__not__int__iff,axiom,
    ! [K: int,N3: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N3 )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) ).

% bit_not_int_iff
thf(fact_9595_not__int__def,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K3: int] : ( minus_minus_int @ ( uminus_uminus_int @ K3 ) @ one_one_int ) ) ) ).

% not_int_def
thf(fact_9596_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_9597_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_9598_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_9599_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_9600_and__not__numerals_I2_J,axiom,
    ! [N3: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) )
      = one_one_int ) ).

% and_not_numerals(2)
thf(fact_9601_bit__minus__int__iff,axiom,
    ! [K: int,N3: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N3 )
      = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N3 ) ) ).

% bit_minus_int_iff
thf(fact_9602_and__not__numerals_I5_J,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).

% and_not_numerals(5)
thf(fact_9603_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_9604_and__not__numerals_I3_J,axiom,
    ! [N3: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) )
      = zero_zero_int ) ).

% and_not_numerals(3)
thf(fact_9605_and__not__numerals_I9_J,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).

% and_not_numerals(9)
thf(fact_9606_and__not__numerals_I6_J,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).

% and_not_numerals(6)
thf(fact_9607_and__not__numerals_I8_J,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) )
      = ( 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 @ N3 ) ) ) ) ) ) ).

% and_not_numerals(8)
thf(fact_9608_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_9609_bitNOT__integer__code,axiom,
    ( bit_ri7632146776885996613nteger
    = ( ^ [I2: code_integer] : ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ I2 ) @ one_one_Code_integer ) ) ) ).

% bitNOT_integer_code
thf(fact_9610_xor__int__unfold,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L: int] :
          ( if_int
          @ ( K3
            = ( uminus_uminus_int @ one_one_int ) )
          @ ( bit_ri7919022796975470100ot_int @ L )
          @ ( if_int
            @ ( L
              = ( uminus_uminus_int @ one_one_int ) )
            @ ( bit_ri7919022796975470100ot_int @ K3 )
            @ ( if_int @ ( K3 = zero_zero_int ) @ L @ ( if_int @ ( L = 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 @ L @ ( 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 @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
thf(fact_9611_or__nonnegative__int__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).

% or_nonnegative_int_iff
thf(fact_9612_or__negative__int__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        | ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).

% or_negative_int_iff
thf(fact_9613_xor__nonnegative__int__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        = ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).

% xor_nonnegative_int_iff
thf(fact_9614_xor__negative__int__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
       != ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).

% xor_negative_int_iff
thf(fact_9615_or__minus__numerals_I6_J,axiom,
    ! [N3: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) ) ).

% or_minus_numerals(6)
thf(fact_9616_or__minus__numerals_I2_J,axiom,
    ! [N3: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) ) ).

% or_minus_numerals(2)
thf(fact_9617_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_9618_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_9619_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_9620_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_9621_and__minus__minus__numerals,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N3 ) @ one_one_int ) ) ) ) ).

% and_minus_minus_numerals
thf(fact_9622_or__minus__minus__numerals,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N3 ) @ one_one_int ) ) ) ) ).

% or_minus_minus_numerals
thf(fact_9623_xor__int__def,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ L ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ L ) ) ) ) ).

% xor_int_def
thf(fact_9624_bit__or__int__iff,axiom,
    ! [K: int,L2: int,N3: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) @ N3 )
      = ( ( bit_se1146084159140164899it_int @ K @ N3 )
        | ( bit_se1146084159140164899it_int @ L2 @ N3 ) ) ) ).

% bit_or_int_iff
thf(fact_9625_bit__xor__int__iff,axiom,
    ! [K: int,L2: int,N3: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) @ N3 )
      = ( ( bit_se1146084159140164899it_int @ K @ N3 )
       != ( bit_se1146084159140164899it_int @ L2 @ N3 ) ) ) ).

% bit_xor_int_iff
thf(fact_9626_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_9627_or__nat__def,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M5: nat,N2: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% or_nat_def
thf(fact_9628_or__greater__eq,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L2 )
     => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L2 ) ) ) ).

% or_greater_eq
thf(fact_9629_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_9630_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_9631_or__int__def,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ ( bit_ri7919022796975470100ot_int @ L ) ) ) ) ) ).

% or_int_def
thf(fact_9632_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_9633_or__not__numerals_I2_J,axiom,
    ! [N3: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) ) ).

% or_not_numerals(2)
thf(fact_9634_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_9635_or__not__numerals_I3_J,axiom,
    ! [N3: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) ) ).

% or_not_numerals(3)
thf(fact_9636_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_9637_or__not__numerals_I6_J,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).

% or_not_numerals(6)
thf(fact_9638_XOR__upper,axiom,
    ! [X: int,N3: nat,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
       => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
         => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% XOR_upper
thf(fact_9639_OR__upper,axiom,
    ! [X: int,N3: nat,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
       => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) )
         => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% OR_upper
thf(fact_9640_or__not__numerals_I5_J,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) )
      = ( 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 @ N3 ) ) ) ) ) ) ).

% or_not_numerals(5)
thf(fact_9641_or__Suc__0__eq,axiom,
    ! [N3: nat] :
      ( ( bit_se1412395901928357646or_nat @ N3 @ ( suc @ zero_zero_nat ) )
      = ( plus_plus_nat @ N3 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% or_Suc_0_eq
thf(fact_9642_Suc__0__or__eq,axiom,
    ! [N3: nat] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N3 )
      = ( plus_plus_nat @ N3 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% Suc_0_or_eq
thf(fact_9643_or__nat__rec,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M5: nat,N2: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 )
              | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_9644_or__not__numerals_I9_J,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) )
      = ( 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 @ N3 ) ) ) ) ) ) ).

% or_not_numerals(9)
thf(fact_9645_or__not__numerals_I8_J,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) )
      = ( 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 @ N3 ) ) ) ) ) ) ).

% or_not_numerals(8)
thf(fact_9646_xor__int__rec,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) )
             != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_int_rec
thf(fact_9647_or__int__rec,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
              | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_int_rec
thf(fact_9648_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M5: nat,N2: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M5 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_9649_or__int__unfold,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L: int] :
          ( if_int
          @ ( ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            | ( L
              = ( uminus_uminus_int @ one_one_int ) ) )
          @ ( uminus_uminus_int @ one_one_int )
          @ ( if_int @ ( K3 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( 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 @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% or_int_unfold
thf(fact_9650_Bit__integer_Oabs__eq,axiom,
    ! [Xa: int,X: $o] :
      ( ( bits_Bit_integer @ ( code_integer_of_int @ Xa ) @ X )
      = ( code_integer_of_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ X ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ).

% Bit_integer.abs_eq
thf(fact_9651_or__minus__numerals_I5_J,axiom,
    ! [N3: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N3 ) ) ) ) ) ).

% or_minus_numerals(5)
thf(fact_9652_or__minus__numerals_I1_J,axiom,
    ! [N3: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N3 ) ) ) ) ) ).

% or_minus_numerals(1)
thf(fact_9653_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_9654_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_9655_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_9656_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_9657_or__minus__numerals_I4_J,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N3 ) ) ) ) ) ).

% or_minus_numerals(4)
thf(fact_9658_or__minus__numerals_I8_J,axiom,
    ! [N3: num,M: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N3 ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N3 ) ) ) ) ) ).

% or_minus_numerals(8)
thf(fact_9659_or__minus__numerals_I7_J,axiom,
    ! [N3: num,M: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N3 ) ) ) ) ) ).

% or_minus_numerals(7)
thf(fact_9660_or__minus__numerals_I3_J,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N3 ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N3 ) ) ) ) ) ).

% or_minus_numerals(3)
thf(fact_9661_or__not__num__neg_Osimps_I1_J,axiom,
    ( ( bit_or_not_num_neg @ one @ one )
    = one ) ).

% or_not_num_neg.simps(1)
thf(fact_9662_or__not__num__neg_Osimps_I4_J,axiom,
    ! [N3: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N3 ) @ one )
      = ( bit0 @ one ) ) ).

% or_not_num_neg.simps(4)
thf(fact_9663_or__not__num__neg_Osimps_I6_J,axiom,
    ! [N3: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N3 ) @ ( bit1 @ M ) )
      = ( bit0 @ ( bit_or_not_num_neg @ N3 @ M ) ) ) ).

% or_not_num_neg.simps(6)
thf(fact_9664_or__not__num__neg_Osimps_I7_J,axiom,
    ! [N3: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N3 ) @ one )
      = one ) ).

% or_not_num_neg.simps(7)
thf(fact_9665_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_9666_or__not__num__neg_Osimps_I5_J,axiom,
    ! [N3: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N3 ) @ ( bit0 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N3 @ M ) ) ) ).

% or_not_num_neg.simps(5)
thf(fact_9667_or__not__num__neg_Osimps_I9_J,axiom,
    ! [N3: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N3 ) @ ( bit1 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N3 @ M ) ) ) ).

% or_not_num_neg.simps(9)
thf(fact_9668_xor__nat__def,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M5: nat,N2: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M5 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% xor_nat_def
thf(fact_9669_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_9670_or__not__num__neg_Osimps_I8_J,axiom,
    ! [N3: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N3 ) @ ( bit0 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N3 @ M ) ) ) ).

% or_not_num_neg.simps(8)
thf(fact_9671_or__not__num__neg_Oelims,axiom,
    ! [X: num,Xa: num,Y: num] :
      ( ( ( bit_or_not_num_neg @ X @ Xa )
        = Y )
     => ( ( ( X = one )
         => ( ( Xa = one )
           => ( Y != one ) ) )
       => ( ( ( X = one )
           => ! [M4: num] :
                ( ( Xa
                  = ( bit0 @ M4 ) )
               => ( Y
                 != ( bit1 @ M4 ) ) ) )
         => ( ( ( X = one )
             => ! [M4: num] :
                  ( ( Xa
                    = ( bit1 @ M4 ) )
                 => ( Y
                   != ( bit1 @ M4 ) ) ) )
           => ( ( ? [N: num] :
                    ( X
                    = ( bit0 @ N ) )
               => ( ( Xa = one )
                 => ( Y
                   != ( bit0 @ one ) ) ) )
             => ( ! [N: num] :
                    ( ( X
                      = ( bit0 @ N ) )
                   => ! [M4: num] :
                        ( ( Xa
                          = ( bit0 @ M4 ) )
                       => ( Y
                         != ( bitM @ ( bit_or_not_num_neg @ N @ M4 ) ) ) ) )
               => ( ! [N: num] :
                      ( ( X
                        = ( bit0 @ N ) )
                     => ! [M4: num] :
                          ( ( Xa
                            = ( bit1 @ M4 ) )
                         => ( Y
                           != ( bit0 @ ( bit_or_not_num_neg @ N @ M4 ) ) ) ) )
                 => ( ( ? [N: num] :
                          ( X
                          = ( bit1 @ N ) )
                     => ( ( Xa = one )
                       => ( Y != one ) ) )
                   => ( ! [N: num] :
                          ( ( X
                            = ( bit1 @ N ) )
                         => ! [M4: num] :
                              ( ( Xa
                                = ( bit0 @ M4 ) )
                             => ( Y
                               != ( bitM @ ( bit_or_not_num_neg @ N @ M4 ) ) ) ) )
                     => ~ ! [N: num] :
                            ( ( X
                              = ( bit1 @ N ) )
                           => ! [M4: num] :
                                ( ( Xa
                                  = ( bit1 @ M4 ) )
                               => ( Y
                                 != ( bitM @ ( bit_or_not_num_neg @ N @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
thf(fact_9672_numeral__or__not__num__eq,axiom,
    ! [M: num,N3: num] :
      ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N3 ) )
      = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).

% numeral_or_not_num_eq
thf(fact_9673_int__numeral__not__or__num__neg,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N3 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N3 @ M ) ) ) ) ).

% int_numeral_not_or_num_neg
thf(fact_9674_int__numeral__or__not__num__neg,axiom,
    ! [M: num,N3: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N3 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N3 ) ) ) ) ).

% int_numeral_or_not_num_neg
thf(fact_9675_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M5: nat,N2: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M5 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_9676_xor__nat__rec,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M5: nat,N2: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
             != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_9677_Suc__0__xor__eq,axiom,
    ! [N3: nat] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N3 )
      = ( minus_minus_nat @ ( plus_plus_nat @ N3 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% Suc_0_xor_eq
thf(fact_9678_xor__Suc__0__eq,axiom,
    ! [N3: nat] :
      ( ( bit_se6528837805403552850or_nat @ N3 @ ( suc @ zero_zero_nat ) )
      = ( minus_minus_nat @ ( plus_plus_nat @ N3 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% xor_Suc_0_eq
thf(fact_9679_int__lsb__numeral_I6_J,axiom,
    ! [W: num] :
      ~ ( least_4859182151741483524sb_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) ).

% int_lsb_numeral(6)
thf(fact_9680_int__lsb__numeral_I3_J,axiom,
    least_4859182151741483524sb_int @ ( numeral_numeral_int @ one ) ).

% int_lsb_numeral(3)
thf(fact_9681_int__lsb__numeral_I8_J,axiom,
    ! [W: num] :
      ~ ( least_4859182151741483524sb_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) ) ).

% int_lsb_numeral(8)
thf(fact_9682_int__lsb__numeral_I5_J,axiom,
    least_4859182151741483524sb_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) ).

% int_lsb_numeral(5)
thf(fact_9683_lsb__integer__code,axiom,
    ( least_7544222001954398261nteger
    = ( ^ [X3: code_integer] : ( bit_se9216721137139052372nteger @ X3 @ zero_zero_nat ) ) ) ).

% lsb_integer_code
thf(fact_9684_lsb__int__def,axiom,
    ( least_4859182151741483524sb_int
    = ( ^ [I2: int] : ( bit_se1146084159140164899it_int @ I2 @ zero_zero_nat ) ) ) ).

% lsb_int_def
thf(fact_9685_bin__last__conv__lsb,axiom,
    ( ( ^ [A7: int] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A7 ) )
    = least_4859182151741483524sb_int ) ).

% bin_last_conv_lsb
thf(fact_9686_dup__1,axiom,
    ( ( code_dup @ one_one_Code_integer )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% dup_1
thf(fact_9687_pow_Osimps_I1_J,axiom,
    ! [X: num] :
      ( ( pow @ X @ one )
      = X ) ).

% pow.simps(1)
thf(fact_9688_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
    ! [X: nat,Y: int] :
      ( ( ( vEBT_V9176841429113362141ildupi @ X )
        = Y )
     => ( ( accp_nat @ vEBT_V3352910403632780892pi_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = one_one_int )
             => ~ ( accp_nat @ vEBT_V3352910403632780892pi_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y = one_one_int )
               => ~ ( accp_nat @ vEBT_V3352910403632780892pi_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [N: nat] :
                  ( ( X
                    = ( suc @ ( suc @ N ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                       => ( Y
                          = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                       => ( Y
                          = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V3352910403632780892pi_rel @ ( suc @ ( suc @ N ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.pelims
thf(fact_9689_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 ) ) ) )
           => ~ ! [Va3: nat] :
                  ( ( X
                    = ( suc @ ( suc @ Va3 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.pelims
thf(fact_9690_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_V8646137997579335489_i_l_d @ X )
        = Y )
     => ( ( accp_nat @ vEBT_V5144397997797733112_d_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
             => ~ ( accp_nat @ vEBT_V5144397997797733112_d_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
               => ~ ( accp_nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va3: nat] :
                  ( ( X
                    = ( suc @ ( suc @ Va3 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y
                          = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y
                          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
thf(fact_9691_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_V8346862874174094_d_u_p @ X )
        = Y )
     => ( ( accp_nat @ vEBT_V1247956027447740395_p_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
             => ~ ( accp_nat @ vEBT_V1247956027447740395_p_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
               => ~ ( accp_nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va3: nat] :
                  ( ( X
                    = ( suc @ ( suc @ Va3 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y
                          = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y
                          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
thf(fact_9692_VEBT__internal_OTb_Opelims,axiom,
    ! [X: nat,Y: int] :
      ( ( ( vEBT_VEBT_Tb @ X )
        = Y )
     => ( ( accp_nat @ vEBT_VEBT_Tb_rel2 @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( numeral_numeral_int @ ( bit1 @ one ) ) )
             => ~ ( accp_nat @ vEBT_VEBT_Tb_rel2 @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( numeral_numeral_int @ ( bit1 @ one ) ) )
               => ~ ( accp_nat @ vEBT_VEBT_Tb_rel2 @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [N: nat] :
                  ( ( X
                    = ( suc @ ( suc @ N ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                       => ( Y
                          = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                       => ( Y
                          = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_VEBT_Tb_rel2 @ ( suc @ ( suc @ N ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb.pelims
thf(fact_9693_VEBT__internal_OTb_H_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_VEBT_Tb2 @ X )
        = Y )
     => ( ( accp_nat @ vEBT_VEBT_Tb_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
             => ~ ( accp_nat @ vEBT_VEBT_Tb_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
               => ~ ( accp_nat @ vEBT_VEBT_Tb_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [N: nat] :
                  ( ( X
                    = ( suc @ ( suc @ N ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                       => ( Y
                          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                       => ( Y
                          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_VEBT_Tb_rel @ ( suc @ ( suc @ N ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.pelims
thf(fact_9694_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( vEBT_V441764108873111860ildupi @ X )
        = Y )
     => ( ( accp_nat @ vEBT_V2957053500504383685pi_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( suc @ zero_zero_nat ) )
             => ~ ( accp_nat @ vEBT_V2957053500504383685pi_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( suc @ zero_zero_nat ) )
               => ~ ( accp_nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [N: nat] :
                  ( ( X
                    = ( suc @ ( suc @ N ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                       => ( Y
                          = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
                       => ( Y
                          = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ ( suc @ N ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
thf(fact_9695_cis__multiple__2pi,axiom,
    ! [N3: real] :
      ( ( member_real @ N3 @ ring_1_Ints_real )
     => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N3 ) )
        = one_one_complex ) ) ).

% cis_multiple_2pi
thf(fact_9696_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_9697_sin__integer__2pi,axiom,
    ! [N3: real] :
      ( ( member_real @ N3 @ ring_1_Ints_real )
     => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N3 ) )
        = zero_zero_real ) ) ).

% sin_integer_2pi
thf(fact_9698_cos__integer__2pi,axiom,
    ! [N3: real] :
      ( ( member_real @ N3 @ ring_1_Ints_real )
     => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N3 ) )
        = one_one_real ) ) ).

% cos_integer_2pi
thf(fact_9699_VEBT__internal_Ospace_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_space @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( Y
                    = ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_space @ Summary2 ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space @ TreeList3 ) @ zero_zero_nat ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ).

% VEBT_internal.space.pelims
thf(fact_9700_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_space2 @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y
                  = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( Y
                    = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_space2 @ Summary2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space2 @ TreeList3 ) @ zero_zero_nat ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ).

% VEBT_internal.space'.pelims
thf(fact_9701_VEBT__internal_Ocnt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: real] :
      ( ( ( vEBT_VEBT_cnt @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y = one_one_real )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( Y
                    = ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( vEBT_VEBT_cnt @ Summary2 ) ) @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ vEBT_VEBT_cnt @ TreeList3 ) @ zero_zero_real ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ).

% VEBT_internal.cnt.pelims
thf(fact_9702_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_cnt2 @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y = one_one_nat )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( Y
                    = ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_cnt2 @ Summary2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_cnt2 @ TreeList3 ) @ zero_zero_nat ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ).

% VEBT_internal.cnt'.pelims
thf(fact_9703_vebt__maxt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( ( B3
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B3
                   => ( ( A3
                       => ( Y
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A3
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) )
               => ( ( Y = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
                 => ( ( Y
                      = ( some_nat @ Ma2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) ) ) ) ) ) ) ) ).

% vebt_maxt.pelims
thf(fact_9704_vebt__mint_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( ( A3
                   => ( Y
                      = ( some_nat @ zero_zero_nat ) ) )
                  & ( ~ A3
                   => ( ( B3
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B3
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) )
               => ( ( Y = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
                 => ( ( Y
                      = ( some_nat @ Mi2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) ) ) ) ) ) ) ) ).

% vebt_mint.pelims
thf(fact_9705_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_i_n_t @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y
                  = ( plus_plus_nat @ one_one_nat @ ( if_nat @ A3 @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
thf(fact_9706_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_a_x_t @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y
                  = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B3 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ( ! [Uu2: nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
thf(fact_9707_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_T_m_i_n_N_u_l_l @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ X )
       => ( ( ( X
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( ( Y = one_one_nat )
             => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv: $o] :
                ( ( X
                  = ( vEBT_Leaf @ $true @ Uv ) )
               => ( ( Y = one_one_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $true @ Uv ) ) ) )
           => ( ! [Uu2: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ Uu2 @ $true ) )
                 => ( ( Y = one_one_nat )
                   => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
             => ( ! [Uw2: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux @ Uy ) )
                   => ( ( Y = one_one_nat )
                     => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux @ Uy ) ) ) )
               => ~ ! [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va2 @ Vb @ Vc ) )
                     => ( ( Y = one_one_nat )
                       => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va2 @ Vb @ Vc ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
thf(fact_9708_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 ) ) ) )
         => ( ! [Uv: $o] :
                ( ( X
                  = ( vEBT_Leaf @ $true @ Uv ) )
               => ( ~ Y
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv ) ) ) )
           => ( ! [Uu2: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ Uu2 @ $true ) )
                 => ( ~ Y
                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
             => ( ! [Uw2: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux @ Uy ) )
                   => ( Y
                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux @ Uy ) ) ) )
               => ~ ! [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va2 @ Vb @ Vc ) )
                     => ( ~ Y
                       => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va2 @ Vb @ Vc ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
thf(fact_9709_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ! [Uv: $o] :
              ( ( X
                = ( vEBT_Leaf @ $true @ Uv ) )
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv ) ) )
         => ( ! [Uu2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
           => ~ ! [Uz: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va2 @ Vb @ Vc ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va2 @ Vb @ Vc ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
thf(fact_9710_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,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux @ Uy ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux @ Uy ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
thf(fact_9711_setceilmax,axiom,
    ! [S2: vEBT_VEBT,M: nat,Listy: list_VEBT_VEBT,N3: nat] :
      ( ( vEBT_invar_vebt @ S2 @ M )
     => ( ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Listy ) )
           => ( vEBT_invar_vebt @ X4 @ N3 ) )
       => ( ( M
            = ( suc @ N3 ) )
         => ( ! [X4: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Listy ) )
               => ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ X4 ) )
                  = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ) )
           => ( ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ S2 ) )
                = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) )
             => ( ( semiri1314217659103216013at_int @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ S2 @ ( set_VEBT_VEBT2 @ Listy ) ) ) ) )
                = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ) ) ) ).

% setceilmax
thf(fact_9712_bin__last__integer__nbe,axiom,
    ( bits_b8758750999018896077nteger
    = ( ^ [I2: code_integer] :
          ( ( modulo364778990260209775nteger @ I2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
         != zero_z3403309356797280102nteger ) ) ) ).

% bin_last_integer_nbe
thf(fact_9713_height__compose__list,axiom,
    ! [T: vEBT_VEBT,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ T @ ( set_VEBT_VEBT2 @ TreeList ) )
     => ( ord_less_eq_nat @ ( vEBT_VEBT_height @ T ) @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary @ ( set_VEBT_VEBT2 @ TreeList ) ) ) ) ) ) ).

% height_compose_list
thf(fact_9714_max__ins__scaled,axiom,
    ! [N3: nat,X14: vEBT_VEBT,M: nat,X13: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( times_times_nat @ N3 @ ( vEBT_VEBT_height @ X14 ) ) @ ( plus_plus_nat @ M @ ( times_times_nat @ N3 @ ( lattic8265883725875713057ax_nat @ ( insert_nat @ ( vEBT_VEBT_height @ X14 ) @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( set_VEBT_VEBT2 @ X13 ) ) ) ) ) ) ) ).

% max_ins_scaled
thf(fact_9715_height__i__max,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,Foo: nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ X13 ) )
     => ( ord_less_eq_nat @ ( vEBT_VEBT_height @ ( nth_VEBT_VEBT @ X13 @ I ) ) @ ( ord_max_nat @ Foo @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( set_VEBT_VEBT2 @ X13 ) ) ) ) ) ) ).

% height_i_max
thf(fact_9716_max__idx__list,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,N3: nat,X14: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ X13 ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ N3 @ ( vEBT_VEBT_height @ ( nth_VEBT_VEBT @ X13 @ I ) ) ) @ ( suc @ ( suc @ ( times_times_nat @ N3 @ ( ord_max_nat @ ( vEBT_VEBT_height @ X14 ) @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( set_VEBT_VEBT2 @ X13 ) ) ) ) ) ) ) ) ) ).

% max_idx_list
thf(fact_9717_Max__divisors__self__nat,axiom,
    ! [N3: nat] :
      ( ( N3 != zero_zero_nat )
     => ( ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D3: nat] : ( dvd_dvd_nat @ D3 @ N3 ) ) )
        = N3 ) ) ).

% Max_divisors_self_nat
thf(fact_9718_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_height @ ( vEBT_Node @ Uu @ Deg @ TreeList @ Summary ) )
      = ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary @ ( set_VEBT_VEBT2 @ TreeList ) ) ) ) ) ) ).

% VEBT_internal.height.simps(2)
thf(fact_9719_VEBT__internal_Oheight_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_height @ X )
        = Y )
     => ( ( ? [A3: $o,B3: $o] :
              ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( Y != zero_zero_nat ) )
       => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Uu2 @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( Y
               != ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary2 @ ( set_VEBT_VEBT2 @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.height.elims
thf(fact_9720_divide__nat__def,axiom,
    ( divide_divide_nat
    = ( ^ [M5: nat,N2: nat] :
          ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
          @ ( lattic8265883725875713057ax_nat
            @ ( collect_nat
              @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N2 ) @ M5 ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_9721_bin__last__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( bits_b8758750999018896077nteger @ ( code_integer_of_int @ X ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) ) ) ).

% bin_last_integer.abs_eq
thf(fact_9722_VEBT__internal_Oheight_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: nat] :
      ( ( ( vEBT_VEBT_height @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y = zero_zero_nat )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ~ ! [Uu2: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu2 @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( Y
                    = ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary2 @ ( set_VEBT_VEBT2 @ TreeList3 ) ) ) ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Node @ Uu2 @ Deg2 @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ).

% VEBT_internal.height.pelims
thf(fact_9723_bij__betw__Suc,axiom,
    ! [M3: set_nat,N7: set_nat] :
      ( ( bij_betw_nat_nat @ suc @ M3 @ N7 )
      = ( ( image_nat_nat @ suc @ M3 )
        = N7 ) ) ).

% bij_betw_Suc
thf(fact_9724_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J2: nat] :
      ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J2 ) )
      = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J2 ) ) ) ).

% image_Suc_atLeastLessThan
thf(fact_9725_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J2: nat] :
      ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J2 ) )
      = ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J2 ) ) ) ).

% image_Suc_atLeastAtMost
thf(fact_9726_range__mult,axiom,
    ! [A2: real] :
      ( ( ( A2 = zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A2 ) @ top_top_set_real )
          = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
      & ( ( A2 != zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A2 ) @ top_top_set_real )
          = top_top_set_real ) ) ) ).

% range_mult
thf(fact_9727_zero__notin__Suc__image,axiom,
    ! [A: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A ) ) ).

% zero_notin_Suc_image
thf(fact_9728_image__int__atLeastAtMost,axiom,
    ! [A2: nat,B2: nat] :
      ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A2 @ B2 ) )
      = ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).

% image_int_atLeastAtMost
thf(fact_9729_image__int__atLeastLessThan,axiom,
    ! [A2: nat,B2: nat] :
      ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A2 @ B2 ) )
      = ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).

% image_int_atLeastLessThan
thf(fact_9730_image__Suc__lessThan,axiom,
    ! [N3: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N3 ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ).

% image_Suc_lessThan
thf(fact_9731_image__Suc__atMost,axiom,
    ! [N3: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N3 ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N3 ) ) ) ).

% image_Suc_atMost
thf(fact_9732_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N3: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N3 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) ) ) ).

% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_9733_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N3: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N3 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N3 ) ) ) ) ).

% atLeast0_atMost_Suc_eq_insert_0
thf(fact_9734_lessThan__Suc__eq__insert__0,axiom,
    ! [N3: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ N3 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).

% lessThan_Suc_eq_insert_0
thf(fact_9735_atMost__Suc__eq__insert__0,axiom,
    ! [N3: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ N3 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N3 ) ) ) ) ).

% atMost_Suc_eq_insert_0
thf(fact_9736_range__mod,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( image_nat_nat
          @ ^ [M5: nat] : ( modulo_modulo_nat @ M5 @ N3 )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) ) ) ).

% range_mod
thf(fact_9737_image__add__int__atLeastLessThan,axiom,
    ! [L2: int,U: int] :
      ( ( image_int_int
        @ ^ [X3: int] : ( plus_plus_int @ X3 @ L2 )
        @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L2 ) ) )
      = ( set_or4662586982721622107an_int @ L2 @ U ) ) ).

% image_add_int_atLeastLessThan
thf(fact_9738_image__add__integer__atLeastLessThan,axiom,
    ! [L2: code_integer,U: code_integer] :
      ( ( image_4470545334726330049nteger
        @ ^ [X3: code_integer] : ( plus_p5714425477246183910nteger @ X3 @ L2 )
        @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ U @ L2 ) ) )
      = ( set_or8404916559141939852nteger @ L2 @ U ) ) ).

% image_add_integer_atLeastLessThan
thf(fact_9739_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_9740_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Y: nat,X: nat] :
      ( ( ( ord_less_nat @ C2 @ Y )
       => ( ( image_nat_nat
            @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C2 )
            @ ( set_or4665077453230672383an_nat @ X @ Y ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C2 ) @ ( minus_minus_nat @ Y @ C2 ) ) ) )
      & ( ~ ( ord_less_nat @ C2 @ Y )
       => ( ( ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C2 )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C2 )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_9741_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_9742_drop__bit__nonnegative__int__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N3 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% drop_bit_nonnegative_int_iff
thf(fact_9743_drop__bit__negative__int__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N3 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% drop_bit_negative_int_iff
thf(fact_9744_drop__bit__minus__one,axiom,
    ! [N3: nat] :
      ( ( bit_se8568078237143864401it_int @ N3 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% drop_bit_minus_one
thf(fact_9745_drop__bit__Suc__minus__bit0,axiom,
    ! [N3: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N3 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% drop_bit_Suc_minus_bit0
thf(fact_9746_drop__bit__of__Suc__0,axiom,
    ! [N3: nat] :
      ( ( bit_se8570568707652914677it_nat @ N3 @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( N3 = zero_zero_nat ) ) ) ).

% drop_bit_of_Suc_0
thf(fact_9747_drop__bit__numeral__minus__bit0,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% drop_bit_numeral_minus_bit0
thf(fact_9748_drop__bit__Suc__minus__bit1,axiom,
    ! [N3: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N3 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_Suc_minus_bit1
thf(fact_9749_drop__bit__numeral__minus__bit1,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_numeral_minus_bit1
thf(fact_9750_drop__bit__nat__eq,axiom,
    ! [N3: nat,K: int] :
      ( ( bit_se8570568707652914677it_nat @ N3 @ ( nat2 @ K ) )
      = ( nat2 @ ( bit_se8568078237143864401it_int @ N3 @ K ) ) ) ).

% drop_bit_nat_eq
thf(fact_9751_shiftr__integer__conv__div__pow2,axiom,
    ( bit_se3928097537394005634nteger
    = ( ^ [N2: nat,X3: code_integer] : ( divide6298287555418463151nteger @ X3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% shiftr_integer_conv_div_pow2
thf(fact_9752_drop__bit__int__def,axiom,
    ( bit_se8568078237143864401it_int
    = ( ^ [N2: nat,K3: int] : ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% drop_bit_int_def
thf(fact_9753_drop__bit__nat__def,axiom,
    ( bit_se8570568707652914677it_nat
    = ( ^ [N2: nat,M5: nat] : ( divide_divide_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% drop_bit_nat_def
thf(fact_9754_bin__rest__code,axiom,
    ! [I: int] :
      ( ( divide_divide_int @ I @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( bit_se8568078237143864401it_int @ one_one_nat @ I ) ) ).

% bin_rest_code
thf(fact_9755_push__bit__nonnegative__int__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N3 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% push_bit_nonnegative_int_iff
thf(fact_9756_push__bit__negative__int__iff,axiom,
    ! [N3: nat,K: int] :
      ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N3 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% push_bit_negative_int_iff
thf(fact_9757_push__bit__of__Suc__0,axiom,
    ! [N3: nat] :
      ( ( bit_se547839408752420682it_nat @ N3 @ ( suc @ zero_zero_nat ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% push_bit_of_Suc_0
thf(fact_9758_drop__bit__push__bit__int,axiom,
    ! [M: nat,N3: nat,K: int] :
      ( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N3 @ K ) )
      = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N3 ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N3 @ M ) @ K ) ) ) ).

% drop_bit_push_bit_int
thf(fact_9759_flip__bit__nat__def,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [M5: nat,N2: nat] : ( bit_se6528837805403552850or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M5 @ one_one_nat ) ) ) ) ).

% flip_bit_nat_def
thf(fact_9760_set__bit__nat__def,axiom,
    ( bit_se7882103937844011126it_nat
    = ( ^ [M5: nat,N2: nat] : ( bit_se1412395901928357646or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M5 @ one_one_nat ) ) ) ) ).

% set_bit_nat_def
thf(fact_9761_push__bit__nat__eq,axiom,
    ! [N3: nat,K: int] :
      ( ( bit_se547839408752420682it_nat @ N3 @ ( nat2 @ K ) )
      = ( nat2 @ ( bit_se545348938243370406it_int @ N3 @ K ) ) ) ).

% push_bit_nat_eq
thf(fact_9762_push__bit__int__code_I1_J,axiom,
    ! [I: int] :
      ( ( bit_se545348938243370406it_int @ zero_zero_nat @ I )
      = I ) ).

% push_bit_int_code(1)
thf(fact_9763_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N3: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N3 )
      = ( ( ord_less_eq_nat @ M @ N3 )
        & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N3 @ M ) ) ) ) ).

% bit_push_bit_iff_int
thf(fact_9764_Bit__Operations_Oset__bit__int__def,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N2: nat,K3: int] : ( bit_se1409905431419307370or_int @ K3 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).

% Bit_Operations.set_bit_int_def
thf(fact_9765_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q3: nat,N3: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q3 ) @ N3 )
      = ( ( ord_less_eq_nat @ M @ N3 )
        & ( bit_se1148574629649215175it_nat @ Q3 @ ( minus_minus_nat @ N3 @ M ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_9766_drop__bit__int__code_I1_J,axiom,
    ! [I: int] :
      ( ( bit_se8568078237143864401it_int @ zero_zero_nat @ I )
      = I ) ).

% drop_bit_int_code(1)
thf(fact_9767_flip__bit__int__def,axiom,
    ( bit_se2159334234014336723it_int
    = ( ^ [N2: nat,K3: int] : ( bit_se6526347334894502574or_int @ K3 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).

% flip_bit_int_def
thf(fact_9768_shiftl__integer__conv__mult__pow2,axiom,
    ( bit_se7788150548672797655nteger
    = ( ^ [N2: nat,X3: code_integer] : ( times_3573771949741848930nteger @ X3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% shiftl_integer_conv_mult_pow2
thf(fact_9769_unset__bit__int__def,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N2: nat,K3: int] : ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ) ).

% unset_bit_int_def
thf(fact_9770_push__bit__int__def,axiom,
    ( bit_se545348938243370406it_int
    = ( ^ [N2: nat,K3: int] : ( times_times_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% push_bit_int_def
thf(fact_9771_push__bit__nat__def,axiom,
    ( bit_se547839408752420682it_nat
    = ( ^ [N2: nat,M5: nat] : ( times_times_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% push_bit_nat_def
thf(fact_9772_push__bit__minus__one,axiom,
    ! [N3: nat] :
      ( ( bit_se545348938243370406it_int @ N3 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ).

% push_bit_minus_one
thf(fact_9773_drop__bit__int__code_I2_J,axiom,
    ! [N3: nat] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N3 ) @ zero_zero_int )
      = zero_zero_int ) ).

% drop_bit_int_code(2)
thf(fact_9774_bin__rest__integer_Oabs__eq,axiom,
    ! [X: int] :
      ( ( bits_b2549910563261871055nteger @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% bin_rest_integer.abs_eq
thf(fact_9775_bin__rest__integer__code,axiom,
    ( bits_b2549910563261871055nteger
    = ( ^ [I2: code_integer] : ( divide6298287555418463151nteger @ I2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% bin_rest_integer_code
thf(fact_9776_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I2: int,J: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J @ I2 ) @ Js @ ( upto_aux @ I2 @ ( minus_minus_int @ J @ one_one_int ) @ ( cons_int @ J @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_9777_floor__real__def,axiom,
    ( archim6058952711729229775r_real
    = ( ^ [X3: real] :
          ( the_int
          @ ^ [Z5: int] :
              ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z5 ) @ X3 )
              & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).

% floor_real_def
thf(fact_9778_concat__bit__Suc,axiom,
    ! [N3: nat,K: int,L2: int] :
      ( ( bit_concat_bit @ ( suc @ N3 ) @ K @ L2 )
      = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N3 @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L2 ) ) ) ) ).

% concat_bit_Suc
thf(fact_9779_concat__bit__0,axiom,
    ! [K: int,L2: int] :
      ( ( bit_concat_bit @ zero_zero_nat @ K @ L2 )
      = L2 ) ).

% concat_bit_0
thf(fact_9780_concat__bit__of__zero__2,axiom,
    ! [N3: nat,K: int] :
      ( ( bit_concat_bit @ N3 @ K @ zero_zero_int )
      = ( bit_se2923211474154528505it_int @ N3 @ K ) ) ).

% concat_bit_of_zero_2
thf(fact_9781_concat__bit__nonnegative__iff,axiom,
    ! [N3: nat,K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N3 @ K @ L2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ).

% concat_bit_nonnegative_iff
thf(fact_9782_concat__bit__negative__iff,axiom,
    ! [N3: nat,K: int,L2: int] :
      ( ( ord_less_int @ ( bit_concat_bit @ N3 @ K @ L2 ) @ zero_zero_int )
      = ( ord_less_int @ L2 @ zero_zero_int ) ) ).

% concat_bit_negative_iff
thf(fact_9783_concat__bit__of__zero__1,axiom,
    ! [N3: nat,L2: int] :
      ( ( bit_concat_bit @ N3 @ zero_zero_int @ L2 )
      = ( bit_se545348938243370406it_int @ N3 @ L2 ) ) ).

% concat_bit_of_zero_1
thf(fact_9784_concat__bit__assoc,axiom,
    ! [N3: nat,K: int,M: nat,L2: int,R3: int] :
      ( ( bit_concat_bit @ N3 @ K @ ( bit_concat_bit @ M @ L2 @ R3 ) )
      = ( bit_concat_bit @ ( plus_plus_nat @ M @ N3 ) @ ( bit_concat_bit @ N3 @ K @ L2 ) @ R3 ) ) ).

% concat_bit_assoc
thf(fact_9785_concat__bit__eq__iff,axiom,
    ! [N3: nat,K: int,L2: int,R3: int,S2: int] :
      ( ( ( bit_concat_bit @ N3 @ K @ L2 )
        = ( bit_concat_bit @ N3 @ R3 @ S2 ) )
      = ( ( ( bit_se2923211474154528505it_int @ N3 @ K )
          = ( bit_se2923211474154528505it_int @ N3 @ R3 ) )
        & ( L2 = S2 ) ) ) ).

% concat_bit_eq_iff
thf(fact_9786_concat__bit__take__bit__eq,axiom,
    ! [N3: nat,B2: int] :
      ( ( bit_concat_bit @ N3 @ ( bit_se2923211474154528505it_int @ N3 @ B2 ) )
      = ( bit_concat_bit @ N3 @ B2 ) ) ).

% concat_bit_take_bit_eq
thf(fact_9787_concat__bit__eq,axiom,
    ( bit_concat_bit
    = ( ^ [N2: nat,K3: int,L: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N2 @ K3 ) @ ( bit_se545348938243370406it_int @ N2 @ L ) ) ) ) ).

% concat_bit_eq
thf(fact_9788_concat__bit__def,axiom,
    ( bit_concat_bit
    = ( ^ [N2: nat,K3: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N2 @ K3 ) @ ( bit_se545348938243370406it_int @ N2 @ L ) ) ) ) ).

% concat_bit_def
thf(fact_9789_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L2: int,N3: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L2 ) @ N3 )
      = ( ( ( ord_less_nat @ N3 @ M )
          & ( bit_se1146084159140164899it_int @ K @ N3 ) )
        | ( ( ord_less_eq_nat @ M @ N3 )
          & ( bit_se1146084159140164899it_int @ L2 @ ( minus_minus_nat @ N3 @ M ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_9790_signed__take__bit__eq__concat__bit,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N2: nat,K3: int] : ( bit_concat_bit @ N2 @ K3 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N2 ) ) ) ) ) ) ).

% signed_take_bit_eq_concat_bit
thf(fact_9791_floor__rat__def,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [X3: rat] :
          ( the_int
          @ ^ [Z5: int] :
              ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z5 ) @ X3 )
              & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).

% floor_rat_def
thf(fact_9792_vebt__minti_Opelims,axiom,
    ! [X: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( ( vEBT_vebt_minti @ X )
        = Y )
     => ( ( accp_VEBT_VEBTi @ vEBT_vebt_minti_rel @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leafi @ A3 @ B3 ) )
             => ( ( ( A3
                   => ( Y
                      = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
                  & ( ~ A3
                   => ( ( B3
                       => ( Y
                          = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
                      & ( ~ B3
                       => ( Y
                          = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) )
               => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Leafi @ A3 @ B3 ) ) ) )
         => ( ! [Uu2: nat,Uv: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
                ( ( X
                  = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) )
               => ( ( Y
                    = ( heap_T3487192422709364219on_nat @ none_nat ) )
                 => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] :
                  ( ( X
                    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
                 => ( ( Y
                      = ( heap_T3487192422709364219on_nat @ ( some_nat @ Mi2 ) ) )
                   => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) ) ) ) ) ) ) ) ).

% vebt_minti.pelims
thf(fact_9793_less__eq__rat__def,axiom,
    ( ord_less_eq_rat
    = ( ^ [X3: rat,Y2: rat] :
          ( ( ord_less_rat @ X3 @ Y2 )
          | ( X3 = Y2 ) ) ) ) ).

% less_eq_rat_def
thf(fact_9794_abs__rat__def,axiom,
    ( abs_abs_rat
    = ( ^ [A7: rat] : ( if_rat @ ( ord_less_rat @ A7 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A7 ) @ A7 ) ) ) ).

% abs_rat_def
thf(fact_9795_sgn__rat__def,axiom,
    ( sgn_sgn_rat
    = ( ^ [A7: rat] : ( if_rat @ ( A7 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A7 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_rat_def
thf(fact_9796_obtain__pos__sum,axiom,
    ! [R3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ R3 )
     => ~ ! [S3: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ S3 )
           => ! [T6: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ T6 )
               => ( R3
                 != ( plus_plus_rat @ S3 @ T6 ) ) ) ) ) ).

% obtain_pos_sum
thf(fact_9797_vebt__maxti_Opelims,axiom,
    ! [X: vEBT_VEBTi,Y: heap_T2636463487746394924on_nat] :
      ( ( ( vEBT_vebt_maxti @ X )
        = Y )
     => ( ( accp_VEBT_VEBTi @ vEBT_vebt_maxti_rel @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leafi @ A3 @ B3 ) )
             => ( ( ( B3
                   => ( Y
                      = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
                  & ( ~ B3
                   => ( ( A3
                       => ( Y
                          = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
                      & ( ~ A3
                       => ( Y
                          = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) )
               => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Leafi @ A3 @ B3 ) ) ) )
         => ( ! [Uu2: nat,Uv: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
                ( ( X
                  = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) )
               => ( ( Y
                    = ( heap_T3487192422709364219on_nat @ none_nat ) )
                 => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu2 @ Uv @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: array_VEBT_VEBTi,Uz: vEBT_VEBTi] :
                  ( ( X
                    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) )
                 => ( ( Y
                      = ( heap_T3487192422709364219on_nat @ ( some_nat @ Ma2 ) ) )
                   => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz ) ) ) ) ) ) ) ) ).

% vebt_maxti.pelims
thf(fact_9798_VEBT__internal_OminNulli_Opelims,axiom,
    ! [X: vEBT_VEBTi,Y: heap_Time_Heap_o] :
      ( ( ( vEBT_VEBT_minNulli @ X )
        = Y )
     => ( ( accp_VEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ X )
       => ( ( ( X
              = ( vEBT_Leafi @ $false @ $false ) )
           => ( ( Y
                = ( heap_Time_return_o @ $true ) )
             => ~ ( accp_VEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Leafi @ $false @ $false ) ) ) )
         => ( ! [Uv: $o] :
                ( ( X
                  = ( vEBT_Leafi @ $true @ Uv ) )
               => ( ( Y
                    = ( heap_Time_return_o @ $false ) )
                 => ~ ( accp_VEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Leafi @ $true @ Uv ) ) ) )
           => ( ! [Uu2: $o] :
                  ( ( X
                    = ( vEBT_Leafi @ Uu2 @ $true ) )
                 => ( ( Y
                      = ( heap_Time_return_o @ $false ) )
                   => ~ ( accp_VEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Leafi @ Uu2 @ $true ) ) ) )
             => ( ! [Uw2: nat,Ux: array_VEBT_VEBTi,Uy: vEBT_VEBTi] :
                    ( ( X
                      = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uw2 @ Ux @ Uy ) )
                   => ( ( Y
                        = ( heap_Time_return_o @ $true ) )
                     => ~ ( accp_VEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uw2 @ Ux @ Uy ) ) ) )
               => ~ ! [Uz: product_prod_nat_nat,Va2: nat,Vb: array_VEBT_VEBTi,Vc: vEBT_VEBTi] :
                      ( ( X
                        = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ Uz ) @ Va2 @ Vb @ Vc ) )
                     => ( ( Y
                          = ( heap_Time_return_o @ $false ) )
                       => ~ ( accp_VEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ Uz ) @ Va2 @ Vb @ Vc ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNulli.pelims
thf(fact_9799_diff__rat__def,axiom,
    ( minus_minus_rat
    = ( ^ [Q4: rat,R5: rat] : ( plus_plus_rat @ Q4 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).

% diff_rat_def
thf(fact_9800_normalize__negative,axiom,
    ! [Q3: int,P6: int] :
      ( ( ord_less_int @ Q3 @ zero_zero_int )
     => ( ( normalize @ ( product_Pair_int_int @ P6 @ Q3 ) )
        = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P6 ) @ ( uminus_uminus_int @ Q3 ) ) ) ) ) ).

% normalize_negative
thf(fact_9801_normalize__denom__pos,axiom,
    ! [R3: product_prod_int_int,P6: int,Q3: int] :
      ( ( ( normalize @ R3 )
        = ( product_Pair_int_int @ P6 @ Q3 ) )
     => ( ord_less_int @ zero_zero_int @ Q3 ) ) ).

% normalize_denom_pos
thf(fact_9802_rat__minus__code,axiom,
    ! [P6: rat,Q3: rat] :
      ( ( quotient_of @ ( minus_minus_rat @ P6 @ Q3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A7: int,C5: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B7: int,D3: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A7 @ D3 ) @ ( times_times_int @ B7 @ C5 ) ) @ ( times_times_int @ C5 @ D3 ) ) )
            @ ( quotient_of @ Q3 ) )
        @ ( quotient_of @ P6 ) ) ) ).

% rat_minus_code
thf(fact_9803_quotient__of__div,axiom,
    ! [R3: rat,N3: int,D2: int] :
      ( ( ( quotient_of @ R3 )
        = ( product_Pair_int_int @ N3 @ D2 ) )
     => ( R3
        = ( divide_divide_rat @ ( ring_1_of_int_rat @ N3 ) @ ( ring_1_of_int_rat @ D2 ) ) ) ) ).

% quotient_of_div
thf(fact_9804_rat__floor__code,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [P4: rat] : ( produc8211389475949308722nt_int @ divide_divide_int @ ( quotient_of @ P4 ) ) ) ) ).

% rat_floor_code
thf(fact_9805_quotient__of__denom__pos,axiom,
    ! [R3: rat,P6: int,Q3: int] :
      ( ( ( quotient_of @ R3 )
        = ( product_Pair_int_int @ P6 @ Q3 ) )
     => ( ord_less_int @ zero_zero_int @ Q3 ) ) ).

% quotient_of_denom_pos
thf(fact_9806_rat__less__code,axiom,
    ( ord_less_rat
    = ( ^ [P4: rat,Q4: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A7: int,C5: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B7: int,D3: int] : ( ord_less_int @ ( times_times_int @ A7 @ D3 ) @ ( times_times_int @ C5 @ B7 ) )
              @ ( quotient_of @ Q4 ) )
          @ ( quotient_of @ P4 ) ) ) ) ).

% rat_less_code
thf(fact_9807_rat__divide__code,axiom,
    ! [P6: rat,Q3: rat] :
      ( ( quotient_of @ ( divide_divide_rat @ P6 @ Q3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A7: int,C5: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B7: int,D3: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A7 @ D3 ) @ ( times_times_int @ C5 @ B7 ) ) )
            @ ( quotient_of @ Q3 ) )
        @ ( quotient_of @ P6 ) ) ) ).

% rat_divide_code
thf(fact_9808_divide__rat__def,axiom,
    ( divide_divide_rat
    = ( ^ [Q4: rat,R5: rat] : ( times_times_rat @ Q4 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).

% divide_rat_def
thf(fact_9809_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_9810_Frct__code__post_I5_J,axiom,
    ! [K: num] :
      ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
      = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).

% Frct_code_post(5)
thf(fact_9811_Frct__code__post_I6_J,axiom,
    ! [K: num,L2: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L2 ) ) )
      = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L2 ) ) ) ).

% Frct_code_post(6)
thf(fact_9812_Cauchy__iff2,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X8: nat > real] :
        ! [J: nat] :
        ? [M9: nat] :
        ! [M5: nat] :
          ( ( ord_less_eq_nat @ M9 @ M5 )
         => ! [N2: nat] :
              ( ( ord_less_eq_nat @ M9 @ N2 )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X8 @ M5 ) @ ( X8 @ N2 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_9813_sdiv__int__numeral__numeral,axiom,
    ! [M: num,N3: num] :
      ( ( signed6714573509424544716de_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N3 ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N3 ) ) ) ).

% sdiv_int_numeral_numeral
thf(fact_9814_signed__divide__int__def,axiom,
    ( signed6714573509424544716de_int
    = ( ^ [K3: int,L: int] : ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( sgn_sgn_int @ L ) ) @ ( divide_divide_int @ ( abs_abs_int @ K3 ) @ ( abs_abs_int @ L ) ) ) ) ) ).

% signed_divide_int_def
thf(fact_9815_entails__solve__init_I1_J,axiom,
    ! [P: assn,Q: assn] :
      ( ( fI_QUERY @ P @ Q @ top_top_assn )
     => ( entails @ P @ ( times_times_assn @ Q @ top_top_assn ) ) ) ).

% entails_solve_init(1)
thf(fact_9816_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_9817_FI__QUERY__def,axiom,
    ( fI_QUERY
    = ( ^ [P3: assn,Q6: assn,F7: assn] : ( entails @ P3 @ ( times_times_assn @ Q6 @ F7 ) ) ) ) ).

% FI_QUERY_def
thf(fact_9818_frame__inference__init,axiom,
    ! [P: assn,Q: assn,F: assn] :
      ( ( fI_QUERY @ P @ Q @ F )
     => ( entails @ P @ ( times_times_assn @ Q @ F ) ) ) ).

% frame_inference_init
thf(fact_9819_entails__solve__init_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( ( fI_QUERY @ P @ Q @ one_one_assn )
     => ( entails @ P @ Q ) ) ).

% entails_solve_init(2)
thf(fact_9820_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_9821_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] :
      ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
      = zero_zero_nat ) ).

% VEBT.size_gen(2)
thf(fact_9822_smod__int__range,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero_int )
     => ( member_int @ ( signed6292675348222524329lo_int @ A2 @ B2 ) @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ ( uminus_uminus_int @ ( abs_abs_int @ B2 ) ) @ one_one_int ) @ ( minus_minus_int @ ( abs_abs_int @ B2 ) @ one_one_int ) ) ) ) ).

% smod_int_range
thf(fact_9823_smod__int__compares_I8_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ B2 @ ( signed6292675348222524329lo_int @ A2 @ B2 ) ) ) ) ).

% smod_int_compares(8)
thf(fact_9824_smod__int__compares_I7_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( signed6292675348222524329lo_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% smod_int_compares(7)
thf(fact_9825_smod__int__compares_I6_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_eq_int @ zero_zero_int @ ( signed6292675348222524329lo_int @ A2 @ B2 ) ) ) ) ).

% smod_int_compares(6)
thf(fact_9826_smod__int__compares_I4_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ ( signed6292675348222524329lo_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% smod_int_compares(4)
thf(fact_9827_smod__int__compares_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_eq_int @ zero_zero_int @ ( signed6292675348222524329lo_int @ A2 @ B2 ) ) ) ) ).

% smod_int_compares(2)
thf(fact_9828_smod__int__compares_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ ( signed6292675348222524329lo_int @ A2 @ B2 ) @ B2 ) ) ) ).

% smod_int_compares(1)
thf(fact_9829_signed__modulo__int__def,axiom,
    ( signed6292675348222524329lo_int
    = ( ^ [K3: int,L: int] : ( minus_minus_int @ K3 @ ( times_times_int @ ( signed6714573509424544716de_int @ K3 @ L ) @ L ) ) ) ) ).

% signed_modulo_int_def
thf(fact_9830_smod__int__compares_I5_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A2 )
     => ( ( ord_less_int @ B2 @ zero_zero_int )
       => ( ord_less_int @ ( signed6292675348222524329lo_int @ A2 @ B2 ) @ ( uminus_uminus_int @ B2 ) ) ) ) ).

% smod_int_compares(5)
thf(fact_9831_smod__int__compares_I3_J,axiom,
    ! [A2: int,B2: int] :
      ( ( ord_less_eq_int @ A2 @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( signed6292675348222524329lo_int @ A2 @ B2 ) ) ) ) ).

% smod_int_compares(3)
thf(fact_9832_uint32_Osize__eq__length,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
    = ( type_l796852477590012082l_num1 @ type_N8448461349408098053l_num1 ) ) ).

% uint32.size_eq_length
thf(fact_9833_len__num0,axiom,
    ( type_l4264026598287037464l_num0
    = ( ^ [Uu4: itself_Numeral_num0] : zero_zero_nat ) ) ).

% len_num0
thf(fact_9834_len__of__finite__2__def,axiom,
    ( type_l31302759751748492nite_2
    = ( ^ [X3: itself_finite_2] : ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% len_of_finite_2_def
thf(fact_9835_len__of__finite__3__def,axiom,
    ( type_l31302759751748493nite_3
    = ( ^ [X3: itself_finite_3] : ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% len_of_finite_3_def
thf(fact_9836_min__Suc__Suc,axiom,
    ! [M: nat,N3: nat] :
      ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N3 ) )
      = ( suc @ ( ord_min_nat @ M @ N3 ) ) ) ).

% min_Suc_Suc
thf(fact_9837_min__0R,axiom,
    ! [N3: nat] :
      ( ( ord_min_nat @ N3 @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_9838_min__0L,axiom,
    ! [N3: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N3 )
      = zero_zero_nat ) ).

% min_0L
thf(fact_9839_min__minus_H,axiom,
    ! [M: nat,K: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M @ K ) @ M )
      = ( minus_minus_nat @ M @ K ) ) ).

% min_minus'
thf(fact_9840_min__minus,axiom,
    ! [M: nat,K: nat] :
      ( ( ord_min_nat @ M @ ( minus_minus_nat @ M @ K ) )
      = ( minus_minus_nat @ M @ K ) ) ).

% min_minus
thf(fact_9841_min__Suc__gt_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_min_nat @ B2 @ ( suc @ A2 ) )
        = ( suc @ A2 ) ) ) ).

% min_Suc_gt(2)
thf(fact_9842_min__Suc__gt_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_min_nat @ ( suc @ A2 ) @ B2 )
        = ( suc @ A2 ) ) ) ).

% min_Suc_gt(1)
thf(fact_9843_rev__min__pm1,axiom,
    ! [A2: nat,B2: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ ( ord_min_nat @ B2 @ A2 ) )
      = A2 ) ).

% rev_min_pm1
thf(fact_9844_rev__min__pm,axiom,
    ! [B2: nat,A2: nat] :
      ( ( plus_plus_nat @ ( ord_min_nat @ B2 @ A2 ) @ ( minus_minus_nat @ A2 @ B2 ) )
      = A2 ) ).

% rev_min_pm
thf(fact_9845_min__pm1,axiom,
    ! [A2: nat,B2: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ ( ord_min_nat @ A2 @ B2 ) )
      = A2 ) ).

% min_pm1
thf(fact_9846_min__pm,axiom,
    ! [A2: nat,B2: nat] :
      ( ( plus_plus_nat @ ( ord_min_nat @ A2 @ B2 ) @ ( minus_minus_nat @ A2 @ B2 ) )
      = A2 ) ).

% min_pm
thf(fact_9847_min__numeral__Suc,axiom,
    ! [K: num,N3: nat] :
      ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N3 ) )
      = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N3 ) ) ) ).

% min_numeral_Suc
thf(fact_9848_min__Suc__numeral,axiom,
    ! [N3: nat,K: num] :
      ( ( ord_min_nat @ ( suc @ N3 ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_min_nat @ N3 @ ( pred_numeral @ K ) ) ) ) ).

% min_Suc_numeral
thf(fact_9849_concat__bit__assoc__sym,axiom,
    ! [M: nat,N3: nat,K: int,L2: int,R3: int] :
      ( ( bit_concat_bit @ M @ ( bit_concat_bit @ N3 @ K @ L2 ) @ R3 )
      = ( bit_concat_bit @ ( ord_min_nat @ M @ N3 ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N3 ) @ L2 @ R3 ) ) ) ).

% concat_bit_assoc_sym
thf(fact_9850_min__diff,axiom,
    ! [M: nat,I: nat,N3: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N3 @ I ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M @ N3 ) @ I ) ) ).

% min_diff
thf(fact_9851_nat__mult__min__right,axiom,
    ! [M: nat,N3: nat,Q3: nat] :
      ( ( times_times_nat @ M @ ( ord_min_nat @ N3 @ Q3 ) )
      = ( ord_min_nat @ ( times_times_nat @ M @ N3 ) @ ( times_times_nat @ M @ Q3 ) ) ) ).

% nat_mult_min_right
thf(fact_9852_nat__mult__min__left,axiom,
    ! [M: nat,N3: nat,Q3: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M @ N3 ) @ Q3 )
      = ( ord_min_nat @ ( times_times_nat @ M @ Q3 ) @ ( times_times_nat @ N3 @ Q3 ) ) ) ).

% nat_mult_min_left
thf(fact_9853_take__bit__concat__bit__eq,axiom,
    ! [M: nat,N3: nat,K: int,L2: int] :
      ( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N3 @ K @ L2 ) )
      = ( bit_concat_bit @ ( ord_min_nat @ M @ N3 ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N3 ) @ L2 ) ) ) ).

% take_bit_concat_bit_eq
thf(fact_9854_mod__mod__power,axiom,
    ! [K: nat,M: nat,N3: nat] :
      ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ K @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
      = ( modulo_modulo_nat @ K @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( ord_min_nat @ M @ N3 ) ) ) ) ).

% mod_mod_power
thf(fact_9855_shiftl__Suc__0,axiom,
    ! [N3: nat] :
      ( ( bit_Sh3965577149348748681tl_nat @ ( suc @ zero_zero_nat ) @ N3 )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ).

% shiftl_Suc_0
thf(fact_9856_shiftr__Suc__0,axiom,
    ! [N3: nat] :
      ( ( bit_Sh2154871086232339855tr_nat @ ( suc @ zero_zero_nat ) @ N3 )
      = ( zero_n2687167440665602831ol_nat @ ( N3 = zero_zero_nat ) ) ) ).

% shiftr_Suc_0
thf(fact_9857_inj__on__diff__nat,axiom,
    ! [N7: set_nat,K: nat] :
      ( ! [N: nat] :
          ( ( member_nat @ N @ N7 )
         => ( ord_less_eq_nat @ K @ N ) )
     => ( inj_on_nat_nat
        @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K )
        @ N7 ) ) ).

% inj_on_diff_nat
thf(fact_9858_inj__Suc,axiom,
    ! [N7: set_nat] : ( inj_on_nat_nat @ suc @ N7 ) ).

% inj_Suc
thf(fact_9859_inj__sgn__power,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( inj_on_real_real
        @ ^ [Y2: real] : ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N3 ) )
        @ top_top_set_real ) ) ).

% inj_sgn_power
thf(fact_9860_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D2: nat] :
      ( ( vEBT_VEBT_valid @ T @ D2 )
     => ( vEBT_invar_vebt @ T @ D2 ) ) ).

% valid_eq2
thf(fact_9861_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D2: nat] :
      ( ( vEBT_invar_vebt @ T @ D2 )
     => ( vEBT_VEBT_valid @ T @ D2 ) ) ).

% valid_eq1
thf(fact_9862_valid__eq,axiom,
    vEBT_VEBT_valid = vEBT_invar_vebt ).

% valid_eq
thf(fact_9863_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv2: $o,D2: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv2 ) @ D2 )
      = ( D2 = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
thf(fact_9864_DERIV__real__root__generic,axiom,
    ! [N3: nat,X: real,D: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( X != zero_zero_real )
       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
           => ( ( ord_less_real @ zero_zero_real @ X )
             => ( D
                = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( power_power_real @ ( root @ N3 @ X ) @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
             => ( ( ord_less_real @ X @ zero_zero_real )
               => ( D
                  = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( power_power_real @ ( root @ N3 @ X ) @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
           => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
               => ( D
                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( power_power_real @ ( root @ N3 @ X ) @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
             => ( has_fi5821293074295781190e_real @ ( root @ N3 ) @ D @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).

% DERIV_real_root_generic
thf(fact_9865_DERIV__even__real__root,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
       => ( ( ord_less_real @ X @ zero_zero_real )
         => ( has_fi5821293074295781190e_real @ ( root @ N3 ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ ( power_power_real @ ( root @ N3 @ X ) @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_even_real_root
thf(fact_9866_DERIV__const__average,axiom,
    ! [A2: real,B2: real,V: real > real,K: real] :
      ( ( A2 != B2 )
     => ( ! [X4: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
       => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A2 @ B2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( V @ A2 ) @ ( V @ B2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% DERIV_const_average
thf(fact_9867_DERIV__const__ratio__const,axiom,
    ! [A2: real,B2: real,F2: real > real,K: real] :
      ( ( A2 != B2 )
     => ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F2 @ K @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
       => ( ( minus_minus_real @ ( F2 @ B2 ) @ ( F2 @ A2 ) )
          = ( times_times_real @ ( minus_minus_real @ B2 @ A2 ) @ K ) ) ) ) ).

% DERIV_const_ratio_const
thf(fact_9868_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_9869_DERIV__neg__dec__right,axiom,
    ! [F2: real > real,L2: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F2 @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ L2 @ zero_zero_real )
       => ? [D5: real] :
            ( ( ord_less_real @ zero_zero_real @ D5 )
            & ! [H6: real] :
                ( ( ord_less_real @ zero_zero_real @ H6 )
               => ( ( ord_less_real @ H6 @ D5 )
                 => ( ord_less_real @ ( F2 @ ( plus_plus_real @ X @ H6 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ).

% DERIV_neg_dec_right
thf(fact_9870_DERIV__pos__inc__right,axiom,
    ! [F2: real > real,L2: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F2 @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [D5: real] :
            ( ( ord_less_real @ zero_zero_real @ D5 )
            & ! [H6: real] :
                ( ( ord_less_real @ zero_zero_real @ H6 )
               => ( ( ord_less_real @ H6 @ D5 )
                 => ( ord_less_real @ ( F2 @ X ) @ ( F2 @ ( plus_plus_real @ X @ H6 ) ) ) ) ) ) ) ) ).

% DERIV_pos_inc_right
thf(fact_9871_DERIV__isconst__all,axiom,
    ! [F2: real > real,X: real,Y: real] :
      ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F2 @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( ( F2 @ X )
        = ( F2 @ Y ) ) ) ).

% DERIV_isconst_all
thf(fact_9872_DERIV__mirror,axiom,
    ! [F2: real > real,Y: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F2 @ Y @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ X ) @ top_top_set_real ) )
      = ( has_fi5821293074295781190e_real
        @ ^ [X3: real] : ( F2 @ ( uminus_uminus_real @ X3 ) )
        @ ( uminus_uminus_real @ Y )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_mirror
thf(fact_9873_DERIV__local__const,axiom,
    ! [F2: real > real,L2: real,X: real,D2: real] :
      ( ( has_fi5821293074295781190e_real @ F2 @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D2 )
       => ( ! [Y3: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D2 )
             => ( ( F2 @ X )
                = ( F2 @ Y3 ) ) )
         => ( L2 = zero_zero_real ) ) ) ) ).

% DERIV_local_const
thf(fact_9874_DERIV__pos__inc__left,axiom,
    ! [F2: real > real,L2: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F2 @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [D5: real] :
            ( ( ord_less_real @ zero_zero_real @ D5 )
            & ! [H6: real] :
                ( ( ord_less_real @ zero_zero_real @ H6 )
               => ( ( ord_less_real @ H6 @ D5 )
                 => ( ord_less_real @ ( F2 @ ( minus_minus_real @ X @ H6 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ).

% DERIV_pos_inc_left
thf(fact_9875_DERIV__neg__dec__left,axiom,
    ! [F2: real > real,L2: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F2 @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ L2 @ zero_zero_real )
       => ? [D5: real] :
            ( ( ord_less_real @ zero_zero_real @ D5 )
            & ! [H6: real] :
                ( ( ord_less_real @ zero_zero_real @ H6 )
               => ( ( ord_less_real @ H6 @ D5 )
                 => ( ord_less_real @ ( F2 @ X ) @ ( F2 @ ( minus_minus_real @ X @ H6 ) ) ) ) ) ) ) ) ).

% DERIV_neg_dec_left
thf(fact_9876_DERIV__const__ratio__const2,axiom,
    ! [A2: real,B2: real,F2: real > real,K: real] :
      ( ( A2 != B2 )
     => ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F2 @ K @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
       => ( ( divide_divide_real @ ( minus_minus_real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) @ ( minus_minus_real @ B2 @ A2 ) )
          = K ) ) ) ).

% DERIV_const_ratio_const2
thf(fact_9877_has__real__derivative__neg__dec__left,axiom,
    ! [F2: real > real,L2: real,X: real,S: set_real] :
      ( ( has_fi5821293074295781190e_real @ F2 @ L2 @ ( topolo2177554685111907308n_real @ X @ S ) )
     => ( ( ord_less_real @ L2 @ zero_zero_real )
       => ? [D5: real] :
            ( ( ord_less_real @ zero_zero_real @ D5 )
            & ! [H6: real] :
                ( ( ord_less_real @ zero_zero_real @ H6 )
               => ( ( member_real @ ( minus_minus_real @ X @ H6 ) @ S )
                 => ( ( ord_less_real @ H6 @ D5 )
                   => ( ord_less_real @ ( F2 @ X ) @ ( F2 @ ( minus_minus_real @ X @ H6 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
thf(fact_9878_has__real__derivative__pos__inc__left,axiom,
    ! [F2: real > real,L2: real,X: real,S: set_real] :
      ( ( has_fi5821293074295781190e_real @ F2 @ L2 @ ( topolo2177554685111907308n_real @ X @ S ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [D5: real] :
            ( ( ord_less_real @ zero_zero_real @ D5 )
            & ! [H6: real] :
                ( ( ord_less_real @ zero_zero_real @ H6 )
               => ( ( member_real @ ( minus_minus_real @ X @ H6 ) @ S )
                 => ( ( ord_less_real @ H6 @ D5 )
                   => ( ord_less_real @ ( F2 @ ( minus_minus_real @ X @ H6 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
thf(fact_9879_has__real__derivative__pos__inc__right,axiom,
    ! [F2: real > real,L2: real,X: real,S: set_real] :
      ( ( has_fi5821293074295781190e_real @ F2 @ L2 @ ( topolo2177554685111907308n_real @ X @ S ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [D5: real] :
            ( ( ord_less_real @ zero_zero_real @ D5 )
            & ! [H6: real] :
                ( ( ord_less_real @ zero_zero_real @ H6 )
               => ( ( member_real @ ( plus_plus_real @ X @ H6 ) @ S )
                 => ( ( ord_less_real @ H6 @ D5 )
                   => ( ord_less_real @ ( F2 @ X ) @ ( F2 @ ( plus_plus_real @ X @ H6 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
thf(fact_9880_has__real__derivative__neg__dec__right,axiom,
    ! [F2: real > real,L2: real,X: real,S: set_real] :
      ( ( has_fi5821293074295781190e_real @ F2 @ L2 @ ( topolo2177554685111907308n_real @ X @ S ) )
     => ( ( ord_less_real @ L2 @ zero_zero_real )
       => ? [D5: real] :
            ( ( ord_less_real @ zero_zero_real @ D5 )
            & ! [H6: real] :
                ( ( ord_less_real @ zero_zero_real @ H6 )
               => ( ( member_real @ ( plus_plus_real @ X @ H6 ) @ S )
                 => ( ( ord_less_real @ H6 @ D5 )
                   => ( ord_less_real @ ( F2 @ ( plus_plus_real @ X @ H6 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
thf(fact_9881_deriv__nonneg__imp__mono,axiom,
    ! [A2: real,B2: real,G: real > real,G2: real > real] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ A2 @ B2 ) )
         => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ A2 @ B2 ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X4 ) ) )
       => ( ( ord_less_eq_real @ A2 @ B2 )
         => ( ord_less_eq_real @ ( G @ A2 ) @ ( G @ B2 ) ) ) ) ) ).

% deriv_nonneg_imp_mono
thf(fact_9882_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A2: real,B2: real,F2: real > real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ! [X4: real] :
            ( ( ord_less_eq_real @ A2 @ X4 )
           => ( ( ord_less_eq_real @ X4 @ B2 )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F2 @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ Y4 @ zero_zero_real ) ) ) )
       => ( ord_less_eq_real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) ) ) ).

% DERIV_nonpos_imp_nonincreasing
thf(fact_9883_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A2: real,B2: real,F2: real > real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ! [X4: real] :
            ( ( ord_less_eq_real @ A2 @ X4 )
           => ( ( ord_less_eq_real @ X4 @ B2 )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F2 @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ zero_zero_real @ Y4 ) ) ) )
       => ( ord_less_eq_real @ ( F2 @ A2 ) @ ( F2 @ B2 ) ) ) ) ).

% DERIV_nonneg_imp_nondecreasing
thf(fact_9884_DERIV__neg__imp__decreasing,axiom,
    ! [A2: real,B2: real,F2: real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ! [X4: real] :
            ( ( ord_less_eq_real @ A2 @ X4 )
           => ( ( ord_less_eq_real @ X4 @ B2 )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F2 @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
                  & ( ord_less_real @ Y4 @ zero_zero_real ) ) ) )
       => ( ord_less_real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) ) ) ).

% DERIV_neg_imp_decreasing
thf(fact_9885_DERIV__pos__imp__increasing,axiom,
    ! [A2: real,B2: real,F2: real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ! [X4: real] :
            ( ( ord_less_eq_real @ A2 @ X4 )
           => ( ( ord_less_eq_real @ X4 @ B2 )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F2 @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y4 ) ) ) )
       => ( ord_less_real @ ( F2 @ A2 ) @ ( F2 @ B2 ) ) ) ) ).

% DERIV_pos_imp_increasing
thf(fact_9886_MVT2,axiom,
    ! [A2: real,B2: real,F2: real > real,F5: real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ! [X4: real] :
            ( ( ord_less_eq_real @ A2 @ X4 )
           => ( ( ord_less_eq_real @ X4 @ B2 )
             => ( has_fi5821293074295781190e_real @ F2 @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
       => ? [Z2: real] :
            ( ( ord_less_real @ A2 @ Z2 )
            & ( ord_less_real @ Z2 @ B2 )
            & ( ( minus_minus_real @ ( F2 @ B2 ) @ ( F2 @ A2 ) )
              = ( times_times_real @ ( minus_minus_real @ B2 @ A2 ) @ ( F5 @ Z2 ) ) ) ) ) ) ).

% MVT2
thf(fact_9887_DERIV__local__max,axiom,
    ! [F2: real > real,L2: real,X: real,D2: real] :
      ( ( has_fi5821293074295781190e_real @ F2 @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D2 )
       => ( ! [Y3: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D2 )
             => ( ord_less_eq_real @ ( F2 @ Y3 ) @ ( F2 @ X ) ) )
         => ( L2 = zero_zero_real ) ) ) ) ).

% DERIV_local_max
thf(fact_9888_DERIV__local__min,axiom,
    ! [F2: real > real,L2: real,X: real,D2: real] :
      ( ( has_fi5821293074295781190e_real @ F2 @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D2 )
       => ( ! [Y3: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y3 ) ) @ D2 )
             => ( ord_less_eq_real @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
         => ( L2 = zero_zero_real ) ) ) ) ).

% DERIV_local_min
thf(fact_9889_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_9890_DERIV__pow,axiom,
    ! [N3: nat,X: real,S2: set_real] :
      ( has_fi5821293074295781190e_real
      @ ^ [X3: real] : ( power_power_real @ X3 @ N3 )
      @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( power_power_real @ X @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) )
      @ ( topolo2177554685111907308n_real @ X @ S2 ) ) ).

% DERIV_pow
thf(fact_9891_DERIV__fun__pow,axiom,
    ! [G: real > real,M: real,X: real,N3: nat] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( has_fi5821293074295781190e_real
        @ ^ [X3: real] : ( power_power_real @ ( G @ X3 ) @ N3 )
        @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( power_power_real @ ( G @ X ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) @ M )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_fun_pow
thf(fact_9892_has__real__derivative__powr,axiom,
    ! [Z: real,R3: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( has_fi5821293074295781190e_real
        @ ^ [Z5: real] : ( powr_real @ Z5 @ R3 )
        @ ( times_times_real @ R3 @ ( powr_real @ Z @ ( minus_minus_real @ R3 @ one_one_real ) ) )
        @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).

% has_real_derivative_powr
thf(fact_9893_DERIV__fun__powr,axiom,
    ! [G: real > real,M: real,X: real,R3: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
       => ( has_fi5821293074295781190e_real
          @ ^ [X3: real] : ( powr_real @ ( G @ X3 ) @ R3 )
          @ ( times_times_real @ ( times_times_real @ R3 @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R3 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_fun_powr
thf(fact_9894_DERIV__log,axiom,
    ! [X: real,B2: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ ( log @ B2 ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B2 ) @ X ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_log
thf(fact_9895_DERIV__powr,axiom,
    ! [G: real > real,M: real,X: real,F2: real > real,R3: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
       => ( ( has_fi5821293074295781190e_real @ F2 @ R3 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( has_fi5821293074295781190e_real
            @ ^ [X3: real] : ( powr_real @ ( G @ X3 ) @ ( F2 @ X3 ) )
            @ ( times_times_real @ ( powr_real @ ( G @ X ) @ ( F2 @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R3 @ ( ln_ln_real @ ( G @ X ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F2 @ X ) ) @ ( G @ X ) ) ) )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_powr
thf(fact_9896_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_9897_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_9898_arsinh__real__has__field__derivative,axiom,
    ! [X: real,A: 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 @ A ) ) ).

% arsinh_real_has_field_derivative
thf(fact_9899_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D: real] :
      ( ( X != zero_zero_real )
     => ( ( ( ord_less_real @ zero_zero_real @ X )
         => ( D
            = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( ( ord_less_real @ X @ zero_zero_real )
           => ( D
              = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
         => ( has_fi5821293074295781190e_real @ sqrt @ D @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_real_sqrt_generic
thf(fact_9900_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A: 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 @ A ) ) ) ).

% arcosh_real_has_field_derivative
thf(fact_9901_artanh__real__has__field__derivative,axiom,
    ! [X: real,A: 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 @ A ) ) ) ).

% artanh_real_has_field_derivative
thf(fact_9902_DERIV__real__root,axiom,
    ! [N3: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( has_fi5821293074295781190e_real @ ( root @ N3 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( power_power_real @ ( root @ N3 @ X ) @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_real_root
thf(fact_9903_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_9904_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_9905_Maclaurin__all__le__objl,axiom,
    ! [Diff: nat > real > real,F2: real > real,X: real,N3: nat] :
      ( ( ( ( Diff @ zero_zero_nat )
          = F2 )
        & ! [M4: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
     => ? [T6: real] :
          ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
          & ( ( F2 @ X )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
                @ ( set_ord_lessThan_nat @ N3 ) )
              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N3 @ T6 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ) ).

% Maclaurin_all_le_objl
thf(fact_9906_Maclaurin__all__le,axiom,
    ! [Diff: nat > real > real,F2: real > real,X: real,N3: nat] :
      ( ( ( Diff @ zero_zero_nat )
        = F2 )
     => ( ! [M4: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
       => ? [T6: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
            & ( ( F2 @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N3 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N3 @ T6 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ) ) ).

% Maclaurin_all_le
thf(fact_9907_DERIV__odd__real__root,axiom,
    ! [N3: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
     => ( ( X != zero_zero_real )
       => ( has_fi5821293074295781190e_real @ ( root @ N3 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( power_power_real @ ( root @ N3 @ X ) @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_odd_real_root
thf(fact_9908_Maclaurin,axiom,
    ! [H2: real,N3: nat,Diff: nat > real > real,F2: real > real] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( ( Diff @ zero_zero_nat )
            = F2 )
         => ( ! [M4: nat,T6: real] :
                ( ( ( ord_less_nat @ M4 @ N3 )
                  & ( ord_less_eq_real @ zero_zero_real @ T6 )
                  & ( ord_less_eq_real @ T6 @ H2 ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
           => ? [T6: real] :
                ( ( ord_less_real @ zero_zero_real @ T6 )
                & ( ord_less_real @ T6 @ H2 )
                & ( ( F2 @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H2 @ M5 ) )
                      @ ( set_ord_lessThan_nat @ N3 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N3 @ T6 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ H2 @ N3 ) ) ) ) ) ) ) ) ) ).

% Maclaurin
thf(fact_9909_Maclaurin2,axiom,
    ! [H2: real,Diff: nat > real > real,F2: real > real,N3: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F2 )
       => ( ! [M4: nat,T6: real] :
              ( ( ( ord_less_nat @ M4 @ N3 )
                & ( ord_less_eq_real @ zero_zero_real @ T6 )
                & ( ord_less_eq_real @ T6 @ H2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ? [T6: real] :
              ( ( ord_less_real @ zero_zero_real @ T6 )
              & ( ord_less_eq_real @ T6 @ H2 )
              & ( ( F2 @ H2 )
                = ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H2 @ M5 ) )
                    @ ( set_ord_lessThan_nat @ N3 ) )
                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N3 @ T6 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ H2 @ N3 ) ) ) ) ) ) ) ) ).

% Maclaurin2
thf(fact_9910_Maclaurin__minus,axiom,
    ! [H2: real,N3: nat,Diff: nat > real > real,F2: real > real] :
      ( ( ord_less_real @ H2 @ zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( ( Diff @ zero_zero_nat )
            = F2 )
         => ( ! [M4: nat,T6: real] :
                ( ( ( ord_less_nat @ M4 @ N3 )
                  & ( ord_less_eq_real @ H2 @ T6 )
                  & ( ord_less_eq_real @ T6 @ zero_zero_real ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
           => ? [T6: real] :
                ( ( ord_less_real @ H2 @ T6 )
                & ( ord_less_real @ T6 @ zero_zero_real )
                & ( ( F2 @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H2 @ M5 ) )
                      @ ( set_ord_lessThan_nat @ N3 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N3 @ T6 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ H2 @ N3 ) ) ) ) ) ) ) ) ) ).

% Maclaurin_minus
thf(fact_9911_Maclaurin__all__lt,axiom,
    ! [Diff: nat > real > real,F2: real > real,N3: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( X != zero_zero_real )
         => ( ! [M4: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
           => ? [T6: real] :
                ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
                & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
                & ( ( F2 @ X )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
                      @ ( set_ord_lessThan_nat @ N3 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N3 @ T6 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ) ) ) ) ).

% Maclaurin_all_lt
thf(fact_9912_Maclaurin__bi__le,axiom,
    ! [Diff: nat > real > real,F2: real > real,N3: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F2 )
     => ( ! [M4: nat,T6: real] :
            ( ( ( ord_less_nat @ M4 @ N3 )
              & ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) ) )
           => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
       => ? [T6: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
            & ( ( F2 @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N3 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N3 @ T6 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ X @ N3 ) ) ) ) ) ) ) ).

% Maclaurin_bi_le
thf(fact_9913_Taylor,axiom,
    ! [N3: nat,Diff: nat > real > real,F2: real > real,A2: real,B2: real,C2: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ( Diff @ zero_zero_nat )
          = F2 )
       => ( ! [M4: nat,T6: real] :
              ( ( ( ord_less_nat @ M4 @ N3 )
                & ( ord_less_eq_real @ A2 @ T6 )
                & ( ord_less_eq_real @ T6 @ B2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A2 @ C2 )
           => ( ( ord_less_eq_real @ C2 @ B2 )
             => ( ( ord_less_eq_real @ A2 @ X )
               => ( ( ord_less_eq_real @ X @ B2 )
                 => ( ( X != C2 )
                   => ? [T6: real] :
                        ( ( ( ord_less_real @ X @ C2 )
                         => ( ( ord_less_real @ X @ T6 )
                            & ( ord_less_real @ T6 @ C2 ) ) )
                        & ( ~ ( ord_less_real @ X @ C2 )
                         => ( ( ord_less_real @ C2 @ T6 )
                            & ( ord_less_real @ T6 @ X ) ) )
                        & ( ( F2 @ X )
                          = ( plus_plus_real
                            @ ( groups6591440286371151544t_real
                              @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C2 ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C2 ) @ M5 ) )
                              @ ( set_ord_lessThan_nat @ N3 ) )
                            @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N3 @ T6 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C2 ) @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
thf(fact_9914_Taylor__up,axiom,
    ! [N3: nat,Diff: nat > real > real,F2: real > real,A2: real,B2: real,C2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ( Diff @ zero_zero_nat )
          = F2 )
       => ( ! [M4: nat,T6: real] :
              ( ( ( ord_less_nat @ M4 @ N3 )
                & ( ord_less_eq_real @ A2 @ T6 )
                & ( ord_less_eq_real @ T6 @ B2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A2 @ C2 )
           => ( ( ord_less_real @ C2 @ B2 )
             => ? [T6: real] :
                  ( ( ord_less_real @ C2 @ T6 )
                  & ( ord_less_real @ T6 @ B2 )
                  & ( ( F2 @ B2 )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C2 ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ B2 @ C2 ) @ M5 ) )
                        @ ( set_ord_lessThan_nat @ N3 ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N3 @ T6 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ ( minus_minus_real @ B2 @ C2 ) @ N3 ) ) ) ) ) ) ) ) ) ) ).

% Taylor_up
thf(fact_9915_Taylor__down,axiom,
    ! [N3: nat,Diff: nat > real > real,F2: real > real,A2: real,B2: real,C2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( ( Diff @ zero_zero_nat )
          = F2 )
       => ( ! [M4: nat,T6: real] :
              ( ( ( ord_less_nat @ M4 @ N3 )
                & ( ord_less_eq_real @ A2 @ T6 )
                & ( ord_less_eq_real @ T6 @ B2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_real @ A2 @ C2 )
           => ( ( ord_less_eq_real @ C2 @ B2 )
             => ? [T6: real] :
                  ( ( ord_less_real @ A2 @ T6 )
                  & ( ord_less_real @ T6 @ C2 )
                  & ( ( F2 @ A2 )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C2 ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ A2 @ C2 ) @ M5 ) )
                        @ ( set_ord_lessThan_nat @ N3 ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N3 @ T6 ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ ( power_power_real @ ( minus_minus_real @ A2 @ C2 ) @ N3 ) ) ) ) ) ) ) ) ) ) ).

% Taylor_down
thf(fact_9916_Maclaurin__lemma2,axiom,
    ! [N3: nat,H2: real,Diff: nat > real > real,K: nat,B: real] :
      ( ! [M4: nat,T6: real] :
          ( ( ( ord_less_nat @ M4 @ N3 )
            & ( ord_less_eq_real @ zero_zero_real @ T6 )
            & ( ord_less_eq_real @ T6 @ H2 ) )
         => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
     => ( ( N3
          = ( suc @ K ) )
       => ! [M2: nat,T7: real] :
            ( ( ( ord_less_nat @ M2 @ N3 )
              & ( 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
                      @ ^ [P4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M2 @ P4 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_real @ U2 @ P4 ) )
                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N3 @ M2 ) ) )
                    @ ( times_times_real @ B @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N3 @ M2 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N3 @ M2 ) ) ) ) ) )
              @ ( minus_minus_real @ ( Diff @ ( suc @ M2 ) @ T7 )
                @ ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [P4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M2 ) @ P4 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_real @ T7 @ P4 ) )
                    @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N3 @ ( suc @ M2 ) ) ) )
                  @ ( times_times_real @ B @ ( divide_divide_real @ ( power_power_real @ T7 @ ( minus_minus_nat @ N3 @ ( suc @ M2 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N3 @ ( suc @ M2 ) ) ) ) ) ) )
              @ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) ) ) ) ).

% Maclaurin_lemma2
thf(fact_9917_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_9918_DERIV__power__series_H,axiom,
    ! [R: real,F2: nat > real,X0: real] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
         => ( summable_real
            @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F2 @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X4 @ N2 ) ) ) )
     => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
       => ( ( ord_less_real @ zero_zero_real @ R )
         => ( has_fi5821293074295781190e_real
            @ ^ [X3: real] :
                ( suminf_real
                @ ^ [N2: nat] : ( times_times_real @ ( F2 @ N2 ) @ ( power_power_real @ X3 @ ( suc @ N2 ) ) ) )
            @ ( suminf_real
              @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F2 @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X0 @ N2 ) ) )
            @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).

% DERIV_power_series'
thf(fact_9919_DERIV__isconst3,axiom,
    ! [A2: real,B2: real,X: real,Y: real,F2: real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( member_real @ X @ ( set_or1633881224788618240n_real @ A2 @ B2 ) )
       => ( ( member_real @ Y @ ( set_or1633881224788618240n_real @ A2 @ B2 ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A2 @ B2 ) )
               => ( has_fi5821293074295781190e_real @ F2 @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
           => ( ( F2 @ X )
              = ( F2 @ Y ) ) ) ) ) ) ).

% DERIV_isconst3
thf(fact_9920_DERIV__series_H,axiom,
    ! [F2: real > nat > real,F5: real > nat > real,X0: real,A2: real,B2: real,L5: nat > real] :
      ( ! [N: nat] :
          ( has_fi5821293074295781190e_real
          @ ^ [X3: real] : ( F2 @ X3 @ N )
          @ ( F5 @ X0 @ N )
          @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A2 @ B2 ) )
           => ( summable_real @ ( F2 @ X4 ) ) )
       => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A2 @ B2 ) )
         => ( ( summable_real @ ( F5 @ X0 ) )
           => ( ( summable_real @ L5 )
             => ( ! [N: nat,X4: real,Y3: real] :
                    ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A2 @ B2 ) )
                   => ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A2 @ B2 ) )
                     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F2 @ X4 @ N ) @ ( F2 @ Y3 @ N ) ) ) @ ( times_times_real @ ( L5 @ N ) @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y3 ) ) ) ) ) )
               => ( has_fi5821293074295781190e_real
                  @ ^ [X3: real] : ( suminf_real @ ( F2 @ X3 ) )
                  @ ( suminf_real @ ( F5 @ X0 ) )
                  @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_series'
thf(fact_9921_finite__greaterThanLessThan,axiom,
    ! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L2 @ U ) ) ).

% finite_greaterThanLessThan
thf(fact_9922_finite__greaterThanLessThan__int,axiom,
    ! [L2: int,U: int] : ( finite_finite_int @ ( set_or5832277885323065728an_int @ L2 @ U ) ) ).

% finite_greaterThanLessThan_int
thf(fact_9923_finite__greaterThanLessThan__integer,axiom,
    ! [L2: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or4266950643985792945nteger @ L2 @ U ) ) ).

% finite_greaterThanLessThan_integer
thf(fact_9924_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L2: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ L2 ) @ U )
      = ( set_or5834768355832116004an_nat @ L2 @ U ) ) ).

% atLeastSucLessThan_greaterThanLessThan
thf(fact_9925_isCont__Lb__Ub,axiom,
    ! [A2: real,B2: real,F2: real > real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ! [X4: real] :
            ( ( ( ord_less_eq_real @ A2 @ X4 )
              & ( ord_less_eq_real @ X4 @ B2 ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F2 ) )
       => ? [L6: real,M10: real] :
            ( ! [X5: real] :
                ( ( ( ord_less_eq_real @ A2 @ X5 )
                  & ( ord_less_eq_real @ X5 @ B2 ) )
               => ( ( ord_less_eq_real @ L6 @ ( F2 @ X5 ) )
                  & ( ord_less_eq_real @ ( F2 @ X5 ) @ M10 ) ) )
            & ! [Y4: real] :
                ( ( ( ord_less_eq_real @ L6 @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ M10 ) )
               => ? [X4: real] :
                    ( ( ord_less_eq_real @ A2 @ X4 )
                    & ( ord_less_eq_real @ X4 @ B2 )
                    & ( ( F2 @ X4 )
                      = Y4 ) ) ) ) ) ) ).

% isCont_Lb_Ub
thf(fact_9926_LIM__fun__gt__zero,axiom,
    ! [F2: real > real,L2: real,C2: real] :
      ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [R2: real] :
            ( ( ord_less_real @ zero_zero_real @ R2 )
            & ! [X5: real] :
                ( ( ( X5 != C2 )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C2 @ X5 ) ) @ R2 ) )
               => ( ord_less_real @ zero_zero_real @ ( F2 @ X5 ) ) ) ) ) ) ).

% LIM_fun_gt_zero
thf(fact_9927_LIM__fun__not__zero,axiom,
    ! [F2: real > real,L2: real,C2: real] :
      ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
     => ( ( L2 != zero_zero_real )
       => ? [R2: real] :
            ( ( ord_less_real @ zero_zero_real @ R2 )
            & ! [X5: real] :
                ( ( ( X5 != C2 )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C2 @ X5 ) ) @ R2 ) )
               => ( ( F2 @ X5 )
                 != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
thf(fact_9928_LIM__fun__less__zero,axiom,
    ! [F2: real > real,L2: real,C2: real] :
      ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
     => ( ( ord_less_real @ L2 @ zero_zero_real )
       => ? [R2: real] :
            ( ( ord_less_real @ zero_zero_real @ R2 )
            & ! [X5: real] :
                ( ( ( X5 != C2 )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C2 @ X5 ) ) @ R2 ) )
               => ( ord_less_real @ ( F2 @ X5 ) @ zero_zero_real ) ) ) ) ) ).

% LIM_fun_less_zero
thf(fact_9929_isCont__real__sqrt,axiom,
    ! [X: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ sqrt ) ).

% isCont_real_sqrt
thf(fact_9930_isCont__real__root,axiom,
    ! [X: real,N3: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ( root @ N3 ) ) ).

% isCont_real_root
thf(fact_9931_continuous__frac,axiom,
    ! [X: real] :
      ( ~ ( member_real @ X @ ring_1_Ints_real )
     => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ archim2898591450579166408c_real ) ) ).

% continuous_frac
thf(fact_9932_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L2: int,U: int] :
      ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L2 @ one_one_int ) @ U )
      = ( set_or5832277885323065728an_int @ L2 @ U ) ) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_9933_isCont__inverse__function2,axiom,
    ! [A2: real,X: real,B2: real,G: real > real,F2: real > real] :
      ( ( ord_less_real @ A2 @ X )
     => ( ( ord_less_real @ X @ B2 )
       => ( ! [Z2: real] :
              ( ( ord_less_eq_real @ A2 @ Z2 )
             => ( ( ord_less_eq_real @ Z2 @ B2 )
               => ( ( G @ ( F2 @ Z2 ) )
                  = Z2 ) ) )
         => ( ! [Z2: real] :
                ( ( ord_less_eq_real @ A2 @ Z2 )
               => ( ( ord_less_eq_real @ Z2 @ B2 )
                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F2 ) ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F2 @ X ) @ top_top_set_real ) @ G ) ) ) ) ) ).

% isCont_inverse_function2
thf(fact_9934_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_9935_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L2: code_integer,U: code_integer] :
      ( ( set_or8404916559141939852nteger @ ( plus_p5714425477246183910nteger @ L2 @ one_one_Code_integer ) @ U )
      = ( set_or4266950643985792945nteger @ L2 @ U ) ) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
thf(fact_9936_isCont__arcosh,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcosh_real ) ) ).

% isCont_arcosh
thf(fact_9937_LIM__cos__div__sin,axiom,
    ( filterlim_real_real
    @ ^ [X3: real] : ( divide_divide_real @ ( cos_real @ X3 ) @ ( sin_real @ X3 ) )
    @ ( 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_9938_DERIV__inverse__function,axiom,
    ! [F2: real > real,D: real,G: real > real,X: real,A2: real,B2: real] :
      ( ( has_fi5821293074295781190e_real @ F2 @ D @ ( topolo2177554685111907308n_real @ ( G @ X ) @ top_top_set_real ) )
     => ( ( D != zero_zero_real )
       => ( ( ord_less_real @ A2 @ X )
         => ( ( ord_less_real @ X @ B2 )
           => ( ! [Y3: real] :
                  ( ( ord_less_real @ A2 @ Y3 )
                 => ( ( ord_less_real @ Y3 @ B2 )
                   => ( ( F2 @ ( G @ Y3 ) )
                      = Y3 ) ) )
             => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ G )
               => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_inverse_function
thf(fact_9939_isCont__arccos,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arccos ) ) ) ).

% isCont_arccos
thf(fact_9940_isCont__arcsin,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcsin ) ) ) ).

% isCont_arcsin
thf(fact_9941_LIM__less__bound,axiom,
    ! [B2: real,X: real,F2: real > real] :
      ( ( ord_less_real @ B2 @ X )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ B2 @ X ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X4 ) ) )
       => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X ) ) ) ) ) ).

% LIM_less_bound
thf(fact_9942_isCont__artanh,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ artanh_real ) ) ) ).

% isCont_artanh
thf(fact_9943_isCont__inverse__function,axiom,
    ! [D2: real,X: real,G: real > real,F2: real > real] :
      ( ( ord_less_real @ zero_zero_real @ D2 )
     => ( ! [Z2: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ X ) ) @ D2 )
           => ( ( G @ ( F2 @ Z2 ) )
              = Z2 ) )
       => ( ! [Z2: real] :
              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ X ) ) @ D2 )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F2 ) )
         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F2 @ X ) @ top_top_set_real ) @ G ) ) ) ) ).

% isCont_inverse_function
thf(fact_9944_GMVT_H,axiom,
    ! [A2: real,B2: real,F2: real > real,G: real > real,G2: real > real,F5: real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ! [Z2: real] :
            ( ( ord_less_eq_real @ A2 @ Z2 )
           => ( ( ord_less_eq_real @ Z2 @ B2 )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F2 ) ) )
       => ( ! [Z2: real] :
              ( ( ord_less_eq_real @ A2 @ Z2 )
             => ( ( ord_less_eq_real @ Z2 @ B2 )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ G ) ) )
         => ( ! [Z2: real] :
                ( ( ord_less_real @ A2 @ Z2 )
               => ( ( ord_less_real @ Z2 @ B2 )
                 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z2 ) @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) )
           => ( ! [Z2: real] :
                  ( ( ord_less_real @ A2 @ Z2 )
                 => ( ( ord_less_real @ Z2 @ B2 )
                   => ( has_fi5821293074295781190e_real @ F2 @ ( F5 @ Z2 ) @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) )
             => ? [C4: real] :
                  ( ( ord_less_real @ A2 @ C4 )
                  & ( ord_less_real @ C4 @ B2 )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) @ ( G2 @ C4 ) )
                    = ( times_times_real @ ( minus_minus_real @ ( G @ B2 ) @ ( G @ A2 ) ) @ ( F5 @ C4 ) ) ) ) ) ) ) ) ) ).

% GMVT'
thf(fact_9945_summable__Leibniz_I3_J,axiom,
    ! [A2: nat > real] :
      ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A2 )
       => ( ( ord_less_real @ ( A2 @ zero_zero_nat ) @ zero_zero_real )
         => ! [N10: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A2 @ I2 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A2 @ I2 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N10 ) @ one_one_nat ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A2 @ I2 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N10 ) ) ) ) ) ) ) ) ).

% summable_Leibniz(3)
thf(fact_9946_summable__Leibniz_I2_J,axiom,
    ! [A2: nat > real] :
      ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A2 )
       => ( ( ord_less_real @ zero_zero_real @ ( A2 @ zero_zero_nat ) )
         => ! [N10: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A2 @ I2 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A2 @ I2 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N10 ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A2 @ I2 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N10 ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% summable_Leibniz(2)
thf(fact_9947_filterlim__Suc,axiom,
    filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).

% filterlim_Suc
thf(fact_9948_mult__nat__left__at__top,axiom,
    ! [C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C2 )
     => ( filterlim_nat_nat @ ( times_times_nat @ C2 ) @ at_top_nat @ at_top_nat ) ) ).

% mult_nat_left_at_top
thf(fact_9949_mult__nat__right__at__top,axiom,
    ! [C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C2 )
     => ( filterlim_nat_nat
        @ ^ [X3: nat] : ( times_times_nat @ X3 @ C2 )
        @ at_top_nat
        @ at_top_nat ) ) ).

% mult_nat_right_at_top
thf(fact_9950_monoseq__convergent,axiom,
    ! [X2: nat > real,B: real] :
      ( ( topolo6980174941875973593q_real @ X2 )
     => ( ! [I5: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X2 @ I5 ) ) @ B )
       => ~ ! [L6: real] :
              ~ ( filterlim_nat_real @ X2 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).

% monoseq_convergent
thf(fact_9951_LIMSEQ__root,axiom,
    ( filterlim_nat_real
    @ ^ [N2: nat] : ( root @ N2 @ ( semiri5074537144036343181t_real @ N2 ) )
    @ ( topolo2815343760600316023s_real @ one_one_real )
    @ at_top_nat ) ).

% LIMSEQ_root
thf(fact_9952_nested__sequence__unique,axiom,
    ! [F2: nat > real,G: nat > real] :
      ( ! [N: nat] : ( ord_less_eq_real @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
     => ( ! [N: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N ) ) @ ( G @ N ) )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( F2 @ N ) @ ( G @ N ) )
         => ( ( filterlim_nat_real
              @ ^ [N2: nat] : ( minus_minus_real @ ( F2 @ N2 ) @ ( G @ N2 ) )
              @ ( topolo2815343760600316023s_real @ zero_zero_real )
              @ at_top_nat )
           => ? [L4: real] :
                ( ! [N10: nat] : ( ord_less_eq_real @ ( F2 @ N10 ) @ L4 )
                & ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
                & ! [N10: nat] : ( ord_less_eq_real @ L4 @ ( G @ N10 ) )
                & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).

% nested_sequence_unique
thf(fact_9953_LIMSEQ__inverse__zero,axiom,
    ! [X2: nat > real] :
      ( ! [R2: real] :
        ? [N9: nat] :
        ! [N: nat] :
          ( ( ord_less_eq_nat @ N9 @ N )
         => ( ord_less_real @ R2 @ ( X2 @ N ) ) )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( inverse_inverse_real @ ( X2 @ N2 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_zero
thf(fact_9954_lim__inverse__n_H,axiom,
    ( filterlim_nat_real
    @ ^ [N2: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% lim_inverse_n'
thf(fact_9955_LIMSEQ__inverse__real__of__nat,axiom,
    ( filterlim_nat_real
    @ ^ [N2: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat
thf(fact_9956_LIMSEQ__root__const,axiom,
    ! [C2: real] :
      ( ( ord_less_real @ zero_zero_real @ C2 )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( root @ N2 @ C2 )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat ) ) ).

% LIMSEQ_root_const
thf(fact_9957_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R3: real] :
      ( filterlim_nat_real
      @ ^ [N2: nat] : ( plus_plus_real @ R3 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
      @ ( topolo2815343760600316023s_real @ R3 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add
thf(fact_9958_increasing__LIMSEQ,axiom,
    ! [F2: nat > real,L2: real] :
      ( ! [N: nat] : ( ord_less_eq_real @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
     => ( ! [N: nat] : ( ord_less_eq_real @ ( F2 @ N ) @ L2 )
       => ( ! [E2: real] :
              ( ( ord_less_real @ zero_zero_real @ E2 )
             => ? [N10: nat] : ( ord_less_eq_real @ L2 @ ( plus_plus_real @ ( F2 @ N10 ) @ E2 ) ) )
         => ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat ) ) ) ) ).

% increasing_LIMSEQ
thf(fact_9959_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_9960_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A2: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( divide_divide_real @ A2 @ ( power_power_real @ X @ N2 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_divide_realpow_zero
thf(fact_9961_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ C2 ) @ one_one_real )
     => ( filterlim_nat_real @ ( power_power_real @ C2 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).

% LIMSEQ_abs_realpow_zero2
thf(fact_9962_LIMSEQ__abs__realpow__zero,axiom,
    ! [C2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ C2 ) @ one_one_real )
     => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C2 ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).

% LIMSEQ_abs_realpow_zero
thf(fact_9963_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N2 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_realpow_zero
thf(fact_9964_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R3: real] :
      ( filterlim_nat_real
      @ ^ [N2: nat] : ( plus_plus_real @ R3 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R3 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_9965_tendsto__exp__limit__sequentially,axiom,
    ! [X: real] :
      ( filterlim_nat_real
      @ ^ [N2: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
      @ at_top_nat ) ).

% tendsto_exp_limit_sequentially
thf(fact_9966_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R3: real] :
      ( filterlim_nat_real
      @ ^ [N2: nat] : ( times_times_real @ R3 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R3 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_9967_summable__Leibniz_I1_J,axiom,
    ! [A2: nat > real] :
      ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A2 )
       => ( summable_real
          @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A2 @ N2 ) ) ) ) ) ).

% summable_Leibniz(1)
thf(fact_9968_summable,axiom,
    ! [A2: nat > real] :
      ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N ) )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N ) ) @ ( A2 @ N ) )
         => ( summable_real
            @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A2 @ N2 ) ) ) ) ) ) ).

% summable
thf(fact_9969_cos__diff__limit__1,axiom,
    ! [Theta: nat > real,Theta2: real] :
      ( ( filterlim_nat_real
        @ ^ [J: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J ) @ Theta2 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ~ ! [K2: nat > int] :
            ~ ( filterlim_nat_real
              @ ^ [J: nat] : ( minus_minus_real @ ( Theta @ J ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
              @ ( topolo2815343760600316023s_real @ Theta2 )
              @ at_top_nat ) ) ).

% cos_diff_limit_1
thf(fact_9970_cos__limit__1,axiom,
    ! [Theta: nat > real] :
      ( ( filterlim_nat_real
        @ ^ [J: nat] : ( cos_real @ ( Theta @ J ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ? [K2: nat > int] :
          ( filterlim_nat_real
          @ ^ [J: nat] : ( minus_minus_real @ ( Theta @ J ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
          @ ( topolo2815343760600316023s_real @ zero_zero_real )
          @ at_top_nat ) ) ).

% cos_limit_1
thf(fact_9971_summable__Leibniz_I4_J,axiom,
    ! [A2: nat > real] :
      ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A2 )
       => ( filterlim_nat_real
          @ ^ [N2: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A2 @ I2 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A2 @ I2 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(4)
thf(fact_9972_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% zeroseq_arctan_series
thf(fact_9973_summable__Leibniz_H_I3_J,axiom,
    ! [A2: nat > real] :
      ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N ) )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N ) ) @ ( A2 @ N ) )
         => ( filterlim_nat_real
            @ ^ [N2: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A2 @ I2 ) )
                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A2 @ I2 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(3)
thf(fact_9974_summable__Leibniz_H_I2_J,axiom,
    ! [A2: nat > real,N3: nat] :
      ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N ) )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N ) ) @ ( A2 @ N ) )
         => ( ord_less_eq_real
            @ ( groups6591440286371151544t_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A2 @ I2 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
            @ ( suminf_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A2 @ I2 ) ) ) ) ) ) ) ).

% summable_Leibniz'(2)
thf(fact_9975_sums__alternating__upper__lower,axiom,
    ! [A2: nat > real] :
      ( ! [N: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N ) ) @ ( A2 @ N ) )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N ) )
       => ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
         => ? [L4: real] :
              ( ! [N10: nat] :
                  ( ord_less_eq_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A2 @ I2 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N10 ) ) )
                  @ L4 )
              & ( filterlim_nat_real
                @ ^ [N2: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A2 @ I2 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat )
              & ! [N10: nat] :
                  ( ord_less_eq_real @ L4
                  @ ( groups6591440286371151544t_real
                    @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A2 @ I2 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N10 ) @ one_one_nat ) ) ) )
              & ( filterlim_nat_real
                @ ^ [N2: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A2 @ I2 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat ) ) ) ) ) ).

% sums_alternating_upper_lower
thf(fact_9976_summable__Leibniz_I5_J,axiom,
    ! [A2: nat > real] :
      ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A2 )
       => ( filterlim_nat_real
          @ ^ [N2: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A2 @ I2 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A2 @ I2 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(5)
thf(fact_9977_summable__Leibniz_H_I4_J,axiom,
    ! [A2: nat > real,N3: nat] :
      ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N ) )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N ) ) @ ( A2 @ N ) )
         => ( ord_less_eq_real
            @ ( suminf_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A2 @ I2 ) ) )
            @ ( groups6591440286371151544t_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A2 @ I2 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ) ) ) ) ).

% summable_Leibniz'(4)
thf(fact_9978_summable__Leibniz_H_I5_J,axiom,
    ! [A2: nat > real] :
      ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N ) )
       => ( ! [N: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N ) ) @ ( A2 @ N ) )
         => ( filterlim_nat_real
            @ ^ [N2: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A2 @ I2 ) )
                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A2 @ I2 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(5)
thf(fact_9979_real__bounded__linear,axiom,
    ( real_V5970128139526366754l_real
    = ( ^ [F8: real > real] :
        ? [C5: real] :
          ( F8
          = ( ^ [X3: real] : ( times_times_real @ X3 @ C5 ) ) ) ) ) ).

% real_bounded_linear
thf(fact_9980_dist__complex__def,axiom,
    ( real_V3694042436643373181omplex
    = ( ^ [X3: complex,Y2: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Y2 ) ) ) ) ).

% dist_complex_def
thf(fact_9981_dist__real__def,axiom,
    ( real_V975177566351809787t_real
    = ( ^ [X3: real,Y2: real] : ( abs_abs_real @ ( minus_minus_real @ X3 @ Y2 ) ) ) ) ).

% dist_real_def
thf(fact_9982_tendsto__exp__limit__at__right,axiom,
    ! [X: real] :
      ( filterlim_real_real
      @ ^ [Y2: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X @ Y2 ) ) @ ( divide_divide_real @ one_one_real @ Y2 ) )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
      @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).

% tendsto_exp_limit_at_right
thf(fact_9983_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_9984_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_9985_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_9986_exp__at__bot,axiom,
    filterlim_real_real @ exp_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_bot_real ).

% exp_at_bot
thf(fact_9987_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_9988_log__inj,axiom,
    ! [B2: real] :
      ( ( ord_less_real @ one_one_real @ B2 )
     => ( inj_on_real_real @ ( log @ B2 ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).

% log_inj
thf(fact_9989_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_9990_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B2: real,F2: real > real,Flim: real] :
      ( ! [X4: real] :
          ( ( ord_less_eq_real @ X4 @ B2 )
         => ? [Y4: real] :
              ( ( has_fi5821293074295781190e_real @ F2 @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
              & ( ord_less_real @ zero_zero_real @ Y4 ) ) )
     => ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
       => ( ord_less_real @ Flim @ ( F2 @ B2 ) ) ) ) ).

% DERIV_pos_imp_increasing_at_bot
thf(fact_9991_filterlim__pow__at__bot__odd,axiom,
    ! [N3: nat,F2: real > real,F: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( filterlim_real_real @ F2 @ at_bot_real @ F )
       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
         => ( filterlim_real_real
            @ ^ [X3: real] : ( power_power_real @ ( F2 @ X3 ) @ N3 )
            @ at_bot_real
            @ F ) ) ) ) ).

% filterlim_pow_at_bot_odd
thf(fact_9992_filterlim__pow__at__bot__even,axiom,
    ! [N3: nat,F2: real > real,F: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( filterlim_real_real @ F2 @ at_bot_real @ F )
       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
         => ( filterlim_real_real
            @ ^ [X3: real] : ( power_power_real @ ( F2 @ X3 ) @ N3 )
            @ at_top_real
            @ F ) ) ) ) ).

% filterlim_pow_at_bot_even
thf(fact_9993_sqrt__at__top,axiom,
    filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).

% sqrt_at_top
thf(fact_9994_greaterThan__0,axiom,
    ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
    = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).

% greaterThan_0
thf(fact_9995_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_9996_ln__x__over__x__tendsto__0,axiom,
    ( filterlim_real_real
    @ ^ [X3: real] : ( divide_divide_real @ ( ln_ln_real @ X3 ) @ X3 )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_real ) ).

% ln_x_over_x_tendsto_0
thf(fact_9997_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_9998_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_9999_tendsto__power__div__exp__0,axiom,
    ! [K: nat] :
      ( filterlim_real_real
      @ ^ [X3: real] : ( divide_divide_real @ ( power_power_real @ X3 @ K ) @ ( exp_real @ X3 ) )
      @ ( topolo2815343760600316023s_real @ zero_zero_real )
      @ at_top_real ) ).

% tendsto_power_div_exp_0
thf(fact_10000_tendsto__exp__limit__at__top,axiom,
    ! [X: real] :
      ( filterlim_real_real
      @ ^ [Y2: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ Y2 ) ) @ Y2 )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
      @ at_top_real ) ).

% tendsto_exp_limit_at_top
thf(fact_10001_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_10002_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_10003_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B2: real,F2: real > real,Flim: real] :
      ( ! [X4: real] :
          ( ( ord_less_eq_real @ B2 @ X4 )
         => ? [Y4: real] :
              ( ( has_fi5821293074295781190e_real @ F2 @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
              & ( ord_less_real @ Y4 @ zero_zero_real ) ) )
     => ( ( filterlim_real_real @ F2 @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
       => ( ord_less_real @ Flim @ ( F2 @ B2 ) ) ) ) ).

% DERIV_neg_imp_decreasing_at_top
thf(fact_10004_lhopital__left__at__top,axiom,
    ! [G: real > real,X: real,G2: real > real,F2: real > real,F5: real > real,Y: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
     => ( ( eventually_real
          @ ^ [X3: real] :
              ( ( G2 @ X3 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
           => ( ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
             => ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G @ X3 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top
thf(fact_10005_eventually__sequentially__Suc,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat
        @ ^ [I2: nat] : ( P @ ( suc @ I2 ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_Suc
thf(fact_10006_eventually__sequentially__seg,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat
        @ ^ [N2: nat] : ( P @ ( plus_plus_nat @ N2 @ K ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_seg
thf(fact_10007_sequentially__offset,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat @ P @ at_top_nat )
     => ( eventually_nat
        @ ^ [I2: nat] : ( P @ ( plus_plus_nat @ I2 @ K ) )
        @ at_top_nat ) ) ).

% sequentially_offset
thf(fact_10008_eventually__sequentially,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ at_top_nat )
      = ( ? [N8: nat] :
          ! [N2: nat] :
            ( ( ord_less_eq_nat @ N8 @ N2 )
           => ( P @ N2 ) ) ) ) ).

% eventually_sequentially
thf(fact_10009_eventually__sequentiallyI,axiom,
    ! [C2: nat,P: nat > $o] :
      ( ! [X4: nat] :
          ( ( ord_less_eq_nat @ C2 @ X4 )
         => ( P @ X4 ) )
     => ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentiallyI
thf(fact_10010_le__sequentially,axiom,
    ! [F: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ F @ at_top_nat )
      = ( ! [N8: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N8 ) @ F ) ) ) ).

% le_sequentially
thf(fact_10011_eventually__at__right__to__0,axiom,
    ! [P: real > $o,A2: real] :
      ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
      = ( eventually_real
        @ ^ [X3: real] : ( P @ ( plus_plus_real @ X3 @ A2 ) )
        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% eventually_at_right_to_0
thf(fact_10012_eventually__at__right__real,axiom,
    ! [A2: real,B2: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( eventually_real
        @ ^ [X3: real] : ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A2 @ B2 ) )
        @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) ) ) ).

% eventually_at_right_real
thf(fact_10013_eventually__at__left__real,axiom,
    ! [B2: real,A2: real] :
      ( ( ord_less_real @ B2 @ A2 )
     => ( eventually_real
        @ ^ [X3: real] : ( member_real @ X3 @ ( set_or1633881224788618240n_real @ B2 @ A2 ) )
        @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) ) ) ).

% eventually_at_left_real
thf(fact_10014_eventually__at__top__to__right,axiom,
    ! [P: real > $o] :
      ( ( eventually_real @ P @ at_top_real )
      = ( eventually_real
        @ ^ [X3: real] : ( P @ ( inverse_inverse_real @ X3 ) )
        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% eventually_at_top_to_right
thf(fact_10015_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
        @ ^ [X3: real] : ( P @ ( inverse_inverse_real @ X3 ) )
        @ at_top_real ) ) ).

% eventually_at_right_to_top
thf(fact_10016_lhopital__at__top__at__top,axiom,
    ! [F2: real > real,A2: real,G: real > real,F5: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F2 @ at_top_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
     => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G @ X3 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top_at_top
thf(fact_10017_lhopital,axiom,
    ! [F2: real > real,X: real,G: real > real,G2: real > real,F5: real > real,F: filter_real] :
      ( ( filterlim_real_real @ F2 @ ( 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
            @ ^ [X3: real] :
                ( ( G @ X3 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X3: real] :
                  ( ( G2 @ X3 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
           => ( ( eventually_real
                @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
             => ( ( eventually_real
                  @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
               => ( ( filterlim_real_real
                    @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                    @ F
                    @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                 => ( filterlim_real_real
                    @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G @ X3 ) )
                    @ F
                    @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ) ).

% lhopital
thf(fact_10018_lhopital__right__at__top__at__top,axiom,
    ! [F2: real > real,A2: real,G: real > real,F5: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F2 @ at_top_real @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
     => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
       => ( ( eventually_real
            @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
         => ( ( eventually_real
              @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
           => ( ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
             => ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G @ X3 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) ) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
thf(fact_10019_lhopital__at__top__at__bot,axiom,
    ! [F2: real > real,A2: real,G: real > real,F5: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F2 @ at_top_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
     => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G @ X3 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A2 @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top_at_bot
thf(fact_10020_lhopital__left__at__top__at__top,axiom,
    ! [F2: real > real,A2: real,G: real > real,F5: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F2 @ at_top_real @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
     => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
       => ( ( eventually_real
            @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
         => ( ( eventually_real
              @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
           => ( ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
             => ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G @ X3 ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
thf(fact_10021_lhospital__at__top__at__top,axiom,
    ! [G: real > real,G2: real > real,F2: real > real,F5: real > real,X: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
     => ( ( eventually_real
          @ ^ [X3: real] :
              ( ( G2 @ X3 )
             != zero_zero_real )
          @ at_top_real )
       => ( ( eventually_real
            @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
            @ at_top_real )
         => ( ( eventually_real
              @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              @ at_top_real )
           => ( ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ at_top_real )
             => ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G @ X3 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ at_top_real ) ) ) ) ) ) ).

% lhospital_at_top_at_top
thf(fact_10022_lhopital__at__top,axiom,
    ! [G: real > real,X: real,G2: real > real,F2: real > real,F5: real > real,Y: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( eventually_real
          @ ^ [X3: real] :
              ( ( G2 @ X3 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G @ X3 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top
thf(fact_10023_lhopital__right__0,axiom,
    ! [F0: real > real,G0: real > real,G2: real > real,F5: real > real,F: 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
            @ ^ [X3: real] :
                ( ( G0 @ X3 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
         => ( ( eventually_real
              @ ^ [X3: real] :
                  ( ( G2 @ X3 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
           => ( ( eventually_real
                @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
             => ( ( eventually_real
                  @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                    @ F
                    @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
                 => ( filterlim_real_real
                    @ ^ [X3: real] : ( divide_divide_real @ ( F0 @ X3 ) @ ( G0 @ X3 ) )
                    @ F
                    @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).

% lhopital_right_0
thf(fact_10024_lhopital__right,axiom,
    ! [F2: real > real,X: real,G: real > real,G2: real > real,F5: real > real,F: filter_real] :
      ( ( filterlim_real_real @ F2 @ ( 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
            @ ^ [X3: real] :
                ( ( G @ X3 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X3: real] :
                  ( ( G2 @ X3 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
           => ( ( eventually_real
                @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
             => ( ( eventually_real
                  @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                    @ F
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
                 => ( filterlim_real_real
                    @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G @ X3 ) )
                    @ F
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ) ) ).

% lhopital_right
thf(fact_10025_lhopital__left,axiom,
    ! [F2: real > real,X: real,G: real > real,G2: real > real,F5: real > real,F: filter_real] :
      ( ( filterlim_real_real @ F2 @ ( 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
            @ ^ [X3: real] :
                ( ( G @ X3 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X3: real] :
                  ( ( G2 @ X3 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
           => ( ( eventually_real
                @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
             => ( ( eventually_real
                  @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                    @ F
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
                 => ( filterlim_real_real
                    @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G @ X3 ) )
                    @ F
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ) ) ).

% lhopital_left
thf(fact_10026_lhopital__right__at__top__at__bot,axiom,
    ! [F2: real > real,A2: real,G: real > real,F5: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F2 @ at_top_real @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
     => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
       => ( ( eventually_real
            @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
         => ( ( eventually_real
              @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
           => ( ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) )
             => ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G @ X3 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) ) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
thf(fact_10027_lhopital__left__at__top__at__bot,axiom,
    ! [F2: real > real,A2: real,G: real > real,F5: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F2 @ at_top_real @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
     => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
       => ( ( eventually_real
            @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
         => ( ( eventually_real
              @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
           => ( ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) )
             => ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G @ X3 ) )
                @ at_bot_real
                @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
thf(fact_10028_lhopital__right__0__at__top,axiom,
    ! [G: real > real,G2: real > real,F2: 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
          @ ^ [X3: real] :
              ( ( G2 @ X3 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
       => ( ( eventually_real
            @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
         => ( ( eventually_real
              @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
           => ( ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
             => ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G @ X3 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).

% lhopital_right_0_at_top
thf(fact_10029_lhopital__right__at__top,axiom,
    ! [G: real > real,X: real,G2: real > real,F2: real > real,F5: real > real,Y: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
     => ( ( eventually_real
          @ ^ [X3: real] :
              ( ( G2 @ X3 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F2 @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
           => ( ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
             => ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F2 @ X3 ) @ ( G @ X3 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ).

% lhopital_right_at_top
thf(fact_10030_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_10031_finite__greaterThanAtMost,axiom,
    ! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L2 @ U ) ) ).

% finite_greaterThanAtMost
thf(fact_10032_GreatestI__nat,axiom,
    ! [P: nat > $o,K: nat,B2: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B2 ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_nat
thf(fact_10033_Greatest__le__nat,axiom,
    ! [P: nat > $o,K: nat,B2: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B2 ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).

% Greatest_le_nat
thf(fact_10034_GreatestI__ex__nat,axiom,
    ! [P: nat > $o,B2: nat] :
      ( ? [X_12: nat] : ( P @ X_12 )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B2 ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_10035_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L2: nat,U: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ L2 ) @ U )
      = ( set_or6659071591806873216st_nat @ L2 @ U ) ) ).

% atLeastSucAtMost_greaterThanAtMost
thf(fact_10036_finite__greaterThanAtMost__int,axiom,
    ! [L2: int,U: int] : ( finite_finite_int @ ( set_or6656581121297822940st_int @ L2 @ U ) ) ).

% finite_greaterThanAtMost_int
thf(fact_10037_finite__greaterThanAtMost__integer,axiom,
    ! [L2: code_integer,U: code_integer] : ( finite6017078050557962740nteger @ ( set_or2715278749043346189nteger @ L2 @ U ) ) ).

% finite_greaterThanAtMost_integer
thf(fact_10038_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L2: int,U: int] :
      ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L2 @ one_one_int ) @ U )
      = ( set_or6656581121297822940st_int @ L2 @ U ) ) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_10039_decseq__bounded,axiom,
    ! [X2: nat > real,B: real] :
      ( ( order_9091379641038594480t_real @ X2 )
     => ( ! [I5: nat] : ( ord_less_eq_real @ B @ ( X2 @ I5 ) )
       => ( bfun_nat_real @ X2 @ at_top_nat ) ) ) ).

% decseq_bounded
thf(fact_10040_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L2: code_integer,U: code_integer] :
      ( ( set_or189985376899183464nteger @ ( plus_p5714425477246183910nteger @ L2 @ one_one_Code_integer ) @ U )
      = ( set_or2715278749043346189nteger @ L2 @ U ) ) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
thf(fact_10041_decseq__convergent,axiom,
    ! [X2: nat > real,B: real] :
      ( ( order_9091379641038594480t_real @ X2 )
     => ( ! [I5: nat] : ( ord_less_eq_real @ B @ ( X2 @ I5 ) )
       => ~ ! [L6: real] :
              ( ( filterlim_nat_real @ X2 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
             => ~ ! [I6: nat] : ( ord_less_eq_real @ L6 @ ( X2 @ I6 ) ) ) ) ) ).

% decseq_convergent
thf(fact_10042_GMVT,axiom,
    ! [A2: real,B2: real,F2: real > real,G: real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ! [X4: real] :
            ( ( ( ord_less_eq_real @ A2 @ X4 )
              & ( ord_less_eq_real @ X4 @ B2 ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F2 ) )
       => ( ! [X4: real] :
              ( ( ( ord_less_real @ A2 @ X4 )
                & ( ord_less_real @ X4 @ B2 ) )
             => ( differ6690327859849518006l_real @ F2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
         => ( ! [X4: real] :
                ( ( ( ord_less_eq_real @ A2 @ X4 )
                  & ( ord_less_eq_real @ X4 @ B2 ) )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ G ) )
           => ( ! [X4: real] :
                  ( ( ( ord_less_real @ A2 @ X4 )
                    & ( ord_less_real @ X4 @ B2 ) )
                 => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
             => ? [G_c: real,F_c: real,C4: real] :
                  ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C4 @ top_top_set_real ) )
                  & ( has_fi5821293074295781190e_real @ F2 @ F_c @ ( topolo2177554685111907308n_real @ C4 @ top_top_set_real ) )
                  & ( ord_less_real @ A2 @ C4 )
                  & ( ord_less_real @ C4 @ B2 )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) @ G_c )
                    = ( times_times_real @ ( minus_minus_real @ ( G @ B2 ) @ ( G @ A2 ) ) @ F_c ) ) ) ) ) ) ) ) ).

% GMVT
thf(fact_10043_atLeast__0,axiom,
    ( ( set_ord_atLeast_nat @ zero_zero_nat )
    = top_top_set_nat ) ).

% atLeast_0
thf(fact_10044_real__differentiable__def,axiom,
    ! [F2: real > real,X: real,S2: set_real] :
      ( ( differ6690327859849518006l_real @ F2 @ ( topolo2177554685111907308n_real @ X @ S2 ) )
      = ( ? [D6: real] : ( has_fi5821293074295781190e_real @ F2 @ D6 @ ( topolo2177554685111907308n_real @ X @ S2 ) ) ) ) ).

% real_differentiable_def
thf(fact_10045_real__differentiableE,axiom,
    ! [F2: real > real,X: real,S2: set_real] :
      ( ( differ6690327859849518006l_real @ F2 @ ( topolo2177554685111907308n_real @ X @ S2 ) )
     => ~ ! [Df: real] :
            ~ ( has_fi5821293074295781190e_real @ F2 @ Df @ ( topolo2177554685111907308n_real @ X @ S2 ) ) ) ).

% real_differentiableE
thf(fact_10046_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( set_or1210151606488870762an_nat @ K ) ) ).

% atLeast_Suc_greaterThan
thf(fact_10047_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_10048_MVT,axiom,
    ! [A2: real,B2: real,F2: real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ F2 )
       => ( ! [X4: real] :
              ( ( ord_less_real @ A2 @ X4 )
             => ( ( ord_less_real @ X4 @ B2 )
               => ( differ6690327859849518006l_real @ F2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
         => ? [L4: real,Z2: real] :
              ( ( ord_less_real @ A2 @ Z2 )
              & ( ord_less_real @ Z2 @ B2 )
              & ( has_fi5821293074295781190e_real @ F2 @ L4 @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) )
              & ( ( minus_minus_real @ ( F2 @ B2 ) @ ( F2 @ A2 ) )
                = ( times_times_real @ ( minus_minus_real @ B2 @ A2 ) @ L4 ) ) ) ) ) ) ).

% MVT
thf(fact_10049_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N3: nat,J2: nat,I: nat] :
      ( ( ord_less_nat @ N3 @ ( minus_minus_nat @ J2 @ ( suc @ I ) ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J2 ) ) @ N3 )
        = ( suc @ ( plus_plus_nat @ I @ N3 ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_10050_continuous__on__arcosh_H,axiom,
    ! [A: set_real,F2: real > real] :
      ( ( topolo5044208981011980120l_real @ A @ F2 )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ A )
           => ( ord_less_eq_real @ one_one_real @ ( F2 @ X4 ) ) )
       => ( topolo5044208981011980120l_real @ A
          @ ^ [X3: real] : ( arcosh_real @ ( F2 @ X3 ) ) ) ) ) ).

% continuous_on_arcosh'
thf(fact_10051_continuous__image__closed__interval,axiom,
    ! [A2: real,B2: real,F2: real > real] :
      ( ( ord_less_eq_real @ A2 @ B2 )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ F2 )
       => ? [C4: real,D5: real] :
            ( ( ( image_real_real @ F2 @ ( set_or1222579329274155063t_real @ A2 @ B2 ) )
              = ( set_or1222579329274155063t_real @ C4 @ D5 ) )
            & ( ord_less_eq_real @ C4 @ D5 ) ) ) ) ).

% continuous_image_closed_interval
thf(fact_10052_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ I ) @ J2 )
     => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J2 ) )
        = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J2 ) ) ) ) ) ).

% sorted_list_of_set_greaterThanAtMost
thf(fact_10053_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J2: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ J2 )
     => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J2 ) )
        = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J2 ) ) ) ) ) ).

% sorted_list_of_set_greaterThanLessThan
thf(fact_10054_Rolle__deriv,axiom,
    ! [A2: real,B2: real,F2: real > real,F5: real > real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ( F2 @ A2 )
          = ( F2 @ B2 ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ F2 )
         => ( ! [X4: real] :
                ( ( ord_less_real @ A2 @ X4 )
               => ( ( ord_less_real @ X4 @ B2 )
                 => ( has_de1759254742604945161l_real @ F2 @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
           => ? [Z2: real] :
                ( ( ord_less_real @ A2 @ Z2 )
                & ( ord_less_real @ Z2 @ B2 )
                & ( ( F5 @ Z2 )
                  = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).

% Rolle_deriv
thf(fact_10055_mvt,axiom,
    ! [A2: real,B2: real,F2: real > real,F5: real > real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ F2 )
       => ( ! [X4: real] :
              ( ( ord_less_real @ A2 @ X4 )
             => ( ( ord_less_real @ X4 @ B2 )
               => ( has_de1759254742604945161l_real @ F2 @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
         => ~ ! [Xi3: real] :
                ( ( ord_less_real @ A2 @ Xi3 )
               => ( ( ord_less_real @ Xi3 @ B2 )
                 => ( ( minus_minus_real @ ( F2 @ B2 ) @ ( F2 @ A2 ) )
                   != ( F5 @ Xi3 @ ( minus_minus_real @ B2 @ A2 ) ) ) ) ) ) ) ) ).

% mvt
thf(fact_10056_DERIV__pos__imp__increasing__open,axiom,
    ! [A2: real,B2: real,F2: real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ! [X4: real] :
            ( ( ord_less_real @ A2 @ X4 )
           => ( ( ord_less_real @ X4 @ B2 )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F2 @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y4 ) ) ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ F2 )
         => ( ord_less_real @ ( F2 @ A2 ) @ ( F2 @ B2 ) ) ) ) ) ).

% DERIV_pos_imp_increasing_open
thf(fact_10057_DERIV__neg__imp__decreasing__open,axiom,
    ! [A2: real,B2: real,F2: real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ! [X4: real] :
            ( ( ord_less_real @ A2 @ X4 )
           => ( ( ord_less_real @ X4 @ B2 )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F2 @ Y4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
                  & ( ord_less_real @ Y4 @ zero_zero_real ) ) ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ F2 )
         => ( ord_less_real @ ( F2 @ B2 ) @ ( F2 @ A2 ) ) ) ) ) ).

% DERIV_neg_imp_decreasing_open
thf(fact_10058_DERIV__isconst__end,axiom,
    ! [A2: real,B2: real,F2: real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ F2 )
       => ( ! [X4: real] :
              ( ( ord_less_real @ A2 @ X4 )
             => ( ( ord_less_real @ X4 @ B2 )
               => ( has_fi5821293074295781190e_real @ F2 @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
         => ( ( F2 @ B2 )
            = ( F2 @ A2 ) ) ) ) ) ).

% DERIV_isconst_end
thf(fact_10059_DERIV__isconst2,axiom,
    ! [A2: real,B2: real,F2: real > real,X: real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ F2 )
       => ( ! [X4: real] :
              ( ( ord_less_real @ A2 @ X4 )
             => ( ( ord_less_real @ X4 @ B2 )
               => ( has_fi5821293074295781190e_real @ F2 @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
         => ( ( ord_less_eq_real @ A2 @ X )
           => ( ( ord_less_eq_real @ X @ B2 )
             => ( ( F2 @ X )
                = ( F2 @ A2 ) ) ) ) ) ) ) ).

% DERIV_isconst2
thf(fact_10060_Rolle,axiom,
    ! [A2: real,B2: real,F2: real > real] :
      ( ( ord_less_real @ A2 @ B2 )
     => ( ( ( F2 @ A2 )
          = ( F2 @ B2 ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B2 ) @ F2 )
         => ( ! [X4: real] :
                ( ( ord_less_real @ A2 @ X4 )
               => ( ( ord_less_real @ X4 @ B2 )
                 => ( differ6690327859849518006l_real @ F2 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
           => ? [Z2: real] :
                ( ( ord_less_real @ A2 @ Z2 )
                & ( ord_less_real @ Z2 @ B2 )
                & ( has_fi5821293074295781190e_real @ F2 @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) ) ) ) ) ).

% Rolle
thf(fact_10061_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N3: nat,J2: nat,I: nat] :
      ( ( ord_less_nat @ N3 @ ( minus_minus_nat @ J2 @ I ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J2 ) ) @ N3 )
        = ( suc @ ( plus_plus_nat @ I @ N3 ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_10062_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
              @ ^ [L: code_integer,J: code_integer] : ( if_int @ ( J = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L ) ) @ one_one_int ) )
              @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% int_of_integer_code
thf(fact_10063_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] :
      ( ( re @ ( csqrt @ Z ) )
      = ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% csqrt.simps(1)
thf(fact_10064_minus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( minus_8373710615458151222nteger @ X @ Xa ) )
      = ( minus_minus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).

% minus_integer.rep_eq
thf(fact_10065_complex__Re__numeral,axiom,
    ! [V: num] :
      ( ( re @ ( numera6690914467698888265omplex @ V ) )
      = ( numeral_numeral_real @ V ) ) ).

% complex_Re_numeral
thf(fact_10066_divide__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( divide6298287555418463151nteger @ X @ Xa ) )
      = ( divide_divide_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).

% divide_integer.rep_eq
thf(fact_10067_Re__divide__of__nat,axiom,
    ! [Z: complex,N3: nat] :
      ( ( re @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N3 ) ) )
      = ( divide_divide_real @ ( re @ Z ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ).

% Re_divide_of_nat
thf(fact_10068_Re__divide__of__real,axiom,
    ! [Z: complex,R3: real] :
      ( ( re @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R3 ) ) )
      = ( divide_divide_real @ ( re @ Z ) @ R3 ) ) ).

% Re_divide_of_real
thf(fact_10069_Re__sgn,axiom,
    ! [Z: complex] :
      ( ( re @ ( sgn_sgn_complex @ Z ) )
      = ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).

% Re_sgn
thf(fact_10070_Re__divide__numeral,axiom,
    ! [Z: complex,W: num] :
      ( ( re @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
      = ( divide_divide_real @ ( re @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).

% Re_divide_numeral
thf(fact_10071_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_10072_Re__csqrt,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).

% Re_csqrt
thf(fact_10073_complex__Re__le__cmod,axiom,
    ! [X: complex] : ( ord_less_eq_real @ ( re @ X ) @ ( real_V1022390504157884413omplex @ X ) ) ).

% complex_Re_le_cmod
thf(fact_10074_abs__Re__le__cmod,axiom,
    ! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).

% abs_Re_le_cmod
thf(fact_10075_integer__less__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [K3: code_integer,L: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L ) ) ) ) ).

% integer_less_iff
thf(fact_10076_less__integer_Orep__eq,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [X3: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).

% less_integer.rep_eq
thf(fact_10077_int__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_less_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Y ) )
      = ( ord_le6747313008572928689nteger @ X @ Y ) ) ).

% int_of_integer_less_iff
thf(fact_10078_minus__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( re @ ( minus_minus_complex @ X @ Y ) )
      = ( minus_minus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).

% minus_complex.simps(1)
thf(fact_10079_zero__complex_Osimps_I1_J,axiom,
    ( ( re @ zero_zero_complex )
    = zero_zero_real ) ).

% zero_complex.simps(1)
thf(fact_10080_imaginary__unit_Osimps_I1_J,axiom,
    ( ( re @ imaginary_unit )
    = zero_zero_real ) ).

% imaginary_unit.simps(1)
thf(fact_10081_plus__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( re @ ( plus_plus_complex @ X @ Y ) )
      = ( plus_plus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).

% plus_complex.simps(1)
thf(fact_10082_scaleR__complex_Osimps_I1_J,axiom,
    ! [R3: real,X: complex] :
      ( ( re @ ( real_V2046097035970521341omplex @ R3 @ X ) )
      = ( times_times_real @ R3 @ ( re @ X ) ) ) ).

% scaleR_complex.simps(1)
thf(fact_10083_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
      = ( ( re @ Z )
        = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).

% cmod_plus_Re_le_0_iff
thf(fact_10084_bin__last__integer_Orep__eq,axiom,
    ( bits_b8758750999018896077nteger
    = ( ^ [X3: code_integer] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ X3 ) ) ) ) ).

% bin_last_integer.rep_eq
thf(fact_10085_bin__rest__integer_Orep__eq,axiom,
    ! [X: code_integer] :
      ( ( code_int_of_integer @ ( bits_b2549910563261871055nteger @ X ) )
      = ( divide_divide_int @ ( code_int_of_integer @ X ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% bin_rest_integer.rep_eq
thf(fact_10086_cos__n__Re__cis__pow__n,axiom,
    ! [N3: nat,A2: real] :
      ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ A2 ) )
      = ( re @ ( power_power_complex @ ( cis @ A2 ) @ N3 ) ) ) ).

% cos_n_Re_cis_pow_n
thf(fact_10087_Bit__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: $o] :
      ( ( code_int_of_integer @ ( bits_Bit_integer @ X @ Xa ) )
      = ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ Xa ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ X ) ) ) ) ).

% Bit_integer.rep_eq
thf(fact_10088_divmod__integer__def,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K3 @ L ) @ ( modulo364778990260209775nteger @ K3 @ L ) ) ) ) ).

% divmod_integer_def
thf(fact_10089_num__of__integer__code,axiom,
    ( code_num_of_integer
    = ( ^ [K3: code_integer] :
          ( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
          @ ( produc7336495610019696514er_num
            @ ^ [L: code_integer,J: code_integer] : ( if_num @ ( J = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L ) @ ( code_num_of_integer @ L ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L ) @ ( code_num_of_integer @ L ) ) @ one ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% num_of_integer_code
thf(fact_10090_csqrt_Ocode,axiom,
    ( csqrt
    = ( ^ [Z5: complex] :
          ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( re @ Z5 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          @ ( times_times_real
            @ ( if_real
              @ ( ( im @ Z5 )
                = zero_zero_real )
              @ one_one_real
              @ ( sgn_sgn_real @ ( im @ Z5 ) ) )
            @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( re @ Z5 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% csqrt.code
thf(fact_10091_complex__Im__fact,axiom,
    ! [N3: nat] :
      ( ( im @ ( semiri5044797733671781792omplex @ N3 ) )
      = zero_zero_real ) ).

% complex_Im_fact
thf(fact_10092_complex__Im__of__int,axiom,
    ! [Z: int] :
      ( ( im @ ( ring_17405671764205052669omplex @ Z ) )
      = zero_zero_real ) ).

% complex_Im_of_int
thf(fact_10093_Im__complex__of__real,axiom,
    ! [Z: real] :
      ( ( im @ ( real_V4546457046886955230omplex @ Z ) )
      = zero_zero_real ) ).

% Im_complex_of_real
thf(fact_10094_Im__power__real,axiom,
    ! [X: complex,N3: nat] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( im @ ( power_power_complex @ X @ N3 ) )
        = zero_zero_real ) ) ).

% Im_power_real
thf(fact_10095_complex__Im__numeral,axiom,
    ! [V: num] :
      ( ( im @ ( numera6690914467698888265omplex @ V ) )
      = zero_zero_real ) ).

% complex_Im_numeral
thf(fact_10096_complex__Im__of__nat,axiom,
    ! [N3: nat] :
      ( ( im @ ( semiri8010041392384452111omplex @ N3 ) )
      = zero_zero_real ) ).

% complex_Im_of_nat
thf(fact_10097_Im__divide__of__real,axiom,
    ! [Z: complex,R3: real] :
      ( ( im @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R3 ) ) )
      = ( divide_divide_real @ ( im @ Z ) @ R3 ) ) ).

% Im_divide_of_real
thf(fact_10098_Im__sgn,axiom,
    ! [Z: complex] :
      ( ( im @ ( sgn_sgn_complex @ Z ) )
      = ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).

% Im_sgn
thf(fact_10099_Re__power__real,axiom,
    ! [X: complex,N3: nat] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( re @ ( power_power_complex @ X @ N3 ) )
        = ( power_power_real @ ( re @ X ) @ N3 ) ) ) ).

% Re_power_real
thf(fact_10100_Im__divide__numeral,axiom,
    ! [Z: complex,W: num] :
      ( ( im @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
      = ( divide_divide_real @ ( im @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).

% Im_divide_numeral
thf(fact_10101_Im__divide__of__nat,axiom,
    ! [Z: complex,N3: nat] :
      ( ( im @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N3 ) ) )
      = ( divide_divide_real @ ( im @ Z ) @ ( semiri5074537144036343181t_real @ N3 ) ) ) ).

% Im_divide_of_nat
thf(fact_10102_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_10103_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_10104_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_10105_complex__is__Int__iff,axiom,
    ! [Z: complex] :
      ( ( member_complex @ Z @ ring_1_Ints_complex )
      = ( ( ( im @ Z )
          = zero_zero_real )
        & ? [I2: int] :
            ( ( re @ Z )
            = ( ring_1_of_int_real @ I2 ) ) ) ) ).

% complex_is_Int_iff
thf(fact_10106_zero__complex_Osimps_I2_J,axiom,
    ( ( im @ zero_zero_complex )
    = zero_zero_real ) ).

% zero_complex.simps(2)
thf(fact_10107_one__complex_Osimps_I2_J,axiom,
    ( ( im @ one_one_complex )
    = zero_zero_real ) ).

% one_complex.simps(2)
thf(fact_10108_plus__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( im @ ( plus_plus_complex @ X @ Y ) )
      = ( plus_plus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).

% plus_complex.simps(2)
thf(fact_10109_scaleR__complex_Osimps_I2_J,axiom,
    ! [R3: real,X: complex] :
      ( ( im @ ( real_V2046097035970521341omplex @ R3 @ X ) )
      = ( times_times_real @ R3 @ ( im @ X ) ) ) ).

% scaleR_complex.simps(2)
thf(fact_10110_minus__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( im @ ( minus_minus_complex @ X @ Y ) )
      = ( minus_minus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).

% minus_complex.simps(2)
thf(fact_10111_abs__Im__le__cmod,axiom,
    ! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).

% abs_Im_le_cmod
thf(fact_10112_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_10113_Im__eq__0,axiom,
    ! [Z: complex] :
      ( ( ( abs_abs_real @ ( re @ Z ) )
        = ( real_V1022390504157884413omplex @ Z ) )
     => ( ( im @ Z )
        = zero_zero_real ) ) ).

% Im_eq_0
thf(fact_10114_cmod__eq__Im,axiom,
    ! [Z: complex] :
      ( ( ( re @ Z )
        = zero_zero_real )
     => ( ( real_V1022390504157884413omplex @ Z )
        = ( abs_abs_real @ ( im @ Z ) ) ) ) ).

% cmod_eq_Im
thf(fact_10115_cmod__eq__Re,axiom,
    ! [Z: complex] :
      ( ( ( im @ Z )
        = zero_zero_real )
     => ( ( real_V1022390504157884413omplex @ Z )
        = ( abs_abs_real @ ( re @ Z ) ) ) ) ).

% cmod_eq_Re
thf(fact_10116_cmod__Im__le__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( re @ X )
        = ( re @ Y ) )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( abs_abs_real @ ( im @ Y ) ) ) ) ) ).

% cmod_Im_le_iff
thf(fact_10117_cmod__Re__le__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( im @ X )
        = ( im @ Y ) )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( abs_abs_real @ ( re @ Y ) ) ) ) ) ).

% cmod_Re_le_iff
thf(fact_10118_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_10119_plus__complex_Ocode,axiom,
    ( plus_plus_complex
    = ( ^ [X3: complex,Y2: complex] : ( complex2 @ ( plus_plus_real @ ( re @ X3 ) @ ( re @ Y2 ) ) @ ( plus_plus_real @ ( im @ X3 ) @ ( im @ Y2 ) ) ) ) ) ).

% plus_complex.code
thf(fact_10120_scaleR__complex_Ocode,axiom,
    ( real_V2046097035970521341omplex
    = ( ^ [R5: real,X3: complex] : ( complex2 @ ( times_times_real @ R5 @ ( re @ X3 ) ) @ ( times_times_real @ R5 @ ( im @ X3 ) ) ) ) ) ).

% scaleR_complex.code
thf(fact_10121_minus__complex_Ocode,axiom,
    ( minus_minus_complex
    = ( ^ [X3: complex,Y2: complex] : ( complex2 @ ( minus_minus_real @ ( re @ X3 ) @ ( re @ Y2 ) ) @ ( minus_minus_real @ ( im @ X3 ) @ ( im @ Y2 ) ) ) ) ) ).

% minus_complex.code
thf(fact_10122_csqrt__principal,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
      | ( ( ( re @ ( csqrt @ Z ) )
          = zero_zero_real )
        & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).

% csqrt_principal
thf(fact_10123_cmod__le,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).

% cmod_le
thf(fact_10124_sin__n__Im__cis__pow__n,axiom,
    ! [N3: nat,A2: real] :
      ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ A2 ) )
      = ( im @ ( power_power_complex @ ( cis @ A2 ) @ N3 ) ) ) ).

% sin_n_Im_cis_pow_n
thf(fact_10125_Re__exp,axiom,
    ! [Z: complex] :
      ( ( re @ ( exp_complex @ Z ) )
      = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( cos_real @ ( im @ Z ) ) ) ) ).

% Re_exp
thf(fact_10126_Im__exp,axiom,
    ! [Z: complex] :
      ( ( im @ ( exp_complex @ Z ) )
      = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( sin_real @ ( im @ Z ) ) ) ) ).

% Im_exp
thf(fact_10127_times__complex_Ocode,axiom,
    ( times_times_complex
    = ( ^ [X3: complex,Y2: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X3 ) @ ( re @ Y2 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( im @ Y2 ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X3 ) @ ( im @ Y2 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( re @ Y2 ) ) ) ) ) ) ).

% times_complex.code
thf(fact_10128_cmod__power2,axiom,
    ! [Z: complex] :
      ( ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cmod_power2
thf(fact_10129_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_10130_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_10131_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero_complex )
      = ( ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_real ) ) ).

% complex_eq_0
thf(fact_10132_norm__complex__def,axiom,
    ( real_V1022390504157884413omplex
    = ( ^ [Z5: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% norm_complex_def
thf(fact_10133_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_10134_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
      = ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_neq_0
thf(fact_10135_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_10136_csqrt__unique,axiom,
    ! [W: complex,Z: complex] :
      ( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = Z )
     => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
          | ( ( ( re @ W )
              = zero_zero_real )
            & ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
       => ( ( csqrt @ Z )
          = W ) ) ) ).

% csqrt_unique
thf(fact_10137_csqrt__square,axiom,
    ! [B2: complex] :
      ( ( ( ord_less_real @ zero_zero_real @ ( re @ B2 ) )
        | ( ( ( re @ B2 )
            = zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ ( im @ B2 ) ) ) )
     => ( ( csqrt @ ( power_power_complex @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = B2 ) ) ).

% csqrt_square
thf(fact_10138_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_10139_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_10140_complex__abs__le__norm,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) @ ( times_times_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).

% complex_abs_le_norm
thf(fact_10141_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_real @ ( power_power_real @ ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real ) ) ).

% complex_unit_circle
thf(fact_10142_inverse__complex_Ocode,axiom,
    ( invers8013647133539491842omplex
    = ( ^ [X3: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X3 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X3 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% inverse_complex.code
thf(fact_10143_Complex__divide,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [X3: complex,Y2: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X3 ) @ ( re @ Y2 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( im @ Y2 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X3 ) @ ( re @ Y2 ) ) @ ( times_times_real @ ( re @ X3 ) @ ( im @ Y2 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% Complex_divide
thf(fact_10144_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] :
      ( ( im @ ( csqrt @ Z ) )
      = ( times_times_real
        @ ( if_real
          @ ( ( im @ Z )
            = zero_zero_real )
          @ one_one_real
          @ ( sgn_sgn_real @ ( im @ Z ) ) )
        @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% csqrt.simps(2)
thf(fact_10145_Im__Reals__divide,axiom,
    ! [R3: complex,Z: complex] :
      ( ( member_complex @ R3 @ real_V2521375963428798218omplex )
     => ( ( im @ ( divide1717551699836669952omplex @ R3 @ Z ) )
        = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R3 ) ) @ ( im @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Im_Reals_divide
thf(fact_10146_nat__of__integer__code,axiom,
    ( code_nat_of_integer
    = ( ^ [K3: code_integer] :
          ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
          @ ( produc1555791787009142072er_nat
            @ ^ [L: code_integer,J: code_integer] : ( if_nat @ ( J = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L ) @ ( code_nat_of_integer @ L ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L ) @ ( code_nat_of_integer @ L ) ) @ one_one_nat ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% nat_of_integer_code
thf(fact_10147_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_10148_nat__of__integer__numeral,axiom,
    ! [N3: num] :
      ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ N3 ) )
      = ( numeral_numeral_nat @ N3 ) ) ).

% nat_of_integer_numeral
thf(fact_10149_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_10150_Re__divide__Reals,axiom,
    ! [R3: complex,Z: complex] :
      ( ( member_complex @ R3 @ real_V2521375963428798218omplex )
     => ( ( re @ ( divide1717551699836669952omplex @ Z @ R3 ) )
        = ( divide_divide_real @ ( re @ Z ) @ ( re @ R3 ) ) ) ) ).

% Re_divide_Reals
thf(fact_10151_Im__divide__Reals,axiom,
    ! [R3: complex,Z: complex] :
      ( ( member_complex @ R3 @ real_V2521375963428798218omplex )
     => ( ( im @ ( divide1717551699836669952omplex @ Z @ R3 ) )
        = ( divide_divide_real @ ( im @ Z ) @ ( re @ R3 ) ) ) ) ).

% Im_divide_Reals
thf(fact_10152_complex__is__Real__iff,axiom,
    ! [Z: complex] :
      ( ( member_complex @ Z @ real_V2521375963428798218omplex )
      = ( ( im @ Z )
        = zero_zero_real ) ) ).

% complex_is_Real_iff
thf(fact_10153_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_10154_Complex__in__Reals,axiom,
    ! [X: real] : ( member_complex @ ( complex2 @ X @ zero_zero_real ) @ real_V2521375963428798218omplex ) ).

% Complex_in_Reals
thf(fact_10155_nat__of__integer__less__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ( ord_less_nat @ ( code_nat_of_integer @ X ) @ ( code_nat_of_integer @ Y ) )
          = ( ord_le6747313008572928689nteger @ X @ Y ) ) ) ) ).

% nat_of_integer_less_iff
thf(fact_10156_image__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
     => ( ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ U )
        = ( image_1215581382706833972nteger @ semiri4939895301339042750nteger @ ( set_ord_lessThan_nat @ ( code_nat_of_integer @ U ) ) ) ) ) ).

% image_atLeastZeroLessThan_integer
thf(fact_10157_Re__Reals__divide,axiom,
    ! [R3: complex,Z: complex] :
      ( ( member_complex @ R3 @ real_V2521375963428798218omplex )
     => ( ( re @ ( divide1717551699836669952omplex @ R3 @ Z ) )
        = ( divide_divide_real @ ( times_times_real @ ( re @ R3 ) @ ( re @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Re_Reals_divide
thf(fact_10158_complex__diff__cnj,axiom,
    ! [Z: complex] :
      ( ( minus_minus_complex @ Z @ ( cnj @ Z ) )
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z ) ) ) @ imaginary_unit ) ) ).

% complex_diff_cnj
thf(fact_10159_complex__mult__cnj,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( cnj @ Z ) )
      = ( real_V4546457046886955230omplex @ ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_mult_cnj
thf(fact_10160_complex__cnj__divide,axiom,
    ! [X: complex,Y: complex] :
      ( ( cnj @ ( divide1717551699836669952omplex @ X @ Y ) )
      = ( divide1717551699836669952omplex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).

% complex_cnj_divide
thf(fact_10161_complex__cnj__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( cnj @ ( minus_minus_complex @ X @ Y ) )
      = ( minus_minus_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).

% complex_cnj_diff
thf(fact_10162_complex__In__mult__cnj__zero,axiom,
    ! [Z: complex] :
      ( ( im @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
      = zero_zero_real ) ).

% complex_In_mult_cnj_zero
thf(fact_10163_Re__complex__div__eq__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ( re @ ( divide1717551699836669952omplex @ A2 @ B2 ) )
        = zero_zero_real )
      = ( ( re @ ( times_times_complex @ A2 @ ( cnj @ B2 ) ) )
        = zero_zero_real ) ) ).

% Re_complex_div_eq_0
thf(fact_10164_Im__complex__div__eq__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ( im @ ( divide1717551699836669952omplex @ A2 @ B2 ) )
        = zero_zero_real )
      = ( ( im @ ( times_times_complex @ A2 @ ( cnj @ B2 ) ) )
        = zero_zero_real ) ) ).

% Im_complex_div_eq_0
thf(fact_10165_Re__complex__div__lt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A2 @ B2 ) ) @ zero_zero_real )
      = ( ord_less_real @ ( re @ ( times_times_complex @ A2 @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).

% Re_complex_div_lt_0
thf(fact_10166_Re__complex__div__gt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A2 @ B2 ) ) )
      = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A2 @ ( cnj @ B2 ) ) ) ) ) ).

% Re_complex_div_gt_0
thf(fact_10167_Re__complex__div__le__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A2 @ B2 ) ) @ zero_zero_real )
      = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A2 @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).

% Re_complex_div_le_0
thf(fact_10168_Re__complex__div__ge__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A2 @ B2 ) ) )
      = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A2 @ ( cnj @ B2 ) ) ) ) ) ).

% Re_complex_div_ge_0
thf(fact_10169_Im__complex__div__lt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A2 @ B2 ) ) @ zero_zero_real )
      = ( ord_less_real @ ( im @ ( times_times_complex @ A2 @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).

% Im_complex_div_lt_0
thf(fact_10170_Im__complex__div__gt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A2 @ B2 ) ) )
      = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A2 @ ( cnj @ B2 ) ) ) ) ) ).

% Im_complex_div_gt_0
thf(fact_10171_Im__complex__div__le__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A2 @ B2 ) ) @ zero_zero_real )
      = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A2 @ ( cnj @ B2 ) ) ) @ zero_zero_real ) ) ).

% Im_complex_div_le_0
thf(fact_10172_Im__complex__div__ge__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A2 @ B2 ) ) )
      = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A2 @ ( cnj @ B2 ) ) ) ) ) ).

% Im_complex_div_ge_0
thf(fact_10173_complex__mod__mult__cnj,axiom,
    ! [Z: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
      = ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% complex_mod_mult_cnj
thf(fact_10174_complex__div__gt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A2 @ B2 ) ) )
        = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A2 @ ( cnj @ B2 ) ) ) ) )
      & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A2 @ B2 ) ) )
        = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A2 @ ( cnj @ B2 ) ) ) ) ) ) ).

% complex_div_gt_0
thf(fact_10175_complex__norm__square,axiom,
    ! [Z: complex] :
      ( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( times_times_complex @ Z @ ( cnj @ Z ) ) ) ).

% complex_norm_square
thf(fact_10176_complex__add__cnj,axiom,
    ! [Z: complex] :
      ( ( plus_plus_complex @ Z @ ( cnj @ Z ) )
      = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z ) ) ) ) ).

% complex_add_cnj
thf(fact_10177_complex__div__cnj,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A7: complex,B7: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A7 @ ( cnj @ B7 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B7 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% complex_div_cnj
thf(fact_10178_cnj__add__mult__eq__Re,axiom,
    ! [Z: complex,W: complex] :
      ( ( plus_plus_complex @ ( times_times_complex @ Z @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z ) @ W ) )
      = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z @ ( cnj @ W ) ) ) ) ) ) ).

% cnj_add_mult_eq_Re
thf(fact_10179_card__lessThan,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
      = U ) ).

% card_lessThan
thf(fact_10180_card__Collect__less__nat,axiom,
    ! [N3: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N3 ) ) )
      = N3 ) ).

% card_Collect_less_nat
thf(fact_10181_card__atMost,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
      = ( suc @ U ) ) ).

% card_atMost
thf(fact_10182_card__atLeastLessThan,axiom,
    ! [L2: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L2 @ U ) )
      = ( minus_minus_nat @ U @ L2 ) ) ).

% card_atLeastLessThan
thf(fact_10183_card__Collect__le__nat,axiom,
    ! [N3: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I2: nat] : ( ord_less_eq_nat @ I2 @ N3 ) ) )
      = ( suc @ N3 ) ) ).

% card_Collect_le_nat
thf(fact_10184_card__greaterThanAtMost,axiom,
    ! [L2: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L2 @ U ) )
      = ( minus_minus_nat @ U @ L2 ) ) ).

% card_greaterThanAtMost
thf(fact_10185_card__UNIV__bool,axiom,
    ( ( finite_card_o @ top_top_set_o )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% card_UNIV_bool
thf(fact_10186_card__atLeastAtMost,axiom,
    ! [L2: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L2 @ U ) )
      = ( minus_minus_nat @ ( suc @ U ) @ L2 ) ) ).

% card_atLeastAtMost
thf(fact_10187_card__atLeastLessThan__int,axiom,
    ! [L2: int,U: int] :
      ( ( finite_card_int @ ( set_or4662586982721622107an_int @ L2 @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ L2 ) ) ) ).

% card_atLeastLessThan_int
thf(fact_10188_card__greaterThanLessThan,axiom,
    ! [L2: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L2 @ U ) )
      = ( minus_minus_nat @ U @ ( suc @ L2 ) ) ) ).

% card_greaterThanLessThan
thf(fact_10189_card__greaterThanAtMost__int,axiom,
    ! [L2: int,U: int] :
      ( ( finite_card_int @ ( set_or6656581121297822940st_int @ L2 @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ L2 ) ) ) ).

% card_greaterThanAtMost_int
thf(fact_10190_card__atLeastAtMost__int,axiom,
    ! [L2: int,U: int] :
      ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L2 @ U ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L2 ) @ one_one_int ) ) ) ).

% card_atLeastAtMost_int
thf(fact_10191_card__greaterThanLessThan__int,axiom,
    ! [L2: int,U: int] :
      ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L2 @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L2 @ one_one_int ) ) ) ) ).

% card_greaterThanLessThan_int
thf(fact_10192_subset__card__intvl__is__intvl,axiom,
    ! [A: set_nat,K: nat] :
      ( ( ord_less_eq_set_nat @ A @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A ) ) ) )
     => ( A
        = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A ) ) ) ) ) ).

% subset_card_intvl_is_intvl
thf(fact_10193_card__less,axiom,
    ! [M3: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M3 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M3 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_10194_card__less__Suc,axiom,
    ! [M3: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M3 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K3: nat] :
                  ( ( member_nat @ ( suc @ K3 ) @ M3 )
                  & ( ord_less_nat @ K3 @ I ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M3 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_10195_card__less__Suc2,axiom,
    ! [M3: set_nat,I: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M3 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ ( suc @ K3 ) @ M3 )
                & ( ord_less_nat @ K3 @ I ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M3 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_10196_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
      = ( nat2 @ U ) ) ).

% card_atLeastZeroLessThan_int
thf(fact_10197_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N7: set_nat,N3: nat] :
      ( ( ord_less_eq_set_nat @ N7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N3 ) )
     => ( ord_less_eq_nat @ ( finite_card_nat @ N7 ) @ N3 ) ) ).

% subset_eq_atLeast0_lessThan_card
thf(fact_10198_card__le__Suc__Max,axiom,
    ! [S: set_nat] :
      ( ( finite_finite_nat @ S )
     => ( ord_less_eq_nat @ ( finite_card_nat @ S ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S ) ) ) ) ).

% card_le_Suc_Max
thf(fact_10199_card__sum__le__nat__sum,axiom,
    ! [S: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ S ) ) ).

% card_sum_le_nat_sum
thf(fact_10200_card__nth__roots,axiom,
    ! [C2: complex,N3: nat] :
      ( ( C2 != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N3 )
       => ( ( finite_card_complex
            @ ( collect_complex
              @ ^ [Z5: complex] :
                  ( ( power_power_complex @ Z5 @ N3 )
                  = C2 ) ) )
          = N3 ) ) ) ).

% card_nth_roots
thf(fact_10201_card__roots__unity__eq,axiom,
    ! [N3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N3 )
     => ( ( finite_card_complex
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N3 )
                = one_one_complex ) ) )
        = N3 ) ) ).

% card_roots_unity_eq
thf(fact_10202_card__num0,axiom,
    ( ( finite6454714172617411596l_num0 @ top_to3689904424835650196l_num0 )
    = zero_zero_nat ) ).

% card_num0
thf(fact_10203_card__nat,axiom,
    ( ( finite_card_nat @ top_top_set_nat )
    = zero_zero_nat ) ).

% card_nat
thf(fact_10204_card__literal,axiom,
    ( ( finite_card_literal @ top_top_set_literal )
    = zero_zero_nat ) ).

% card_literal
thf(fact_10205_UNIV__bool,axiom,
    ( top_top_set_o
    = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% UNIV_bool

% Helper facts (34)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( if_num @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( if_num @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( if_rat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( if_rat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( if_real @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( if_real @ $true @ X @ Y )
      = X ) ).

thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: real > $o] :
      ( ( P @ ( fChoice_real @ P ) )
      = ( ? [X8: real] : ( P @ X8 ) ) ) ).

thf(help_If_2_1_If_001t__Assertions__Oassn_T,axiom,
    ! [X: assn,Y: assn] :
      ( ( if_assn @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Assertions__Oassn_T,axiom,
    ! [X: assn,Y: assn] :
      ( ( if_assn @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( if_complex @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( if_complex @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__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__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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
      = X ) ).

% Conjectures (11)
thf(conj_0,hypothesis,
    ( tia
    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) ) ) @ ( suc @ ( suc @ va ) ) @ x13 @ x14 ) ) ).

thf(conj_1,hypothesis,
    ( x11
    = ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) ) ) ) ).

thf(conj_2,hypothesis,
    ~ ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ mi ) ).

thf(conj_3,hypothesis,
    ( ( size_s7982070591426661849_VEBTi @ tree_is )
    = ( size_s6755466524823107622T_VEBT @ treeList ) ) ).

thf(conj_4,hypothesis,
    ( ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) )
   != mi ) ).

thf(conj_5,hypothesis,
    xa = mi ).

thf(conj_6,hypothesis,
    ( ( vEBT_vebt_mint @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
    = ( some_nat @ y ) ) ).

thf(conj_7,hypothesis,
    ( ma
    = ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) ) ).

thf(conj_8,hypothesis,
    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 @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ).

thf(conj_9,hypothesis,
    ~ ( 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 ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_low @ ( 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 @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

thf(conj_10,conjecture,
    entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ tree_is @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ xb ) @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( snga_assn_VEBT_VEBTi @ x13 @ ( list_u6098035379799741383_VEBTi @ tree_is @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ xb ) ) ) @ ( vEBT_vebt_assn_raw @ summary @ x14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ ( insert_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( list_u6098035379799741383_VEBTi @ tree_is @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ xb ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( a_11_ATP @ ( nth_c_11_ATP @ uu_16_ATP @ uua_16_ATP ) @ xi_11_ATP ) @ ( vEBT_L375988980963497884911_ATP @ ( minus_minus_set_nat @ i_11_ATP @ ( insert_nat @ i_11_ATP2 @ bot_bot_set_nat ) ) @ a_11_ATP @ xs_11_ATP @ xsi_11_ATP ) ) @ f_11_ATP ) ).

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